ESO-based and FTDO-based anti-swing control for overhead cranes with external disturbance

Abstract

Overhead cranes play a pivotal role in various industrial applications. This paper presents two methodologies for suppressing the payload’s swing and completing the accurate positioning. The original crane model is initially transformed into a chained form. Two nonlinear controllers are devised to regulate the trolley’s position and achieve the payload’s anti-swing control. The first approach utilizes backstepping, extended state observer, and filter technique to effectively estimate system state and disturbance and make the closed-loop system uniformly ultimately bounded. In contrast, the second method involves sliding mode control and finite-time disturbance observer to ensure the finite-time convergence of the resulting closed-loop system and the accurate disturbance estimation. Finally, simulation results are provided to validate the effectiveness of the proposed methods.

Keywords

Underactuated system overhead crane disturbance observer extended state observer sliding mode control

Introduction

Recently, researchers have given significant attention to the development and application of underactuated systems, leading to important advancements in the field of automatic control.^1–11 Underactuated systems, which have less independent control inputs than their degrees of freedom, offer advantages like simplicity in structure, low energy consumption, and operational flexibility. Moreover, underactuated systems have numerous critical applications in real-life scenarios, including mobile robots, aerospace vehicles, cranes, inverted pendulums, and so on.^1–7 As a vital transportation tool, overhead cranes confront numerous control challenges. One major research challenge is to achieve precise positioning and oscillation suppression during the motion of trolley, especially when subjected to external disturbances.^8–11 Consequently, many researchers have dedicated their efforts to the automatic control of overhead cranes,^12–25 not only to improve the efficiency and safety but also to enhance the control capabilities and promote the utilization.

However, due to the characteristics of underactuated system, most conventional control techniques are no longer relevant for overhead cranes. The progress of industrial technology and the improvement of technical requirements have compelled researchers to develop more complex control strategies.^16–25 A significant portion of previous studies has focused on the regulation control of overhead cranes using optimal control,^16,17 trajectory planning control,^18,19 input shaping control,^20,21 feedback linearization control.^22,23 Moreover, recent years have witnessed the development of numerous innovative and effective control strategies. For instance, in Sun et al.,²⁴ a novel amplitude-saturated output feedback control approach was proposed to achieve the improved performance while considering practical saturation constraints. Additionally, an enhanced-coupling adaptive controller was designed by incorporating more swing information to enhance the swing suppression performance and robustness.²⁵

In practical applications, overhead cranes are deeply affected by parameter uncertainty and unknown disturbances.^26–30 Thus, it becomes very important to select and design suitable anti-disturbance control strategies.^31–40 On the other hand, active anti-disturbance techniques mainly use disturbance observers to actively estimate and compensate for the existing disturbances, such as extended state observers (ESO), finite-time disturbance observers (FTDO), etc. A practical adaptive tracking controller was proposed without velocity measurements by working in conjunction with ESO^32,33 developed a robust tracking controller for a class of uncertain nonlinear systems with unknown external perturbations and system uncertainties by using backstepping technique and disturbance observer. Based on a disturbance observer,³⁴ designed an adaptive tracking controller for a saturated uncertain nonlinear system. In addition, the sliding mode control methods were respectively proposed for overhead cranes in Wu et al.³⁵ and Zhang et al.³⁶ to achieve the rapid and accurate disturbance rejection by combining an FTDO.³⁷ proposed an optimized backstepping controller for tower cranes based on a fixed-time disturbance observer.

Building on previous work, this paper proposes two anti-disturbance control methods for overhead cranes that effectively handle disturbances. To aid the controller design, the original dynamic model is transformed into a chained structure using a mathematical technique.³⁵ Treating the external disturbance as an extended state, the first anti-disturbance controller is proposed by using backstepping method with ESO technique and filter approach.³⁷ Meanwhile, the sliding mode control method is employed in conjunction with FTDO to design the second anti-disturbance controller, guaranteeing the finite-time convergence of the closed-loop system. The stability of a closed-loop system is eventually deduced from the theoretical analysis of both proposed methods. The suggested controllers’ effectiveness and robustness are then demonstrated through simulation results. In summary, the proposed control methods’ contribution are ascertained as follows:

In this article, two anti-disturbance control methods are provided for underactuated overhead cranes. Based on an ESO, the first suggested anti-disturbance method can estimate both the system states and the external disturbance. The second anti-disturbance approach with an FTDO can estimate the external disturbance exactly within a finite time.

