Robust Anti-Swing Control of a 6-DOF Crane via Time-Delay Estimation-Based Adaptive Sliding Mode Control

In recent years, underactuated systems (where the number of variables to be controlled exceeds the number of control inputs) have developed rapidly. In particular, the overhead crane is a typical underactuated system, widely used in various fields such as construction, manufacturing, and transportation due to its high load capacity, efficient material handling, and small footprint. Due to the great applicability of overhead crane systems, they have drawn significant interest among researchers. During operation, several main issues are always of concern: moving the payload to the desired position accurately while simultaneously limiting its swing within an allowable range. Input shaping feedforward control [1, 2], which employs a linear model and swarm optimization, has been used to reduce payload oscillations. This control strategy performs well without the influence of external disturbances or variations in system parameters. However, since open-loop control lacks feedback, it cannot eliminate the effects of external disturbances, making it difficult to achieve control objectives. Feedback control can overcome this limitation. A backstepping controller with tuned parameters is proposed to enhance steady-state performance [3]. A terminal sliding mode controller, combined with the construction of an S-shaped reference trajectory [4], is used to optimize operational efficiency and reduce payload oscillation. The issues of disturbance estimation and rejection are addressed in the studies [5, 6] through the use of a finite-time disturbance observer. The problem of system model uncertainty has been addressed through adaptive control. An online adaptive output shaping controller is effective for a system with an uncertain payload in the study [7]. The controller parameters are tuned by a fuzzy controller, which improved performance as well as controllability [8]. The collision-free motion planning proposed in the study [9], utilizing a second-order sliding mode controller combined with an extended state observer, effectively tackles system uncertainties and external disturbances. The terminal sliding mode controller combined with a fixed-time extended state observer, along with the optimal motion planning based on flatness theory proposed in the study [10], has ensured robust payload transportation in the shortest time.

However, the above studies only focused on single pendulum crane systems without considering the swing of the hook. In practice, the hook swing has a significant impact on payload stability. When the crane system takes hook oscillations into account, the number of degrees of freedom of the system increases, making the suppression of payload swing more challenging. Therefore, anti-sway control for double pendulum crane systems in three-dimensional space is a challenging problem that has attracted considerable attention. Based on system dynamics, an auxiliary control input is introduced to develop a nonlinear anti-sway control method as described in the study [11]. Another approach, an energy-based controller that also considers saturation constraints [12], successfully eliminated payload swing. A hierarchical sliding mode controller with a statedependent switching gain is proposed in the study [13], which effectively suppressed the oscillations of both the hook and the payload. A disturbance observer is developed based on the dynamics of the system, reformulated with an additional auxiliary signal to eliminate the effects of disturbances in the studies [14, 15]. An integral sliding mode controller that constrains system errors, combined with a neural network to estimate unknown terms, is presented in the study [16]. An S-shaped trajectory combined with minimal position error is introduced, and a trajectory tracking adaptive anti sway controller is used to estimate the system parameters online in the study [17]. An adaptive backstepping based hierarchical sliding mode controller is developed by using a simplified dynamic model of the system in the study [18]. The adaptive neural tracking controller presented in the study [19] ensures that the jib and trolley quickly follow the desired trajectory, while eliminating oscillations of the hook and payload. Meanwhile, Yumin et al. [20] introduced an adaptive mutation approach to update the mutation factor in real time, where the controller parameters are tuned using a differential evolution algorithm.

In 6-DOF overhead crane control, challenges arising from strong nonlinearity, the large number of degrees of freedom (DOFs), and significant uncertainties have spurred the development of numerous robust control schemes. Among these, methods based on adaptive sliding mode control (ASMC) are widely applied to ensure robustness and adapt to parametric uncertainties [21, 22]. However, designing the adaptive laws for a 6-DOF system is computationally complex and prone to chattering. Consequently, standalone ASMC often struggles to maintain high performance and durability in real industrial environments.

