Finite-time model-free robust synchronous control of multi-lift overhead cranes based on iterative learning

Abstract

A model-free control method based on iterative learning law combined with adaptive super-twisting is proposed to realize the synchronous coordination control of multi-lift overhead crane system for the problems of inaccurate modeling, system parameter variation, and disturbance uncertainty that exist in multi-lift overhead crane system. First, a load-coupling model of the double-container overhead crane considering the deformation tangential force in the interlocking mode is established. Second, a time-varying sliding mode surface (TSMC) designed using nonlinear functions effectively improves the convergence speed of the system state. The method of iterative learning control is introduced to compensate the system dynamics to achieve model-free control, and the dynamic iterative learning control (DILC) is designed to improve the convergence speed of the error of the system and the steady-state performance. To suppress uncertainty disturbances and avoid control gain overestimation, an adaptive gain is added to the generalized super-twisting algorithm, which has the advantages of both finite-time convergence and chattering suppression, and improves the robustness and tracking performance of the multi-lift overhead crane system. The stability of the controlled system is analyzed using Lyapunov stability theory. The simulation experiments illustrate the effectiveness of the proposed synchronization control scheme.

Keywords

Synchronization control interlocking model model-free control iterative learning time-varying sliding mode adaptive super-twisting algorithm

Introduction

In the current rapid development of port shipping business, to maximize the efficiency of container loading and unloading, multi-lift overhead crane came into being. In the process of loading and unloading operation of multi-lift overhead crane, each spreader needs to maintain synchronized and coordinated operation, but various uncertainties in the operation of multi-lift overhead crane make this synchronized operation very difficult. First of all, the drive motors of the multi-lift overhead crane due to uncertainty perturbation with strong nonlinear, time-varying, and high-order characteristics are in synchronous coordinated operation (Wang et al., 2019). Unlike the general motor drive system, the load of the multi-lift overhead crane system is the potential load (Zhu and Xu, 2023). At the same time, in the interlocking working mode, the containers are connected with each other by interlocking rods, which present strong coupling in the case of different loads (Zhu et al., 2021). Therefore, due to the above reasons, in the actual loading and unloading process, the multi-lift overhead crane system is prone to cause the containers asynchronous situation, and the accurate model is difficult to establish. To meet the synchronization control requirements under such special conditions, it is of great significance to study a multi-lift synchronization control method which does not rely on the model parameter information at all.

At present, most of the loads of multi-axis synchronous systems are inertial loads or kinetic loads, while the loads of multi-lift overhead crane systems are potential loads. The study of synchronous and coordinated control of multi-lift overhead cranes belongs to a special kind of multi-axis synchronous and coordinated control problem (Zhu and Xu, 2023). Scholars have proposed many control methods based on models, such as sliding mode control (Wang et al., 2019), model predictive control (Mousavi et al., 2021), backstepping control (Lin et al., 2020), and optimal control (Wang et al., 2021). The above control methods rely on the known mathematical and nominal models of the system, but it is difficult to obtain the exact model of the multi-lift overhead crane system, so the model-based control methods are not suitable for the multi-lift overhead crane system.

Model-free control (MFC) is an alternative technique for controlling uncertain complex systems that does not require information from the dynamics (Sun et al., 2021). The development of MFC has many rich theoretical results, and some scholars have proposed proportional–integral–derivative (PID) control algorithms (H Yu et al., 2020). However, it is difficult for PID to achieve high synchronization control accuracy relative to overhead crane systems with complex uncertain disturbances. Neural network control (NNC) methods have significant advantages in parameter estimation of complex nonlinear systems (Duan et al., 2020), but NNC is computationally intensive and requires specific training sets for training. The fuzzy rules of fuzzy control (Zhai et al., 2023) rely on manual adjustment. Bessa et al. (2019) combined adaptive fuzzy system with variable structure controller to enhance the control performance of uncertain underactuated mechanical system. Although the simulation verifies the effectiveness of the fuzzy control scheme, the problem of “rule explosion” needs to be solved. While the time delay estimation (TDE) technique helps to compensate for unknown dynamics and external disturbances, delayed estimation errors may lead to instability and reduced robustness (Han et al., 2020). Iterative learning law (ILC) has the advantages of control mechanisms based on uncertain models of unit memory, high adaptability, and simple algorithms, and is widely used in control systems with strong nonlinear coupling, high position repetition accuracy, difficult modeling, and high-precision trajectory tracking control (Xu, 2011).

ILC has established a more complete theoretical system, and the traditional ILCs in the form of P-type (Long et al., 2014) and PD-type (Norouzi and Koch, 2020) have shown some superiority. To improve the control performance, scholars have combined adaptive techniques with iterative learning (Li et al., 2022), which enables the controller to handle parameter uncertainty adaptively while updating parameters over the iterative domain. In the above iterative learning form, the choice of learning gain often uses fixed values, which often affects the control accuracy and convergence speed, while the choice of learning gain plays a key role in the convergence performance of the ILC. To address this problem, optimization algorithms are introduced into the field of iterative learning control, ILC based on optimization algorithms can achieve high tracking control accuracy, but the good performance brought by optimization algorithms is often accompanied by the disadvantage of algorithmic complexity (Tao et al., 2018; Q Yu et al., 2020). In contrast, the information of the sliding mode surface is used in the work by Liu et al. (2022) to design a dynamic learning rate (DLR) that changes with the sliding mode surface, improving the sliding mode dynamics while increasing the control accuracy and convergence speed of the ILC.

In fact, for approaches to date, there is a fundamental trade-off between plant model knowledge and ILC convergence speed, and the robustness of the system is often sacrificed when little information about the study object is known while focusing on the convergence speed (Armstrong et al., 2021). To improve the control performance of iterative learning, many scholars have combined iterative learning with sliding mode control methods. Feng et al. (2021) proposed an iterative learning–enhanced integral terminal sliding mode control method to further improve the performance of precision motion systems under repeated trajectories and disturbances. To deal with nonparametric uncertainties and external disturbances, the adaptive sliding mode method is added to the iterative learning control scheme to suppress the chattering of the control torque (Wang et al., 2022; Zhang et al., 2021). Chattering is a problem that must be solved in the sliding mode control, and the presence of excessive chattering will have a negative impact on the dynamic performance of the multi-lift system (Zhu and Xu, 2023). Current approaches that make extensive use of convergence law techniques (Gu and Xu, 2022; Liu and Xu, 2023) and super-twisting algorithms (Zhu et al., 2021) have shown advantages in suppressing chattering in overhead crane control. However, the super-twisting algorithm often brings unnecessary chattering due to the overestimation of control gain. In the study of Lochan et al. (2020), adaptive super-twisting algorithm (ASTA) is designed to enhance the robustness of the controller against external disturbances and parameter uncertainties, while the chattering is also suppressed.

Motivated by the above work, the main contributions of this paper are as follows: (1) To illustrate the significance of the research problem and simulation validation, a load-coupling model considering the deformation tangential forces in the interlocking mode is developed for multi-lift overhead crane system. (2) Dynamic iterative learning control (DILC)-ASTA-time-varying sliding mode surface (TSMC) as a new model-free framework is proposed for the simultaneous coordinated control of multiple spreaders, which does not require accurate model information relative to the model-based controller design, while providing good suppression of uncertainty. (3) A new adaptive law is added to the generalized super-twisting algorithm (GSTA), which effectively weakens the chattering of the control system, and achieves finite-time convergence. (4) ILC is introduced to compensate for the unknown dynamics of the system, and a DLR is proposed instead of a fixed learning rate to balance the convergence speed and steady-state performance of the learning process.