The first anti-disturbance controller employs backstepping design approach with filter technique to prevent the differential explosion issue and attains the positioning and anti-swing control asymptotically. The second anti-disturbance controller can satisfy the requirement for finite-time convergence, improving the control performance and system stability rapidly.

The feasibility and robustness of the suggested controllers against external disturbance are supported by simulation tests.

The remainder of this paper is organized as follows. In Section “Problem formulation,” the crane system is briefly introduced and the control problem is formulated. Section “Backstepping controller with ESO design and analysis” provides the backstepping controller design with ESO. Section “Sliding mode controller with FTDO design and analysis” provides the sliding mode controller design with FTDO. Simulation results are given in Section “Simulation results.” Finally, Section “Conclusion” summarizes the main work of this paper and discusses our future work.

Notations: In this paper, $| x |$ represents the absolute value of $x$ and $x^{[γ]} = sign (x) | x |^{γ}$ $(γ > 0)$ , where $sign (\cdot)$ is the sign function.

Problem formulation

The dynamic model of the considered underactuated overhead crane system (Figure 1) is expressed as^35,36

(M + m) \overset{\cdot\cdot}{x} + ml \overset{\cdot\cdot}{θ} \cos θ - ml {\overset{\cdot}{θ}}^{2} \sin θ = F + d,

(1)

m l^{2} \overset{\cdot\cdot}{θ} + ml \overset{\cdot\cdot}{x} \cos θ + mgl \sin θ = 0

(2)

where $x (t)$ and $θ (t)$ represent the position of trolley and the angle of payload, respectively; $M$ and $m$ denote the masses of trolley and payload, respectively; $l$ is the length of cable; $g$ denotes the gravity acceleration; $F (t)$ is the control input, and $d (t)$ is the term encompassing frictions, uncertain disturbances, unmodelled dynamics, and so on.

After some mathematical arrangements, one can rearrange it as

F = ϕ_{u} u - ml {\overset{\cdot}{θ}}^{2} \sin θ - (M + m) g \tan θ,

(3)

\overset{\cdot\cdot}{x} = - g \tan θ - l \overset{\cdot\cdot}{θ} \sec θ

(4)

where $u (t)$ , $ϕ_{u} (t)$ are the auxiliary functions whose expressions are given as

\begin{matrix} u (t) = \overset{\cdot\cdot}{θ} - ϕ_{d}, \\ ϕ_{d} (t) = - \frac{\cos θ}{(M + m \sin^{2} θ) l} d, \\ ϕ_{u} (t) = - (M + m \sin^{2} θ) l \sec θ . \end{matrix}

The control goal for the overhead crane system is to move the trolley to the appropriate position while also removing the payload swing, that is

lim_{t \to \infty} x (t) = P_{d}, lim_{t \to \infty} θ (t) = 0

(5)

where $P_{d}$ is the desired location of trolley.

In order to form a chained structure, the following auxiliary variables are introduced as³⁵

{\begin{matrix} ξ_{1} = x + l \ln (\sec θ + \tan θ), \\ ξ_{2} = {\overset{\cdot}{ξ}}_{1}, \\ ξ_{3} = g \tan θ, \\ ξ_{4} = {\overset{\cdot}{ξ}}_{3} . \end{matrix}

(6)

Then, based on (6), one can obtain the following dynamic equations

{\begin{matrix} {\overset{\cdot}{ξ}}_{1} = ξ_{2}, \\ {\overset{\cdot}{ξ}}_{2} = ξ_{3} [1 - ρ (ξ_{3}, ξ_{4})], \\ {\overset{\cdot}{ξ}}_{3} = ξ_{4}, \\ {\overset{\cdot}{ξ}}_{4} = - g \overset{2}{\sec} θ (u + ϕ_{d} + 2 {\overset{\cdot}{θ}}^{2} \tan θ) \end{matrix}

(7)

where $ρ (ξ_{3}, ξ_{4})$ is an auxiliary function with the following expression

ρ (ξ_{3}, ξ_{4}) = \frac{l ξ_{4}^{2}}{{(g^{2} + ξ_{3}^{2})}^{3 / 2}} .