Concurrently, the Time Delay Estimation (TDE) technique is also employed as a model-free approach to estimate lumped uncertainty [23], but its performance is highly dependent on the sampling rate. When the sampling rate is low, estimation errors increase, which degrades the controller's robustness, particularly in fast-dynamic systems such as 6-DOF overhead cranes. Therefore, the independent application of these schemes frequently entails inherent limitations, necessitating a more effective combined solution.

Indeed, most of the controllers for 6-DOF overhead crane systems discussed above are developed based on detailed system dynamics models or possess complex structures that demand significant computational resources. In this research, a time-delay estimation-based adaptive sliding mode controller (ASMC-TDE) is introduced to control the tracking trajectory of the trolley and effectively suppress the oscillations of both the hook and the payload. The proposed controller not only ensures robust stability but also effectively compensates for model uncertainties and external disturbances. Crucially, it does not require precise model information, complex training, or large data collection, thus consuming minimal computational resources.

The objective of controlling the 3D double-pendulum overhead crane system (3DDOC) must achieve two simultaneous goals: (i) the actuated state variables $x(t)$ and $y(t)$ are controlled to follow a trajectory to the desired position $\left(x_d, y_d\right), x(t) \rightarrow x_d, y(t) \rightarrow y_d$; and (ii) the unactuated state variables $\theta_i(i=1 \div 4)$ are suppressed to zero, $\theta_i \rightarrow 0$. To facilitate the controller design for the 3DDOC system, the system is divided into two subsystems: the actuated subsystem and the unactuated subsystem, corresponding to the state variables $\mathbf{q}_a=[x, y]^T$ and $\mathbf{q}_u= \left[\theta_1, \theta_2, \theta_3, \theta_4\right]^T$, respectively. The system dynamics (2) can then be rewritten as follows:

$\begin{aligned} & \mathbf{M}_{a a}(\mathbf{q}) \ddot{\mathbf{q}}_a+\mathbf{M}_{a u}(\mathbf{q}) \ddot{\mathbf{q}}_u+\mathbf{C}_{a a}(\mathbf{q}, \dot{\mathbf{q}}) \dot{\mathbf{q}}_a+\mathbf{C}_{a u}(\mathbf{q}, \dot{\mathbf{q}}) \dot{\mathbf{q}}_u +\mathbf{G}_{a a}(\mathbf{q})+\mathbf{D}_{a a}=\mathbf{F}_a-\mathbf{U}_{f a}\end{aligned}$             (3)

$\begin{aligned} & \mathbf{M}_{u a}(\mathbf{q}) \ddot{\mathbf{q}}_a+\mathbf{M}_{u u}(\mathbf{q}) \ddot{\mathbf{q}}_u+\mathbf{C}_{u a}(\mathbf{q}, \dot{\mathbf{q}}) \dot{\mathbf{q}}_a+\mathbf{C}_{u u}(\mathbf{q}, \dot{\mathbf{q}}) \dot{\mathbf{q}}_u +\mathbf{G}_{u u}(\mathbf{q})+\mathbf{D}_{u u}=-\mathbf{U}_{f u}\end{aligned}$            (4)

where:

$\begin{aligned} & \mathbf{M}_{a a}(\mathbf{q})=\left[\begin{array}{ll}M_{11} & M_{12} \\ M_{21} & M_{22}\end{array}\right] ; \mathbf{M}_{a u}=\left[\begin{array}{llll}M_{13} & M_{14} & M_{15} & M_{16} \\ M_{23} & M_{24} & M_{25} & M_{26}\end{array}\right] \\ & \mathbf{M}_{u a}(\mathbf{q})=\left[\begin{array}{ll}M_{31} & M_{32} \\ M_{41} & M_{42} \\ M_{51} & M_{52} \\ M_{61} & M_{62}\end{array}\right] ; \mathbf{M}_{u u}=\left[\begin{array}{llll}M_{33} & M_{34} & M_{35} & M_{36} \\ M_{43} & M_{44} & M_{45} & M_{46} \\ M_{53} & M_{54} & M_{55} & M_{56} \\ M_{63} & M_{64} & M_{65} & M_{66}\end{array}\right] ;\end{aligned}$