The rest of the paper is organized as follows: In the “System model and problem formulation” section, the mathematical model of the multi-lift overhead crane system and the control requirements to be achieved are established. In the “Main results” section, the overall framework of the MFC scheme for the multi-lift overhead crane system is constructed and the stability analysis of the closed-loop system is carried out. The simulation results verify the effectiveness of the method. Finally, the paper is summarized in the “Conclusion” section.

Problem descriptions

Coupling model of crane in interlocking mode

Here, taking double-lift overhead crane system as an example, the load-coupling model of double-lift overhead crane in interlocking mode is established. When the double spreaders work in interlocking mode, there will be coupling between the spreaders. The mechanical interlocking device is a linkage device. Since the mass of the mechanical interlocking device is much smaller than the mass of the spreader and its load, the mass of the linkage can be neglected.

The rectangular coordinate system xoz in Figure 1 is the two-dimensional plane of the double spreader coupling system, the x-axis is the horizontal direction, the z-axis is the vertical direction, and the mass points A and B are in asynchronous state. Suppose the masses of the loads corresponding to mass points A and B are $m_{A}, m_{B} \in R$ , and the length of the connecting rod between mass points is fixed as $l_{AB} \in R$ . The fixed distance between the driving motors of the double spreaders is $l_{CD} \in R$ . The lengths of rope A and rope B are $l_{A}, l_{B} \in R$ . $θ_{1}$ is the angle between rope A and the vertical direction, $θ_{2}$ is the angle between rope B and the vertical direction. In Figure 1, $F_{A}, F_{B}$ are the corresponding tensions of the two ropes, and $g$ is the acceleration of gravity. The instantaneous velocity of mass point A is decomposed into the normal velocity $u_{1}$ of rope 1 and the tangential velocity $v_{A}$ perpendicular to the rope. Mass point B is decomposed into the normal velocity $u_{2}$ of rope 2 and the tangential velocity $v_{B}$ perpendicular to the rope.

Figure 1.

Asynchronous state in interlock mode and force analysis of double containers.

The geometric relationship can be obtained from Figure 1 as follows

{\begin{matrix} CD = l_{A} \sin θ_{1} + l_{B} \sin θ_{2} + Lsin α \\ l_{A} \cos θ_{1} = Lcos α + l_{B} \cos θ_{2} \end{matrix}

(1)

Deformation is inevitable in the course of practical engineering operations. Therefore, according to Dou (2019), we can treat the motion of the connecting rod as a component of its rigid motion and elastic deformation. Assuming that the deformation of the connecting rod is small, for the purpose of deriving the coupling model, assume that there is no deformation at end A of the connecting rod and deformation at end B. Consider the following boundary conditions as shown in equation (2)

ω (0, t) = 0, \overset{\cdot}{ω} (0, t) = 0, \overset{\cdot\cdot}{ω} (l_{AB}, t) = 0

(2)

where $ω (l_{AB}, t)$ is the transverse deformation of the beam with respect to frame o-xz. Here, $\overset{\cdot}{ω} (0, t)$ , $\overset{\cdot}{ω} (l_{AB}, t)$ is the derivative of the deformation variable with respect to time. $EI$ is the uniform flexural rigidity, and $EI . . . ω (l_{AB}, t)$ is the shear force at end $l_{AB}$ .

The following expression can be obtained from the relationship between linear velocity and angle

{\begin{matrix} u_{A} = {\overset{\cdot}{l}}_{A}, v_{A} = l_{A} {\overset{\cdot}{θ}}_{1} - \cos φ * \overset{\cdot}{ω} (0, t) = 0 \\ u_{B} = {\overset{\cdot}{l}}_{B}, v_{B} = l_{B} {\overset{\cdot}{θ}}_{2} - \cos φ * \overset{\cdot}{ω} (l_{AB}, t) = 0 \end{matrix}

(3)

Based on the force analysis of mass points A and B, the expressions of the action forces $F_{A}$ , $F_{B}$ obtained are expressed as follows

{\begin{matrix} F_{A} = m_{A} \frac{i_{A}^{2}}{l_{A}} + m_{A} gcos θ_{1} - \frac{m_{B} gsin θ_{2} + m_{B} {\overset{\cdot}{v}}_{B} + EI . . . ω (0, t) \cos φ}{\sin (φ - θ_{2})} \cos (θ_{1} + φ) \\ F_{B} = m_{B} \frac{i_{B}^{2}}{l_{B}} + m_{B} gcos θ_{2} - \frac{m_{A} gsin θ_{1} + m_{A} {\overset{\cdot}{v}}_{A} - EI . . . ω (l_{AB}, t) \cos φ}{\sin (θ_{1} + φ)} \cos (φ - θ_{2}) \end{matrix}

(4)

The functional forms of $F_{A}$ and $F_{B}$ can be obtained from equation (4) as follows

{\begin{matrix} F_{A} = F (θ_{1}, l_{A}, l_{B}, l_{AB}, t, φ, θ_{2}) \\ F_{B} = F_{1} (θ_{1}, l_{A}, l_{B}, l_{AB}, t, φ, θ_{2}) \end{matrix}

(5)

In the ideal state, the double spreaders exist in a perfectly synchronized state, when $l_{A} = l_{B}, F_{A} = F_{B}, φ = Π / 2$ , and the force analysis is performed on the mass points A, B

{\begin{matrix} F_{A} = m_{A} \frac{{u_{A}}^{2}}{l_{A}} + m_{A} g \\ F_{B} = m_{B} \frac{{u_{B}}^{2}}{l_{B}} + m_{B} g \end{matrix}

(6)

It is also known that if the radius of the winding wheel of the drive motor is $r ϵ R$ , we can obtain the potential load moment of the double spreaders as follows

{\begin{matrix} G_{A} = F_{A} r \\ G_{B} = F_{B} r \end{matrix}

(7)

Remark 1. In the independent operation mode, $G_{i} = G_{li}$ , that is, only the force generated by the load itself is considered. The coupling model in this section is used for the simulation verification of the controller in interlock mode in “Simulations and results” section. At the same time, the stability of the established interlocking model can be guaranteed in the case of $θ_{1}, θ_{2} \in [0, \frac{π}{2}], φ \in [0, π], θ_{1} + φ \in [\frac{π}{2}, \frac{3 π}{2}],$ $θ_{1} + θ_{2} + φ \in [\frac{3 π}{2}, 2 π]$ .

System model and problem formulation

The lifting motor system consists of a rotational speed mechanical subsystem and an electromagnetic subsystem system; consider the following mathematical dynamic model of the lifting motor system

{\begin{matrix} {\overset{\cdot}{θ}}_{i} = w_{i} \\ J_{i} {\overset{\cdot\cdot}{θ}}_{i} + B_{i} {\overset{\cdot}{θ}}_{i} = T_{i} + d_{i} \end{matrix}

(8)

where $d_{i} = τ_{e} + G_{i} + Δ J_{i} {\overset{\cdot\cdot}{θ}}_{i} + Δ B_{i} {\dot{θ}}_{i}, Δ J_{i} = J_{i} - J_{r i}, Δ B_{i} = B_{i} - B_{r i}$ , $J_{i} \in R$ is moment of inertia of the lifting motor, $B_{i} \in R$ is the actual viscous friction coefficient of the lifting motor system, $τ_{e} \in R$ is the sum of the external unknown disturbances of the system, $G_{i}$ is the potential load of the containers, the matched disturbance $d_{i}$ is the lumped disturbance, $θ_{i} \in R$ is the actual angular position of the lifting motor rotation, ${\overset{\cdot}{θ}}_{i} \in R$ is the actual angular velocity of the lifting motor, and $T_{i}$ is the actual control input torque of the lifting motor. $i = 1, 2 \dots \dots n$ .