Moreover, after some mathematical arrangements, one has that

\frac{l ξ_{4}^{2}}{{(g^{2} + ξ_{3}^{2})}^{3 / 2}} \leq \frac{l ξ_{4}^{2}}{g^{3}} \approx 0.001 l ξ_{4}^{2} << 1 .

According to $ξ_{3} \approx ξ_{3} [1 - ρ (ξ_{3}, ξ_{4})]$ , (7) can be rewritten as

{\begin{matrix} {\overset{\cdot}{ξ}}_{1} = ξ_{2}, \\ {\overset{\cdot}{ξ}}_{2} = ξ_{3}, \\ {\overset{\cdot}{ξ}}_{3} = ξ_{4}, \\ {\overset{\cdot}{ξ}}_{4} = - g \overset{2}{\sec} θ (u + ϕ_{d} + 2 {\overset{\cdot}{θ}}^{2} \tan θ) . \end{matrix}

(8)

As a result, the original control goal stated above, namely, $lim_{t \to \infty} {[\begin{matrix} x & \overset{\cdot}{x} & θ & \overset{\cdot}{θ} \end{matrix}]}^{T} = {[\begin{matrix} p_{d} & 0 & 0 & 0 \end{matrix}]}^{T}$ is equivalently transformed into $lim_{t \to \infty} {[\begin{matrix} ξ_{1} & ξ_{2} & ξ_{3} & ξ_{4} \end{matrix}]}^{T} = {[\begin{matrix} P_{d} & 0 & 0 & 0 \end{matrix}]}^{T}$ .

For convenience, one defines

\begin{matrix} b_{0} = - g \overset{2}{\sec} θ, \\ D = b_{0} ϕ_{d}, \\ f = 2 b_{0} {\overset{\cdot}{θ}}^{2} \tan θ . \end{matrix}

In order to simplify the control system design, it is necessary to make some general assumptions as follows

Assumption 1. The swing angle $θ (t)$ satisfies that $| θ (t) | \leq \frac{π}{9}$ and $| \overset{\cdot}{θ} (t) | \leq N$ where $N$ is an unknown positive constant.

Assumption 2. Define $h (t) = \overset{\cdot}{D}$ , and it is assumed that there is a positive constant $L$ such that $| h (t) | \leq L$ .

Remark 1. In the practical crane control system, the payload swing amplitude is usually limited to $| θ (t) | \leq \frac{π}{9}$ and Assumption 1 has also been used in the literature such as Wu et al.³⁵ This allows us to conclude that both (8) and Assumption 1 are valid. According to Assumption 1, we can establish that $b_{0}$ , $f$ , and ${\overset{\cdot}{b}}_{0}$ , are all bounded.

Backstepping controller with ESO design and analysis

In this section, we propose a backstepping control scheme that utilizes a constructed extended state observer, aiming to enhance the robustness. Furthermore, we employ a nonlinear one-order filter to address the issue of differential explosion and simplify the computation.

ESO design and analysis

Introduce $ξ_{5} = D$ , then system (8) can be rewritten into the following augmented form

{\begin{matrix} {\overset{\cdot}{ξ}}_{1} = ξ_{2}, \\ {\overset{\cdot}{ξ}}_{2} = ξ_{3}, \\ {\overset{\cdot}{ξ}}_{3} = ξ_{4}, \\ {\overset{\cdot}{ξ}}_{4} = b_{0} u + f + ξ_{5}, \\ {\overset{\cdot}{ξ}}_{5} = h (t), \\ y = ξ_{1} . \end{matrix}

(9)

In view of the structure of system (9), the ESO is introduced as

{\begin{matrix} {\overset{\cdot}{\hat{ξ}}}_{1} = {\hat{ξ}}_{2} + l_{1} {\tilde{ξ}}_{1}, \\ {\overset{\cdot}{\hat{ξ}}}_{2} = {\hat{ξ}}_{3} + l_{2} {\tilde{ξ}}_{1}, \\ {\overset{\cdot}{\hat{ξ}}}_{3} = {\hat{ξ}}_{4} + l_{3} {\tilde{ξ}}_{1}, \\ {\overset{\cdot}{\hat{ξ}}}_{4} = b_{0} u + f + {\hat{ξ}}_{5} + l_{4} {\tilde{ξ}}_{1}, \\ {\overset{\cdot}{\hat{ξ}}}_{5} = l_{5} {\tilde{ξ}}_{1} \end{matrix}