$\mathbf{C}_{a a}(\mathbf{q}, \dot{\mathbf{q}})=\left[\begin{array}{ll}C_{11} & C_{12} \\ C_{21} & C_{22}\end{array}\right] ;$

$\mathbf{C}_{a u}(\mathbf{q}, \dot{\mathbf{q}})=\left[\begin{array}{llll}C_{13} & C_{14} & C_{15} & C_{16} \\ C_{23} & C_{24} & C_{25} & C_{26}\end{array}\right]$

$\mathbf{C}_{u a}(\mathbf{q}, \dot{\mathbf{q}})=\left[\begin{array}{ll}C_{31} & C_{32} \\ C_{41} & C_{42} \\ C_{51} & C_{52} \\ C_{61} & C_{62}\end{array}\right]$;

$\mathbf{C}_{u u}(\mathbf{q}, \dot{\mathbf{q}})=\left[\begin{array}{llll}C_{33} & C_{34} & C_{35} & C_{36} \\ C_{43} & C_{44} & C_{45} & C_{46} \\ C_{53} & C_{54} & C_{55} & C_{56} \\ C_{63} & C_{64} & C_{65} & C_{66}\end{array}\right]$

$\mathbf{G}_{a a}=[0,0]^T ; \mathbf{G}_{u u}=\left[G_{33}, G_{44}, G_{55}, G_{66}\right]^T ; \mathbf{D}_{a a} \in R^{2,1} ;$

$\mathbf{D}_{u u} \in R^{4,1} ; \mathbf{F}_a=\left[F_x, F_y\right]^T ; \mathbf{U}_{f a}=\left[F_{r x}+d_x \dot{x}, F_{r y}+d_y \dot{y}\right]^T ;$

$\mathbf{U}_{f l}=\left[\begin{array}{llll}d_1 \dot{\theta}_1 & d_2 \dot{\theta}_2 & d_3 \dot{\theta}_3 & d_4 \dot{\theta}_4\end{array}\right]^T$

From Eq. (4), it follows that $\ddot{\mathbf{q}}_u=\mathbf{h}_u(\mathbf{q}, \dot{\mathbf{q}})$, which is then substituted into Eq. (3). The system dynamics can be rewritten as a fully actuated system as follows:

$\mathbf{M}_a(\mathbf{q}) \ddot{\mathbf{q}}_a+\mathbf{h}_a(\mathbf{q}, \dot{\mathbf{q}})=\mathbf{F}_a$              (5)

$\begin{aligned} & \text { with } \mathbf{M}_a=\mathbf{M}_{a a}(\mathbf{q})-\mathbf{M}_{a u}(\mathbf{q}) \mathbf{M}_{u u}^{-1}(\mathbf{q}) \mathbf{M}_{u a}(\mathbf{q}) ; \mathbf{h}_a(\mathbf{q}, \dot{\mathbf{q}})=\left(\mathbf{C}_{a a}(\mathbf{q}, \dot{\mathbf{q}})-\mathbf{M}_{a u}(\mathbf{q}) \mathbf{M}_{u u}^{-1}(\mathbf{q}) \mathbf{C}_{u a}(\mathbf{q}, \dot{\mathbf{q}})\right) \dot{\mathbf{q}}_a \\ & +\left(\mathbf{C}_{a u}(\mathbf{q}, \dot{\mathbf{q}})-\mathbf{M}_{a u}(\mathbf{q}) \mathbf{M}_{u u}^{-1}(\mathbf{q}) \mathbf{C}_{u u}(\mathbf{q}, \dot{\mathbf{q}})\right) \dot{\mathbf{q}}_u+\mathbf{G}_{a a}+\mathbf{D}_{a a}+\mathbf{U}_{f a} -\mathbf{M}_{a u}(\mathbf{q}) \mathbf{M}_{u u}^{-1}(\mathbf{q})\left(\mathbf{G}_{u u}+\mathbf{D}_{u u}+\mathbf{U}_{f u}\right) .\end{aligned}$.