Simplifying equations (7) and (8) yields equation (9)

{\begin{matrix} {\overset{\cdot}{θ}}_{ı} = w_{i} \\ {\overset{\cdot\cdot}{θ}}_{i} = ζ T_{i} + D_{i} (t) \end{matrix}

(9)

where $ζ = \frac{1}{J_{i}}, D_{i} (t) = ζ (\overset{\cdot}{θ_{ı}} + τ_{e} + G_{i} + Δ J_{i} {\overset{\cdot\cdot}{θ}}_{i} + Δ B_{i} {\overset{\cdot}{θ}}_{ı})$ denotes the lumped uncertainties. $i = 1, 2 \dots \dots n$ .

In summary, in this work, the main objective is to design a model-free synchronous controller for the multi-lift overhead crane system where disturbances exist and accurate models are difficult to establish, so that the spreaders can track the position signals simultaneously in a finite time, achieve synchronous and coordinated control among spreaders, and effectively suppress the influence of unknown disturbances on the control performance of the system.

Assumption 1 The initial state of the system is the same for each iteration. In other words, the desired position $θ_{d}$ is the same for each iteration, then the tracking error ( $e_{i} = θ_{d} - θ_{i}$ ) satisfies $e_{i} (0) = e_{i + 1} (0)$ .

Main results

The basic idea of this section is to develop a DLR-based ILC combined with an adaptive super-twisting model-free controller for the synchronous and coordinated control of multiple spreaders for the multi-lift overhead crane system containing disturbances. The block diagram of the control system is shown in Figure 2.

Figure 2.

Block diagram of the control system.

Nonlinear time-varying sliding mode surface design

First, the tracking error of the spreader is defined as follows

e_{i} = θ_{d} - θ_{i}

(10)

where $θ_{d}$ is the desired angular position of the lifting motor. $i = 1, 2 \dots \dots n$ .

The synchronization error is defined as follows

ε_{i} = e_{i} - e_{i + 1}

(11)

The difference of the synchronization error between the coupled motors is used as a compensation term to couple with the synchronization error of the controlled motors, and the coupling error is obtained as the following equation

e_{i}^{*} = ε_{i} + β \int (ε_{i} (τ) - ε_{i + 1} (τ)) d τ

(12)

where $β \in R^{+}$ .

The time-varying sliding mode surface using nonlinear functions is proposed as follows

s_{i} = {\overset{\cdot}{e}}_{i} + f_{1} (t) e_{i}^{*} + f_{2} (t) e_{i}

(13)

The nonlinear function is defined as follows

f_{1} (t) = \frac{2 a_{1}}{e^{b_{1} {| e_{i}^{*} |}^{c_{1}}} + 1}

(14)

f_{2} (t) = \frac{2 a_{2}}{e^{b_{2} {| e_{i} |}^{c_{2}}} + 1}

(15)

where $a_{i}, b_{i}, c_{i} > 0$ , $i = 1, 2$ . Sliding surface (13) can reduce the convergence time of the system state when multi-lift overhead crane system reaches the desired position.

Remark 2. The designed nonlinear function $f_{i} (t)$ varies with the tracking error $e_{i}$ and the compound error $e_{i}^{*}$ based on the synchronization error of the system, respectively, and when the error increases, the exponential part increases and $f_{i} (t)$ decreases overall, and when the error decreases, the exponential part decreases and $f_{i} (t)$ increases overall, so it is always possible to make the initial state move on the sliding mode surface quickly. Eventually, $f_{i} (t)$ stabilizes to a constant value. The application of the nonlinear function changes the invariant characteristic of the traditional linear sliding surface coefficients and effectively improves the response speed of the system error.

Differentiation of equation (13) with respect to time yields

{\overset{\cdot}{s}}_{i} = {\overset{\cdot\cdot}{θ}}_{i} - {\overset{\cdot}{θ}}_{d} + {\overset{\cdot}{f}}_{1} (t) e_{i}^{*} + {\overset{\cdot}{f}}_{2} (t) e_{i} + f_{1} (t) {\overset{\cdot}{e}}_{i}^{*} + f_{2} (t) {\overset{\cdot}{e}}_{i}

(16)

Combining equation (13) with equation (16), ${\overset{\cdot}{s}}_{i}$ is rewritten as

{\overset{\cdot}{s}}_{i} = Ψ + ζ T_{i}

(17)

where $Ψ = {\overset{\cdot}{f}}_{1} (t) e_{i}^{*} + {\overset{\cdot}{f}}_{2} (t) e_{i} + f_{1} (t) {\overset{\cdot}{e}}_{i}^{*} + f_{2} (t) {\overset{\cdot}{e}}_{i} - {\overset{\cdot\cdot}{θ}}_{d} + D_{i}$ .

The DILC-ASTA-TSMC scheme is proposed as follows, whose structure is shown in Figure 2. In this scheme, each repetition of the repetition task is called an iteration, denoted by $j$ . The total controller is designed as follows, and subscripts i are omitted for ease of writing. $j = 1, 2 \cdot \cdot \cdot \cdot \cdot n$

T_{j} = u_{j}^{asta} + u_{j}^{ilc}

(18)

Substitution of equation (18) into equation (17) yields

{\overset{\cdot}{s}}_{j} = Ψ + ζ u_{j}^{asta} + ζ u_{j}^{ilc}

(19)

where $u_{j}^{asta}$ denotes adaptive super-twisting controller and $u_{j}^{ilc}$ denotes iterative learning controller. The specific design of each term in the DILC-ASTA-TSMC scheme is discussed in the following subsections.

ASTA-TSMC controller design

Severe chattering will cause an increase in the system losses and affect the control performance and stability of the multi-lift overhead crane system (Zhu and Xu, 2023). Therefore, to improve the system control performance, a linear correction term is introduced in this paper and an adaptive generalized super-twisting algorithm (ASTA) is designed, and the expression of the ASTA method is

u_{j}^{asta} = - k_{1} ({| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j}) - ρ

(20)

\overset{\cdot}{ρ} = k_{2} (\frac{1}{2} sgn (s_{j}) + \frac{3}{2} {| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j}) = k_{2} (\frac{d g_{j + 1}}{d s_{j}} g_{j + 1})

(21)

where $g_{j + 1} = {| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j}$ and the control gains $k_{1}$ and $k_{2}$ are adjusted by adaptive updating law

{\overset{\cdot}{k}}_{1} = \frac{K \tanh (| s_{j} |) {| s_{j} |}^{\frac{1}{2}}}{α + (1 - α) e^{(- α_{1} {| s_{j} |}^{σ})}}

(22)

k_{2} = 2 β_{1} k_{1}

(23)

where $K > 0, K_{1} > 0, α > \frac{1}{2}, α_{1} > 0, σ > 0, β_{1} > 0$ , $ϖ > 0$ .

Remark 3. The designed adaptive rate uses the sliding mode information, as shown in equation (22), when the sliding mode variable $s_{j}$ is extremely large, $e^{(- α_{1} {| s_{j} |}^{σ})} \to 0$ , $\tanh (| s_{j} |) \to 1$ , the adaptive law tends to $\frac{K {| s_{j} |}^{σ}}{α}$ , at which time the control gain increases fast, thus making the sliding surface to converge to zero quickly. When the sliding surface converges to 0, ${\overset{\cdot}{k}}_{1} = 0$ . Noting that the adaptive gain form proposed in the work by Shen et al. (2022) allows the tracking error to converge to a small domain in a finite time, while the ASTA designed in this paper can achieve asymptotic convergence in a finite time, as demonstrated in “Simulations and results” section.

DILC-ASTA-TSMC controller design

The specific model information of the multi-lift overhead crane system is difficult to obtain, while there are uncertain disturbances, to further suppress the disturbances and unknown dynamics, an iterative learning method based on switching terms is proposed

u_{j + 1}^{ilc} = u_{j}^{ilc} - q (ρ + s_{j})

(24)

where $u_{j}^{ilc}$ is the ILC input at the jth iteration, $j = 1, 2 \dots \dots n, q$ is the learning rate.

To balance the convergence speed and steady-state accuracy, the proposed nonlinear function–based DLR instead of fixed learning rate is as follows

Q = {\begin{matrix} ϱ (1 + μ \tanh (| s_{j} | - σ)), | s_{j} | > σ \\ ϱ, | s_{j} | \leq σ \end{matrix}

(25)

where $σ > 0$ is the threshold value of the nonlinear function, $0 < μ < 1$ , $ϱ > 0, Q \in [ϱ, ϱ (1 + μ)]$ . The final iterative learning controller can be written in the following form

u_{j + 1}^{ilc} = u_{j}^{ilc} - Q (ρ + s_{j})

(26)

Remark 4. The constant learning rate (CLR) is typically used in the work by Wang et al. (2022) and Zhang et al. (2021), but ILC requires a trade-off between the convergence speed and robustness, which may sacrifice the iterative learning process for transient performance or steady-state performance during iterative learning. The good property of tanh continuous smoothing is exploited to design a DLR that changes according to the information of the sliding surface. When the sliding surface is far from $σ, Q$ adjusts the value of the nonlinear function independent variable according to the difference of the sliding surface reaching the domain of $σ$ , so that $μ \cdot \tanh (| s_{j} | - σ)$ varies in the range of [0, $μ$ ], $Q$ constantly tends to $ϱ (1 + μ)$ . When the sliding surface is in the neighborhood of $σ$ , the learning rate is given a fixed value of $ϱ$ .

The iterative learning controller generates an estimated signal in the iterative domain $- \hat{Ψ_{j}}$ , so there is

u_{j}^{ilc} = - ζ^{- 1} \hat{Ψ_{j}}

(27)

Then, the sliding variable derivative is expressed as

{\overset{\cdot}{s}}_{j} = φ_{j} + ζ u_{j}^{asta}

(28)

where $φ_{j} = Ψ - \hat{Ψ_{j}}$ , from the above analysis, it is known that $ζ^{- 1} \hat{Ψ_{j}}$ containing unknown dynamics and uncertainty disturbances will be compensated during the iterative process, and the remaining uncertainties $φ_{j}$ are compensated using the ASTA, thus driving the sliding variable to 0. In addition, the uncertainty coefficient $ζ$ does not affect the system convergence even if it cancels each other with the adaptive gain during the compensation process.

Stability analysis

Lemma 1: Considering gain adaptive laws (22) and (23), the gains $k_{1}$ and $k_{2}$ are positive functions and have upper bounds, that is, there are positive constants $\hat{k_{1}}$ and $\hat{k_{2}}$ , such that $k_{1} \leq \hat{k_{1}}$ and $k_{2} \leq \hat{k_{2}}$ .

Assumption 2: In a practical tracking problem, the length of the reference signal is always limited. It is assumed that the uncertainty disturbance $φ_{j}$ is bounded. Furthermore, the derivative of the uncertainty satisfies $| {\overset{\cdot}{φ}}_{i} | \leq l_{\max}$ , where $l_{\max}$ is unknown.

The proposed ASTAs (20) and (21) can guarantee the finite-time tracking performance and avoid the control gain overestimation problem. Now, ASTAs (20) and (21) are substituted into sliding mode dynamics (28)

u_{j}^{asta} = - k_{1} ({| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j}) + η

(29)

\overset{\cdot}{η} = {\overset{\cdot}{φ}}_{j} - k_{2} (\frac{1}{2} sgn (s_{j}) + \frac{3}{2} {| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j})

(30)

Theorem 1. For the multi-lift overhead crane dynamic model in equation (9), under the assumption of unknown disturbance, the control laws in equations (20) and (21) are constructed according to the nonlinear time-varying sliding surface designed in equation (13) and the adaptive laws in equations (22) and (23), when the control and parameter setting conditions are satisfied, the sliding mode surface $s_{j}$ converges to 0 in a finite time $T$ under the action of the controller.

Proof First, a new state is established as

x = [\begin{matrix} x_{1}, & x_{2} \end{matrix}]^{T} = [\begin{matrix} {| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j}, & η \end{matrix}]^{T}

(31)

Taking the derivative of the new state along equation (31) gives that

\overset{\cdot}{x} = [\begin{matrix} (\frac{1}{2} {| s_{j} |}^{- \frac{1}{2}} + 1) (- k_{1} ({| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j}) + η) \\ {\overset{\cdot}{φ}}_{j} - k_{2} (\frac{1}{2} sgn (s_{j}) + \frac{3}{2} {| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j}) \end{matrix}]

(32)

In view of Assumption 2, we can obtain that $| {\overset{\cdot}{φ}}_{j} | \leq l_{\max} \leq l_{\max} (\frac{1}{2} + \frac{3}{2} {| s_{j} |}^{\frac{1}{2}} + | s_{j} |)$ , Therefore, for arbitrary point in $R ∖ S (S = {(s, η) \in R ∣ s = 0})$ , so we have ${\overset{\cdot}{φ}}_{j} = l (t) (\frac{1}{2} sgn (s_{j}) + \frac{3}{2} {| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j})$ , because $l (t)$ is bounded, so the following equation is obtained

\begin{matrix} \overset{\cdot}{x} = [\begin{matrix} (\frac{1}{2} {| s_{j} |}^{- \frac{1}{2}} + 1) (- k_{1} ({| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j}) + η) \\ (\frac{1}{2} {| s_{j} |}^{- \frac{1}{2}} + 1) ({| s_{j} |}^{\frac{1}{2}} sgn (s_{j}) + s_{j}) (- k_{2} + l (t)) \end{matrix}] \\ = (\frac{1}{2} {| s_{j} |}^{- \frac{1}{2}} + 1) (Ax + [\begin{matrix} 0 \\ x_{1} l (t) \end{matrix}]) \end{matrix}

(33)

where $A = [\begin{matrix} - k_{1} & 1 \\ - k_{2} & 0 \end{matrix}]$ . To prove the stability of system, the Lyapunov candidate function is chosen as

V_{a} = V_{1} + \frac{1}{2} {\tilde{k}}_{1}^{2} + \frac{1}{2} {\tilde{k}}_{2}^{2}

(34)

where $V_{1} = x^{T} Px$ , ${\tilde{k}}_{1} = \hat{k_{1}} - k_{1}, {\tilde{k}}_{2} = \hat{k_{2}} - k_{2}$ , and $k_{1} \leq \hat{k_{1}}, k_{2} \leq \hat{k_{2}}$ , so ${\tilde{k}}_{1}, {\tilde{k}}_{2}$ are positive.