The nominal mass matrix of $\mathbf{M}_a(\mathbf{q})$ is defined as $\mathbf{M}_{\text {const}}$. Eq. (5) can be rewritten as follows:

$\mathbf{M}_{\text {const }} \ddot{\mathbf{q}}_a+\mathbf{f}_a(\mathbf{q}, \dot{\mathbf{q}}, \ddot{\mathbf{q}})=\mathbf{F}_a$             (6)

where: $\mathbf{M}_{\text {const }}=\operatorname{diag}\left(M_{\text {const } 1}, M_{\text {const } 2}\right)$, with $M_{\text {const } 1}$ and $M_{\text {const } 2}$ are positive coefficients $\mathbf{f}_a(\mathbf{q}, \dot{\mathbf{q}}, \ddot{\mathbf{q}})=\left(\mathbf{M}_a-\right. \left.\mathbf{M}_{\text {const }}\right) \ddot{\mathbf{q}}_a+\mathbf{h}_a$.

In Eq. (6), the uncertainties and external disturbances of the 3DDOC system are contained in the vector $\mathbf{f}_a(\mathbf{q}, \dot{\mathbf{q}}, \ddot{\mathbf{q}})$.

Based on the relationship between the actuated and unactuated state variables, the signal error is defined as follows:

$\mathbf{e}=\mathbf{q}_a-\mathbf{q}_r-\mathbf{q}_{l u}$               (7)

with $\mathbf{q}_r=\left[x_r, y_r\right]^T$, the reference trajectory of the trolley when moving along the axis $x$, axis $y ; \mathbf{e}_a=\mathbf{q}_a-\mathbf{q}_r$: tracking error of position control; $\mathbf{q}_{l u}=\left[l_1 \theta_1+l_2 \theta_3 \quad l_1 \theta_2+l_2 \theta_4\right]$.

The sliding surface is designed with the following structure:

$\mathbf{s}=\dot{\mathbf{e}}+\Gamma \mathbf{e}$           (8)

with $\boldsymbol{\Gamma}=\operatorname{diag}\left(\Gamma_1, \Gamma_2\right)$ is the matrix of positive coefficients and the sliding surface matrix. The derivative of the sliding surface (8) with respect to time is expressed as:

$\begin{aligned} \dot{\mathbf{s}} & =\ddot{\mathbf{e}}+\boldsymbol{\Gamma} \dot{\mathbf{e}} \\ & =\ddot{\mathbf{q}}_a-\left(\ddot{\mathbf{q}}_r+\ddot{\mathbf{q}}_{l u}-\boldsymbol{\Gamma} \dot{\mathbf{e}}\right) \\ & =\ddot{\mathbf{q}}_a-\ddot{\mathbf{q}}_{a c}\end{aligned}$              (9)

where: $\ddot{\mathbf{q}}_{a c}=\ddot{\mathbf{q}}_r+\ddot{\mathbf{q}}_{l u}-\boldsymbol{\Gamma} \dot{\boldsymbol{e}}$, with $\ddot{\mathbf{q}}_r=\left[\ddot{x}_r, \ddot{y}_r\right]^T ; \ddot{\mathbf{q}}_{l u}=\begin{aligned}& {\left[l_1 \ddot{\theta_1}+l_2 \ddot{\theta_3} \quad l_1 \ddot{\theta_2}+l_2 \ddot{\theta}_4\right]^T ; \dot{\mathbf{e}}=\dot{\mathbf{q}}_a-\dot{\mathbf{q}}_r-\dot{\mathbf{q}}_{l u} ; \dot{\mathbf{q}}_a=[\dot{x},} \dot{y}]^T ; \dot{\mathbf{q}}_r=\left[\dot{x}_r, \dot{y}_r\right]^T ; \dot{\mathbf{q}}_{l u}=\left[l_1 \dot{\theta_1}+l_2 \dot{\theta_3} \quad l_1 \dot{\theta_2}+l_2 \dot{\theta}_4\right]^T\end{aligned}$.   