The positive-definite matrix P is defined as

P = [\begin{matrix} λ + 4 δ^{2} & - 2 δ \\ - 2 δ & 1 \end{matrix}]

(35)

where $λ > 0, δ > \frac{1}{3}$ . By taking the derivative of equation (34), we have

{\dot{V}}_{a} = {\dot{V}}_{1} - {\dot{k}}_{1} | {\tilde{k}}_{1} | - {\dot{k}}_{2} | {\tilde{k}}_{2} |

(36)

Taking into account equations (33) and (35), the first term of above equation is presented as

{\overset{\cdot}{V}}_{1} = {\overset{\cdot}{x}}^{T} Px + x^{T} P \overset{\cdot}{x} = (\frac{1}{2} {| s_{j} |}^{- \frac{1}{2}} + 1) (x^{T} (A^{T} P + PA) x + 2 x^{T} Pb)

(37)

where $x^{T} P b = - 4 l (t) δ x_{1}^{2} + 2 l (t) x_{1} x_{2} \leq - 4 δ l (t) x_{1}^{2} + 2 l^{2} (t) x_{1}^{2} + x_{2}^{2} = x^{T} C x$ , the symmetric matrix $C = diag {2 l^{2} (t) - 4 δ l (t), 1}$ , equation (37) can also be written as

{\overset{\cdot}{V}}_{1} \leq (\frac{1}{2} {| s_{j} |}^{- \frac{1}{2}} + 1) x^{T} (A^{T} P + PA + C) = - (\frac{1}{2} {| s_{j} |}^{- \frac{1}{2}} + 1) x^{T} Ω x

(38)

where $Ω = A^{T} P + PA + C$ . To ensure that $Ω$ is positive definite, invoking Schur’s complement, the matrix shall satisfy the following conditions

k_{1} \geq \frac{- 2 l_{\max}^{2} + 2 l_{\max} δ + 4 δ^{2} + λ}{2 (4 δ β_{1} + β_{1} - λ - δ - 4 δ^{2})}

(39)

Evoking Lemma (Shen et al., 2022) and using equations (38) and (39), we have

{\overset{\cdot}{V}}_{1} \leq - δ (\frac{1}{2} {| s_{j} |}^{- \frac{1}{2}} + 1) x^{T} x \leq - ϑ_{1} {V_{1}}^{\frac{1}{2}} - ϑ_{2} V_{1}

(40)

where $ϑ_{1} = \frac{δ λ \min {(P)}^{\frac{1}{2}}}{2 λ \max (P)}$ , $ϑ_{2} = \frac{δ}{λ \max (P)}$ . Considering equation (36), ${\overset{\cdot}{V}}_{a}$ can be written in the following form

{\dot{V}}_{a} = {\dot{V}}_{1} - \dot{k_{1}} | {\tilde{k}}_{1} | - {\dot{k}}_{2} | {\tilde{k}}_{2} | \leq - ϑ_{1} {V_{1}}^{\frac{1}{2}} - ϑ_{2} V_{1} - \dot{k_{1}} | {\tilde{k}}_{1} | - {\dot{k}}_{2} | {\tilde{k}}_{2} |

(41)

Substituting equations (22) and (23) into equation (41) yields the following equation

\begin{matrix} {\overset{\cdot}{V}}_{a} \leq - ϑ_{1} {V_{1}}^{\frac{1}{2}} - ϑ_{2} V_{1} + ξ_{1} {\tilde{k}}_{1}^{2} - ξ_{1} {\tilde{k}}_{1}^{2} + ξ_{2} {\tilde{k}}_{2}^{2} - ξ_{2} {\tilde{k}}_{2}^{2} \\ + ξ_{1} | {\tilde{k}}_{1} | - ξ_{1} | {\tilde{k}}_{1} | + ξ_{2} | {\tilde{k}}_{2} | - ξ_{2} | {\tilde{k}}_{2} | \\ - k_{1} | {\tilde{k}}_{1} | - 2 β_{1} k_{1} | {\tilde{k}}_{2} | \\ \leq (- ϑ_{1} {V_{1}}^{\frac{1}{2}} - ξ_{1} | {\tilde{k}}_{1} | - ξ_{2} | {\tilde{k}}_{2} |) + (- ϑ_{2} V_{1} - ξ_{1} {\tilde{k}}_{1}^{2} - ξ_{2} {\tilde{k}}_{2}^{2}) \\ + | {\tilde{k}}_{1} | (\begin{matrix} ξ_{1} | {\tilde{k}}_{1} | \\ + ξ_{2} + \frac{Ktanh (| s_{j} |) {| s_{j} |}^{σ}}{α + (1 - α) e^{(- α_{1} {| s_{j} |}^{σ})}} \end{matrix}) \\ + | {\tilde{k}}_{2} | (ξ_{1} | {\tilde{k}}_{2} | + ξ_{2} + \frac{2 β_{1} Ktanh (| s_{j} |) {| s_{j} |}^{σ}}{α + (1 - α) e^{(- α_{1} {| s_{j} |}^{σ})}}) \\ \leq (- ϑ_{1} {V_{1}}^{\frac{1}{2}} - ξ_{1} | {\tilde{k}}_{1} | - ξ_{2} | {\tilde{k}}_{2} |) - ϑ_{2} V_{1} - ξ {k_{1}}^{2} - ξ_{2} {k_{2}}^{2} \\ + | {\tilde{k}}_{1} | (ξ_{1} k_{1} + ξ_{2} - K {| s_{j} |}^{σ}) + | {\tilde{k}}_{2} | (ξ_{2} k_{2} \\ + ξ_{1} 2 β_{1} K {| s_{j} |}^{σ}) \end{matrix}

(42)

when $k_{1} \leq \frac{K {| s_{j} |}^{σ}}{ξ_{1}} - \frac{ξ_{2}}{ξ_{1}}$ , $k_{2} \leq \frac{2 β_{1} K {| s_{j} |}^{σ}}{ξ_{2}} - \frac{ξ_{1}}{ξ_{2}}$ , $ξ_{i} < 0$ , ${\overset{\cdot}{V}}_{a}$ can be written in the following form

\begin{matrix} {\overset{\cdot}{V}}_{a} \leq (- ϑ_{1} {V_{1}}^{\frac{1}{2}} - ξ_{1} | {\tilde{k}}_{1} | - ξ_{2} | {\tilde{k}}_{2} | \end{matrix}) - ϑ_{2} V_{1} - ξ_{1} {k_{1}}^{2} - ξ_{2} {k_{2}}^{2} \leq - m_{1} {V_{1}}^{\frac{1}{2}} - m_{2} V_{1}

(43)

where $m_{1} = \sqrt{2} \min {ϑ_{1}, ξ_{1}, ξ_{2}}, m_{2} = \min {ϑ_{2}, ξ_{1}, ξ_{2}}$ , ${\overset{\cdot}{V}}_{a}$ is negative, according to Lemmas (Lochan et al., 2020; Shen et al., 2022), at this point, the sliding mode variables can converge to zero within $T \leq \frac{2}{m_{1}} In \frac{m_{1} V_{1}^{\frac{1}{2}} (0) + m_{2}}{m_{2}}$ . Regarding the LaSalle’s invariance principle, the system is asymptotically stable. Consequently, the convergence of tracking error and synchronization error is guaranteed.

Theorem 2. For the multi-lift overhead crane dynamic model in equation (9), under the conditions of Assumptions 1 and 2, control law (18) based on equations (20)–(23), (25), and (26) drives the closed-loop system’s sliding variable $s_{j}$ to converge to zero for arbitrary time $t \in [0, T]$ as the iteration approaches infinity. The multi-lift reaches the desired position synchronously under the action of the controller that is, when $j \to \infty, t \to T$ , the $θ_{i} \to θ_{d}, ε_{i} \to 0$ .

Proof:

V_{j} (t) = V_{j}^{1} (t) + V_{j}^{2} (t) + V_{j}^{3} (t) + V_{j}^{4} (t)

(44)

V_{j}^{1} (t) = \frac{1}{2} ϱ g_{j} {(t)}^{2}

(45)

V_{j}^{2} (t) = \frac{1}{2} \int_{0}^{T} \frac{1}{\hat{L}} φ_{j} {(τ)}^{T} φ_{j} (τ) d τ

(46)

V_{j}^{3} (t) = \frac{1}{2} {\tilde{k}}_{j}^{2}

(47)

V_{j}^{4} (t) = \frac{1}{2} ϱ s_{j} (t)^{2}

(48)

For Lyapunov equation (44), it is proved for $V_{j}^{1} (t), V_{j}^{2} (t), V_{j}^{3} (t), V_{j}^{4} (t)$ , respectively. Since $k_{2}$ is a linear function with respect to $k_{1}$ , it is sufficient to prove $k_{1}$ separately. So according to equation (45), we have

Δ V_{j}^{1} (t) = \frac{1}{2} g_{j + 1} {(t)}^{2} - \frac{1}{2} g_{j} {(t)}^{2}

(49)

Substituting equations (28) and (20)–(23) into equation (49), we get

\begin{array}{l} Δ V_{j + 1}^{1} (t) = ϱ \int_{0}^{t} ρ_{j + 1} (t) {\dot{s}}_{j + 1} (t) d τ - \frac{1}{2} g_{j} {(t)}^{2} + \frac{1}{2} g_{j + 1} {(0)}^{2} \\ = ϱ (\int_{0}^{t} ρ_{j + 1} (τ) (φ_{j} + ζ \cdot u_{j + 1}^{a s t a}) d τ - \frac{1}{2} g_{j} {(t)}^{2} + \frac{1}{2} g_{j + 1} {(0)}^{2}) \\ = ϱ (\int_{0}^{t} ρ_{j + 1} (τ) φ_{j + 1} (τ) d τ - ζ \int_{0}^{t} k_{1} {({| s_{j + 1} (τ) |}^{\frac{1}{2}} s g n (s_{j + 1} (τ)) + s_{j + 1} (τ))}^{2} d τ \\ - \frac{1}{2} ζ \int_{0}^{t} k_{1} {| s_{j + 1} (τ) |}^{\frac{1}{2}} {(s g n (s_{j + 1} (τ)))}^{2} d τ - \frac{3}{2} ζ \int_{0}^{t} k_{1} {(| s_{j + 1} (τ {)|}^{\frac{1}{2}} s g n (s_{j + 1} (τ)))}^{2} d τ \\ - \frac{5}{2} ζ \int_{0}^{t} k_{1} ({| s_{j + 1} (τ) |}^{\frac{1}{2}} s g n (s_{j + 1} (τ)) s_{j + 1} (τ)) d τ - \frac{1}{2} ζ \int_{0}^{t} k_{1} (s g n (s_{j + 1} (τ)) s_{j + 1} (τ)) d τ \\ - \frac{1}{2} ζ \int_{0}^{t} k_{1} s_{j}^{2} (τ) d τ - \frac{1}{2} g_{j} {(t)}^{2} + \frac{1}{2} g_{j + 1} {(0)}^{2}) \end{array}

(50)

The difference between two iterations of the equation $V_{j + 1}^{2} (t)$ is

\begin{array}{l} Δ V_{j + 1}^{2} (t) = \frac{1}{2} \int_{0}^{T} φ_{j + 1} {(τ)}^{T} φ_{j + 1} (τ) - φ_{j} {(τ)}^{T} φ_{j} (τ) d τ \\ = - \int_{0}^{T} {({\hat{Ψ}}_{j + 1} (τ) - {\hat{Ψ}}_{j} (τ))}^{T} (Ψ_{j + 1} (τ) - {\hat{Ψ}}_{j + 1} (τ)) d τ \\ - \frac{1}{2} \int_{0}^{T} {({\hat{Ψ}}_{j + 1} (τ) - {\hat{Ψ}}_{j} (τ))}^{T} ({\hat{Ψ}}_{j + 1} (τ) - {\hat{Ψ}}_{j} (τ)) d τ \end{array}

(51)

Substituting equation (26) into equation (51) yields

Δ V_{j + 1}^{2} (t) = - \int_{0}^{t} Q (ρ_{j + 1} (t) + s_{j + 1} (τ)) φ_{j} (τ) d τ - 2 \int_{0}^{t} {(Q (ρ_{j + 1} (t) + s_{j + 1} (τ)))}^{2} d τ

(52)

where $Q \in [ϱ$ , $ϱ (1 + μ)$ ], $ϱ$ is positive, and equation (52) can also be written as

Δ V_{j + 1}^{2} (t) = - \int_{0}^{t} ϱ (ρ_{j + 1} (t) + s_{j + 1} (τ)) φ_{j} (τ) d τ - 2 \int_{0}^{t} {(Q (ρ_{j + 1} (t) + s_{j + 1} (τ)))}^{2} d τ

(53)

According to equations (22), (23), and (47), $Δ V_{j + 1}^{3} (t)$ can be written as

Δ V_{j + 1}^{3} (t) = \frac{1}{2} {\tilde{k}}_{j + 1}^{2} - \frac{1}{2} {\tilde{k}}_{j}^{2} = - \int_{0}^{t} {\overset{\cdot}{k}}_{j + 1} | {\tilde{k}}_{j + 1} | d τ - \frac{1}{2} {\tilde{k}}_{j}^{2} + \frac{1}{2} {\tilde{k}}_{j + 1} {(0)}^{2}

(54)

Also, combining equation (28) with equation (48), we can get $Δ V_{j + 1}^{4} (t)$

\begin{matrix} Δ V_{j + 1}^{4} (t) = ϱ (\int_{0}^{t} {\overset{\cdot}{s}}_{j + 1} (τ) s_{j + 1} (τ) d τ - \frac{1}{2} s_{j} {(t)}^{2} + \frac{1}{2} s_{j + 1} {(0)}^{2}) \\ = ϱ (\int_{0}^{t} φ_{j} (τ) s_{j + 1} (τ) d τ + ζ \int_{0}^{T} u_{j + 1}^{asta} s_{j + 1} (τ) d τ - \frac{1}{2} s_{j} {(t)}^{2} + \frac{1}{2} s_{j + 1} {(0)}^{2}) \end{matrix}

(55)