An adaptive sliding mode controller leveraging time-delay estimation (ASMC-TDE) is proposed as follows:

$\mathbf{F}_a=\hat{\mathbf{f}}_a+\mathbf{M}_{\text {const }}\left(\ddot{\mathbf{q}}_{a c}-\Lambda \operatorname{sign}(\mathbf{s})-\mathbf{K s}\right)$            (10)

$\hat{\mathbf{f}}_a$ is the estimate of $\mathbf{f}_a$. Then, the estimation error is defined as:

$\boldsymbol{\sigma}=\hat{\mathbf{f}}_a-\mathbf{f}_a$           (11)

Meanwhile, $\boldsymbol{\Lambda}=\operatorname{diag}\left(\Lambda_1, \Lambda_2\right)$ is switching gain matrix and matrice $\mathbf{K}=\operatorname{diag}\left(K_1, K_2\right)$ is linear gain matrix, where $\Lambda_1$ and $\Lambda_2$ are chosen such that $\Lambda_{\text {min }}=\min \left(\Lambda_1, \Lambda_2\right)_{\text {}}$, with $\xi= \left\|\mathbf{M}_{\text {const }}^{-1} \boldsymbol{\sigma}\right\|$.

Using the time-delay estimation technique [25] the estimate $\hat{\mathbf{f}}_a$ is structured as follows:

$\hat{\mathbf{f}}_a=\mathbf{f}_{a(t-T)}=\mathbf{F}_{a(t-T)}-\mathbf{M}_{\mathrm{const}} \ddot{\mathbf{q}}_{a(t-T)}$               (12)

where, T is a very small time delay, selected to be the same as the sampling time (TDE).

The parameters in the proposed ASMC-TDE controller are systematically selected to ensure Lyapunov stability and optimize tracking performance, while effectively mitigating chattering:

$\boldsymbol{\Gamma}$ is chosen such that the eigenvalues of $\boldsymbol{\Gamma}$ establish the desired exponential convergence rate of the error to zero.

$\mathbf{M}_{\text {const}}$ must be chosen as a constant, positive definite matrix that is the best approximation of the actual mass matrix $\mathbf{M}_a(\boldsymbol{q})$ to minimize the residual uncertainty.

T is set equal to the smallest possible sampling period of the discrete-time control system to minimize the TDE estimation error.

$\boldsymbol{\Lambda}$ is a positive diagonal matrix determined by the upper bound of the residual uncertainty.

$\mathbf{K}$ is selected through simulation tuning to balance a fast transient response with the need to avoid actuator saturation.

Theorem 3.1. The sliding surface designed in Eq. (8), the proposed controller in Eq. (10), and the uncertainties and disturbances estimated in Eq. (12) are employed for the 3DDOC system. The signal error e will converge to zero. Simultaneously, the actuated state variables $x$ and $y$ converge to the desired positions $x_d$ and $y_d$ $\left(x \rightarrow x_d, y \rightarrow y_d\right)$, and the swing angles of the hook and payload will converge to zero ($\theta_i \rightarrow 0$).

Proof: The following is the chosen candidate Lyapunov function:

$V=\frac{1}{2} \mathbf{s}^T \mathbf{s}$               (13)

By differentiating the Lyapunov function (13) with respect to time and subsequently substituting (9) and (10) into it, we obtain:

$\begin{array}{cc}\dot{V}= & \mathbf{s}^T\left(\mathbf{M}_{\text {const }}^{-1}\left(\hat{\mathbf{f}}_a-\mathbf{f}_a\right)-\boldsymbol{\Lambda} \operatorname{sign}(\mathbf{s})-\mathbf{K} \mathbf{s}\right) \\ = & \mathbf{s}^T\left(\mathbf{M}_{\text {const }}^{-1} \sigma-\boldsymbol{\Lambda} \operatorname{sign}(\mathbf{s})-\mathbf{K} \mathbf{s}\right) \\ \leqslant & -\left(\Lambda_{\text {min }}-\xi\right)| | \mathbf{s}| |-\mathbf{s}^T \mathbf{K} \mathbf{s} \leqslant 0\end{array}$             (14)

By applying the Lyapunov stability principle, it is ensured that the system is stable and the sliding surface $\mathbf{s}$ reaches zero, $\mathbf{s} \rightarrow 0$ as $t \rightarrow \infty$. This implies that $\dot{\mathbf{e}}+\boldsymbol{\Gamma} \rightarrow 0$. For this firstorder differential equation, the solution is expressed as $e_i= e^{-\Gamma_i t} \rightarrow 0$.