The difference of the overall iteration domain is

\begin{array}{l} Δ V_{j + 1} (t) = Δ V_{j + 1}^{1} (t) + Δ V_{j + 1}^{2} (t) + Δ V_{j + 1}^{3} (t) + Δ V_{j + 1}^{4} (t) \\ = ϖ (\int_{0}^{t} ρ_{j + 1} (τ) φ_{j + 1} (τ) d τ - ζ \int_{0}^{t} k_{1} {({| s_{j + 1} (τ) |}^{\frac{1}{2}} s g n (s_{j + 1} (τ)) + s_{j + 1} (τ))}^{2} d τ \\ - \frac{1}{2} ζ \int_{0}^{t} k_{1} {| s_{j + 1} (τ) |}^{\frac{1}{2}} {(s g n (s_{j + 1} (τ)))}^{2} d τ - \frac{3}{2} ζ \int_{0}^{t} k_{1} {({| s_{j + 1} (τ) |}^{\frac{1}{2}} s g n (s_{j + 1} (τ)))}^{2} d τ \\ - \frac{5}{2} ζ \int_{0}^{t} k_{1} ({| s_{j + 1} (τ) |}^{\frac{1}{2}} \cdot s g n (s_{j + 1} (τ)) s_{j + 1} (τ)) d τ - \frac{1}{2} ζ \int_{0}^{t} k_{1} (s g n (s_{j + 1} (τ)) s_{j + 1} (τ)) d τ \\ - \frac{1}{2} ζ \int_{0}^{t} k_{1} s_{j}^{2} (τ) d τ - \frac{1}{2} g_{j} {(t)}^{2} + \frac{1}{2} g_{j + 1} {(0)}^{2}) - \int_{0}^{t} ϱ (ρ_{j + 1} (t) + s_{j + 1} (τ)) φ_{j} (τ) d τ \\ - 2 \int_{0}^{t} {(\hat{Q} \cdot (ρ_{j + 1} (t) + s_{j + 1} (τ)))}^{2} d τ - \int_{0}^{t} | {\dot{\tilde{k}}}_{j + 1} | k_{j + 1} d τ - \frac{1}{2} {\tilde{k}}_{j}^{2} + \frac{1}{2} {\tilde{k}}_{j + 1} {(0)}^{2} \\ + ϱ (\int_{0}^{t} φ_{j} (τ) s_{j + 1} (τ) d + ζ \int_{0}^{T} u_{j + 1}^{asta} s_{j + 1} (τ) d τ - \frac{1}{2} s_{j} {(t)}^{2} + \frac{1}{2} s_{j + 1} {(0)}^{2} \\ \leq - Δ X_{j + 1} (t) + ϱ (\frac{1}{2} g_{j + 1} {(0)}^{2} - \frac{1}{2} g_{j} {(t)}^{2}) + ϱ (\frac{1}{2} s_{j + 1} {(0)}^{2} - \frac{1}{2} s_{j} {(t)}^{2}) - \frac{1}{2} {\tilde{k}}_{j + 1}^{2} + \frac{1}{2} {\tilde{k}}_{j + 1} {(0)}^{2} \end{array}

(56)

$Δ X_{j + 1} (t)$ is defined as the following equation

\begin{matrix} Δ X_{j + 1} (t) = ϱ (\int_{0}^{t} ρ_{j + 1} (τ) φ_{j + 1} (τ) d τ \\ - ζ \int_{0}^{t} k_{1} {({| s_{j + 1} (τ) |}^{\frac{1}{2}} sgn (s_{j + 1} (τ)) + s_{j + 1} (τ))}^{2} d τ \\ - \frac{1}{2} ζ \int_{0}^{t} k_{1} {| s_{j + 1} (τ) |}^{\frac{1}{2}} {(sgn (s_{j + 1} (τ)))}^{2} d τ \\ - \frac{3}{2} ζ \int_{0}^{t} k_{1} {({| s_{j + 1} (τ) |}^{\frac{1}{2}} sgn (s_{j + 1} (τ)))}^{2} d τ \\ - \frac{5}{2} ζ \int_{0}^{t} k_{1} ({| s_{j + 1} (τ) |}^{\frac{1}{2}} sgn (s_{j + 1} (τ)) s_{j + 1} (τ)) d τ \\ - \frac{1}{2} ζ \int_{0}^{t} k_{1} (sgn (s_{j + 1} (τ)) s_{j + 1} (τ)) d τ - \frac{1}{2} ζ \int_{0}^{t} k_{1} s_{j}^{2} (τ) d τ \\ - \int_{0}^{t} ϱ (ρ_{j + 1} (t) + s_{j + 1} (τ)) φ_{j} (τ) d τ \\ - 2 \int_{0}^{t} {(Q (ρ_{j + 1} (t) + s_{j + 1} (τ)))}^{2} d τ + ϱ (\int_{0}^{t} φ_{j} (τ) s_{j + 1} (τ) d τ \\ + ζ \int_{0}^{T} u_{j + 1}^{asta} s_{j + 1} (τ) d τ) \end{matrix}

(57)

From the analysis of Theorem 1, it follows that $K {| s_{j} |}^{σ} \geq {\dot{k}}_{j + 1} > 0$ , recalling Lemma 1 shows that $k_{1}, k_{2}$ are positive functions, so $Δ X_{j + 1} (t) \geq 0$ .

According to Assumption 1, we know that $s_{j} (0) = s_{j + 1} (0)$ , recalling equations (20) and (22), we easily get ${\tilde{k}}_{j + 1} (0) = {\tilde{k}}_{j} (0)$ , so equation (59) can be written as

\begin{matrix} Δ V_{j + 1} (t) \leq - Δ X_{j + 1} (t) - \frac{1}{2} ϱ (g_{j} {(t)}^{2} - g_{k} {(0)}^{2}) \\ - \frac{1}{2} ϱ (s_{j} {(t)}^{2} - s_{j} {(0)}^{2}) - \frac{1}{2} ({\tilde{k}}_{j} {(t)}^{2} - {\tilde{k}}_{j} {(0)}^{2} \\ = - Δ X_{j + 1} (t) - \frac{1}{2} ϱ (\int_{0}^{t} ρ_{j} (τ) {\overset{\cdot}{s}}_{j} (τ) d τ) \\ - \frac{1}{2} ϱ \int_{0}^{t} s_{j} (\overset{\cdot}{τ}) s_{j} (τ) d τ - \frac{1}{2} \int_{0}^{t} {\overset{\cdot}{k}}_{j} (τ) | \hat{k_{j}} - k_{j} (τ) | d τ \end{matrix}

(58)

From Theorem 1, when t tends to a finite time $T$ , $s_{j} (T)$ converges to zero. At this time, $s_{j} (T)$ = 0, $k_{j} (τ)$ = 0, and taking the limit on equation (58) yields

\begin{matrix} lim_{t \to T} Δ V_{j + 1} (t) \leq - lim_{t \to T} Δ X_{j + 1} (t) - \frac{1}{2} ϱ lim_{t \to T} \int_{0}^{t} ρ_{j} (τ) {\overset{\cdot}{s}}_{j} (τ) d τ \\ - \frac{1}{2} ϱ lim_{t \to T} \int_{0}^{t} {\overset{\cdot}{s}}_{j} (τ) s_{j} (τ) d τ \\ - \frac{1}{2} lim_{t \to T} \int_{0}^{t} {\overset{\cdot}{k}}_{j} (τ) | \hat{k_{j}} - k_{j} (τ) | d τ \leq 0 \end{matrix}

(59)

From equation (59), we know that $s_{j + 1} (0) \equiv 0$ when and only when $lim_{t \to T} Δ V_{j + 1} (t) = 0$ . According to LaSalle’s invariance principle, it is known that the sliding mode variables and tracking error will converge to zero when t tends to a finite time $T$ . Essentially, the equality holds when the variables fully converge. Therefore, $V_{j} (t)$ is monotonically decreasing and $Δ V_{j} (t) = V_{j + 1} (t) - V_{j} (t)$ will converge to zero when the iteration $j \to \infty$ .

Simulations and results

In this paper, the model-free synchronous control method for the multi-lift overhead crane system is verified in the MATLAB/Simulink environment. Here, taking the double-lift overhead crane system as an example, the simulation investigates the synchronization performance of the double-lift overhead crane system under uncertain disturbances from the independent operation mode and interlocking operation mode of the double-lift overhead crane system.

Simulation 1: To verify the chattering suppression capability and control performance of the designed ATSA, the above controller was compared with the ASTA in the work by Yan et al. (2019) and the GSTA.

The parameters of the system model are $J_{i} = 0.16 kg \cdot m^{2}$ and $B_{i}$ = 0.1 N/ms/rad, and the desired angular position $θ_{d}$ = 10 rad. The parameters of the sliding surface are $a_{i}$ = 15, $b_{i}$ = 0.01, $c_{i}$ = 2. The parameters of the adaptive terminal sliding mode control (ATSMC) proposed in this paper are $K$ = 3, $σ$ = 2, $α$ = 0.9, $α_{1}$ = 0.5, $β_{1}$ = 0.005; and the simulation results are shown in Figure 3.