When the system reaches stability, the following equation holds:

$\begin{aligned} & x=l_1 \theta_1+l_2 \theta_3+x_d ; \quad \dot{x}=l_1 \dot{\theta}_1+l_2 \dot{\theta}_3 \\ & y=l_1 \theta_2+l_2 \theta_4+y_d ; \quad \dot{y}=l_1 \dot{\theta}_2+l_2 \dot{\theta}_4\end{aligned}$              (15)

where: $\dot{\mathbf{q}}_a=[\dot{x}, \dot{y}]^T=\mathbf{K}_u \dot{\mathbf{q}}_u$, with $\mathbf{K}_u=\left[\begin{array}{cccc}l_1 & 0 & l_2 & 0 \\ 0 & l_1 & 0 & l_2\end{array}\right]$; $\ddot{\mathbf{q}}_a=\mathbf{K}_u \ddot{\mathbf{q}}_u$.

Since the hook and payload swing angles are typically small during the trolley motion, it is reasonable to approximate $\cos \theta_i \approx 1$ and $\sin \theta_i \approx \theta_i$. Consequently:

$\mathbf{M}_{u a} \approx\left[\begin{array}{cc}\left(m_1+m_2\right) l_1 & 0 \\ 0 & \left(m_1+m_2\right) l_1 \\ m_2 l_2 & 0 \\ 0 & m_2 l_2\end{array}\right] ;$

$\mathbf{M}_{u u} \approx\left[\begin{array}{cccc}\left(m_1+m_2\right) l_1^2 & 0 & m_2 l_1 l_2 & 0 \\ 0 & \left(m_1+m_2\right) l_1^2 & 0 & m_2 l_1 l_2 \\ m_2 l_1 l_2 & 0 & m_2 l_2^2 & 0 \\ 0 & m_2 l_1 l_2 & 0 & m_2 l_2^2\end{array}\right] ;$

$\begin{gathered}\mathbf{G}_{u u} \approx\left[\left(m_1+m_2\right) g l_1 \theta_1,\left(m_1+m_2\right) g l_1 \theta_2, m_2 g l_2 \theta_3, m_2 g l_2 \theta_4\right]^T \\ =\mathbf{K}_G \mathbf{q}_u\end{gathered}$

with $\mathbf{K}_G=\operatorname{diag}\left(\left(m_1+m_2\right) g l_1,\left(m_1+m_2\right) g l_1, m_2 g l_2, m_2 g l_2\right)$.

$\mathbf{C}_{u a}(\mathbf{q}, \dot{\mathbf{q}}) \approx\left[\begin{array}{ll}0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0\end{array}\right] ; \mathbf{C}_{u u}(\mathbf{q}, \dot{\mathbf{q}}) \approx\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$

This is because most components of the $\mathbf{C}_{u a}(\mathbf{q}, \dot{\mathbf{q}})$ and $\mathbf{C}_{u u}(\mathbf{q}, \dot{\mathbf{q}})$ are approximately zero, as they contain products of angles $\left(\theta_i\right)$ with angular velocities $\left(\dot{\theta_i}\right)$, which are treated as higher-order terms.

From the above approximation and Eq. (4), we obtain:

$\begin{aligned} & \mathbf{M}_{u a} \mathbf{K}_u \ddot{\mathbf{q}}_u+\mathbf{M}_{u u} \ddot{\mathbf{q}}_u+\mathbf{C}_{u a} \mathbf{K}_u \dot{\mathbf{q}}_u+\mathbf{C}_{u u} \dot{\mathbf{q}}_u +\mathbf{K}_G \mathbf{q}_u+\mathbf{d} \dot{\mathbf{q}}_u=0\end{aligned}$              (16)

with $\mathbf{d}=\operatorname{diag}\left(d_1, d_2, d_3, d_4\right)$.