Figure 3

Tracking errors and control force under comparative ATSA methods: (a) tracking error and (b) control force.

From Figure 3, it can be seen that the tracking error converges to zero around 1.15 seconds using the adaptive gain method in GSTA; the tracking error converges to zero around 0.80 seconds using the adaptive gain method in the literature by Yan et al. (2019); and the tracking error converges to zero around 0.40 seconds using the adaptive gain method proposed in this paper. For the chattering phenomenon, it can be seen by analyzing the results in Figure 3 that the designed ATSMC method effectively reduces the chattering problem. The results show that the proposed controller can accelerate the convergence of the system state and suppress the chattering phenomenon better.

Simulation 2: The DLR strategy employed in the DILC-ASTA-TSMC scheme is further validated. The performance of the CLR and the DLR proposed in this paper is compared. The parameters of DLR proposed in this paper are $ϱ = 0.15, μ$ = 0.5, $σ$ = 5. Then, the values of CLR are chosen as q = 1.5q = 0.5, respectively. The parameters related to the system model are the same as in Simulation 1. The number of iterative learning is set to 6.

In Figure 4, the convergence of the tracking error using the DLR is faster than that of the CLR. At the same time, from Figure 4, it can be seen that the control force contains larger chattering when a fixed learning rate is used. Therefore, the DLR strategy proposed in this paper can improve the convergence speed of the tracking error and synchronization error of the multi-lift overhead crane system, thus improving the steady-state performance and making the controller robust.

Figure 4.

Tracking errors and control force and synchronization errors under different learning gain strategies: (a) tracking error and (b) control force.

Simulation 3: To verify the performance of the designed model-free controller for DILC-ASTA-TSMC, the above controller was compared with the methods of PID-fast terminal sliding mode control (FTSMC) (Zhong et al., 2020) and ILC-adaptive sliding mode control (ASMC) (Wang et al., 2022). The parameters of the DILC-ASTA-TSMC scheme were referred to Simulation 1, respectively.

It can be seen from Figure 5 and Table 1 that the tracking error convergence of the scheme in this paper is better than that of the PID-FTSMC scheme and the ILC-ASMC scheme while controlling torque smoothing. The DILC-ASTA-TSMC scheme proposed in this paper does not require system model information, uses DILC to compensate the system dynamically, and uses a sliding mode scheme to improve the robustness. The above scheme ensures the effectiveness of this controller.

Figure 5.

Tracking errors and control force under comparative controllers: (a) tracking error and (b) control force.

Table 1.

Comparative items of the three methods.

Method	Tracking errorconvergencetime (seconds)	Chatteringamplitude
PID-FTSMC	1.4	Big
ILC-ASMC	1.0	Small
Proposed method	0.3	Small

To better illustrate the error reduction process, the tracking error of the DILC-ASTA-TSMC scheme for different number of iterations is shown in Figure 6. It can be observed that after many iterations, the tracking error of the controlled system’s convergence becomes faster. To observe the synchronization error in each iteration, the potential load of the double-lift overhead crane system was set to $G_{l 1}$ = 4 $N \cdot m$ , $G_{l 2}$ = 2 $N \cdot m$ . According to Figure 6, the DILC-ASTA-TSMC scheme converges faster in terms of tracking error at different iterations. The synchronization error is significantly reduced with iterations.

Simulation 4: To further verify the robustness of the designed controller, two cases of independent operation mode and interlocking operation mode are simulated. The robustness of the dual spreader in independent operation mode is first verified. The potential load $G_{l 1}$ = 2 $N \cdot m$ , $G_{l 2}$ = 4 $N \cdot m$ and the control system parameters are the same as in Simulation 1. In the independent operation mode of the dual containers, the Container 1 drive motors have a sudden change in the rotational inertia $J_{i}$ of three times at 3 seconds. The Container 1 drive motor viscosity factor $B_{i}$ has a sudden change to two times at 2 seconds. $d_{1}$ changes from 0 to 2 at 3 seconds. The tracking error and synchronization error of the system for this simulation are shown in Figure 7.

Figure 6.

Tracking error and synchronization error of DILC-ASTA-TSMC scheme under different iterations: (a) tracking error, (b) tracking error under different potential loads, and (c) synchronization error under different potential loads.

Figure 7.

Tracking error and synchronization error in independent operation mode: (a) synchronization error, (b) tracking error, (c) synchronization error of the proposed method, and (d) tracking error of the proposed method.

Figure 7 shows the tracking error and synchronization error convergence of the dual containers under uncertain perturbations. Compared with the PID-FTSMC control method and the ILC-ASMC control method, it can be seen from Table 2 that the proposed control scheme can keep the dual containers in a better synchronous control state under the influence of a given parameter perturbation, and the external perturbation and the tracking and synchronization errors can still converge quickly.

Table 2.

Comparative items of the three methods.

Method	Tracking error convergence time (seconds)	Chattering amplitude	Synchronized tracking effect
PID-FTSMC	1.3	Big	Poor
ILC-ASMC	1.0	Small	Less well
Proposed method	0.5	Small	Well

The robustness of the proposed control scheme is verified for the interlocking mode as shown in Figure 8. External perturbations mainly include time-varying perturbations $τ_{e 1} = \sin (t) + 1.5$ , $τ_{e 2} = \cos (t) + 2$ and sudden-change perturbations of amplitude 5 at 3 seconds. In the interlocking mode, it is assumed that the spreader load masses m1 = 3 kg, m2 = 5 kg, and the gravitational acceleration $g = 10 m / s^{2}$ . To better adapt to the practical application of the double-lift overhead crane system, it is assumed that the hoisting motor operates at a constant speed or variable speed in the spreader loading and unloading operation.

Figure 8.

Synchronization error and tracking error in the interlocking mode: (a) synchronization error, (b) tracking error, (c) synchronization error of the proposed method, and (d) tracking error of the proposed method.

Assuming that the hoisting motor runs at a constant speed, there is a difference in the running speed of the spreader due to unfavorable factors, where $w 1 = 2 m / s, w 2 = 1.8 m / s$ . At this time, the spreader has the phenomenon of unsynchronized swing angle $θ_{1} = 8 °$ , $θ_{2} = 10 °$ . The convergence of tracking error and synchronization error is shown in Figure 8.

As shown in Figure 8, the tracking error and synchronization error of the double containers tend to converge in the asynchronous state of the constant speed operation. The proposed control scheme can keep the double containers in a better synchronous control state under the influence of a given disturbance, and the tracking error and synchronization error can still converge quickly.

Conclusion

This paper investigates the problem of synchronous coordinated control of the multi-lift overhead crane system for which an accurate mathematical model is not available under uncertain disturbances. The proposed DILC-ATSA-TSMC controller achieves excellent model-free characteristics, which not only speeds up the convergence of the system state error but also better suppresses the chattering phenomenon. The stability of the designed controller is analyzed using the Lyapunov stability theory, and it is demonstrated that the designed controller can converge in finite time in the time domain. Simulation results verify the effectiveness of the scheme and show that the proposed control scheme has good robustness in both independent and interlocking operation modes. However, the work in this paper does not take into account the input saturation problem that will exist in practice, and the network delay problem in signal transmission. In the future, the controller will be improved to address these issues and the effectiveness will be verified through experiments.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Xinming Jin

Weimin Xu

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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