The underactuated dynamic Eq. (16) can be rewritten as follows:

$\ddot{\mathbf{q}}_u=-\mathbf{H}_a \mathbf{K}_G \mathbf{q}_u-\mathbf{H}_a\left(\mathbf{C}_{u a} \mathbf{K}_u+\mathbf{C}_{u u}+\mathbf{d}\right) \dot{\mathbf{q}}_u$                 (17)

where, $\mathbf{H}_a=\left(\mathbf{M}_{u a} \mathbf{K}_u+\mathbf{M}_{u u}\right)^{-1}$.

The state vector $\mathbf{z} \in R^{8 \times 1}$ representing the swing angles and angular velocities of the hook and payload is defined as follows:

$\mathbf{z}=\left[\mathbf{q}_u, \dot{\mathbf{q}}_u\right]^T$            (18)

Based on Eq. (18), Eq. (17) can be rewritten as follows:

$\dot{\mathbf{z}}=\mathbf{B} \mathbf{z}$              (19)

in which the state matrix B is described as follows:

$\begin{aligned} \mathbf{B}= & {\left[\begin{array}{cc}\mathbf{0}_{4 \times 4} & \mathbf{I}_{4 \times 4} \\ \mathbf{B}_1 & \mathbf{B}_2\end{array}\right], \text { with } \mathbf{B}_1=-\mathbf{H}_a \mathbf{K}_G ; } \\ & \mathbf{B}_2=-\mathbf{H}_a\left(\mathbf{C}_{u a} \mathbf{K}_u+\mathbf{C}_{u u}+\mathbf{d}\right)\end{aligned}$

The characteristic equation of B is given as follows:

$\left|\lambda \mathbf{I}_{4 \times 4}-\mathbf{B}\right|=0$              (20)

Considering the parameters of the 3DDOC system, the eigenvalues of Eq. (20) all have real parts located on the left half of the imaginary axis, hence the system Eq. (17) is asymptotically stable. This means that:

$\lim _{t \rightarrow \infty} \mathbf{z}=0$            (21)

Substituting Eq. (21) into Eq. (15), we obtain:

$\lim _{t \rightarrow \infty}\left[\begin{array}{llll}x & y & \dot{x} & \dot{y}\end{array}\right]^T=\left[\begin{array}{llll}x_d & y_d & 0 & 0\end{array}\right]^T$              (22)

2.png

Figure 2. The designed control scheme

Hence, the proof of Theorem 3.1 is complete.

The operation of the 3DDOC system under ASMC-TDE control is illustrated in Figure 2, which presents the closed-loop block diagram.

The proposed ASMC-TDE controller is highly suitable for real-time implementation due to its computational simplicity and inherent robustness to practical constraints.

5.1 Real-time feasibility and sampling time

The control structure is primarily algebraic, involving only matrix additions and multiplications of state variables, thereby avoiding computationally intensive tasks like online Jacobian inversion or complex adaptive laws. This low computational load ensures the controller is highly efficient for real-time execution on standard industrial microcontrollers or PLCs.

Crucially, the TDE technique relies on setting the time delay $T$ equal to twice the sampling period $T_s$. For the overhead crane system, $T_s$ can typically be set very small (e.g., 1-10 ms ), ensuring that the unmodeled dynamics and uncertainty remain approximately constant over $T_s$. This guarantees that the TDE estimation error is minimized, maintaining the high precision of the ASMC-TDE scheme.

5.2 Actuator saturation mitigation

Actuator saturation is a critical practical issue. The ASMC-TDE architecture inherently addresses this:

• Compensation: The TDE term $\hat{\mathbf{f}}_a$ compensates for the majority of the large, nonlinear dynamics of the system, drastically reducing the required control effort from the SMC component.
• Chattering Reduction: The small switching gain used in the ASMC-TDE (as justified in Section 3) ensures that the high-frequency control energy is minimized.

This combination allows the total required control input $\left(\mathbf{F}_a\right)$ to remain well within the physical limits of typical industrial actuators, avoiding saturation under most operational conditions.