(10)

where $\hat{ξ} = {[{\hat{ξ}}_{1}, {\hat{ξ}}_{2}, {\hat{ξ}}_{3}, {\hat{ξ}}_{4}, {\hat{ξ}}_{5}]}^{T}$ is the observer state vector, ${\tilde{ξ}}_{1} = ξ_{1} - {\hat{ξ}}_{1}$ is the estimation error of $ξ_{1}$ , $l_{i} > 0$ , $i = 1, 2, \dots, 5$ , are the observer gains.

Combining (9) and (10), the estimation error system becomes

{\begin{matrix} {\overset{\cdot}{\tilde{ξ}}}_{1} = {\tilde{ξ}}_{2} - l_{1} {\tilde{ξ}}_{1}, \\ {\overset{\cdot}{\tilde{ξ}}}_{2} = {\tilde{ξ}}_{3} - l_{2} {\tilde{ξ}}_{1}, \\ {\overset{\cdot}{\tilde{ξ}}}_{3} = {\tilde{ξ}}_{4} - l_{3} {\tilde{ξ}}_{1}, \\ {\overset{\cdot}{\tilde{ξ}}}_{4} = {\tilde{ξ}}_{5} - l_{4} {\tilde{ξ}}_{1}, \\ {\overset{\cdot}{\tilde{ξ}}}_{5} = - l_{5} {\tilde{ξ}}_{1} + h (t) \end{matrix}

(11)

where ${\tilde{ξ}}_{i} = ξ_{i} - {\hat{ξ}}_{i}$ , $i = 2, \dots, 5$ . The compact form of (11) can described as

\overset{\cdot}{\tilde{ξ}} = A \tilde{ξ} + Eh (t)

(12)

where $\tilde{ξ} = {[{\tilde{ξ}}_{1}, {\tilde{ξ}}_{2}, {\tilde{ξ}}_{3}, {\tilde{ξ}}_{4}, {\tilde{ξ}}_{5}]}^{T}$ , $E = {[0 0 0 0 1]}^{T}$ , and

A = [\begin{matrix} - l_{1} & 1 & 0 & 0 & 0 \\ - l_{2} & 0 & 1 & 0 & 0 \\ - l_{3} & 0 & 0 & 1 & 0 \\ - l_{4} & 0 & 0 & 0 & 1 \\ - l_{5} & 0 & 0 & 0 & 0 \end{matrix}],

which is assumed to be Hurwitz.

Backstepping controller design and analysis

To carry out the control design, some auxiliary transformations are chosen as

\begin{matrix} z_{i} = {\hat{ξ}}_{i} - ϖ_{id}, \\ y_{i} = ϖ_{id} - ϖ_{i}^{*}, i = 2, 3, 4 \end{matrix}

(13)

where $ϖ_{i}^{*}$ represent the virtual controllers and $ϖ_{id}$ are the filter outputs which are generated by the following one-order filters³⁷

τ_{i - 1} {\overset{\cdot}{ϖ}}_{id} = ϖ_{i}^{*} - ϖ_{id} - τ_{i - 1} z_{i - 1}

(14)

where $τ_{i - 1} \in (0, 2)$ are the positive design parameters. According to (10) and (13), we can get that

{\hat{ξ}}_{i} = z_{i} + y_{i} + ϖ_{i}^{*} .

(15)

Step 1: Define the trolley’s tracking error as

z_{1} = y - y_{d}

(16)

where $y_{d} = P_{d}$ denotes the trolley’s reference position, which is differentiable. The derivative of $z_{1}$ with respect to time can be expressed as

{\overset{\cdot}{z}}_{1} = \overset{\cdot}{y} - {\overset{\cdot}{y}}_{d} = ξ_{2} - {\overset{\cdot}{y}}_{d} .

(17)

Then, the virtual control law $ϖ_{2}^{*}$ can be designed as

ϖ_{2}^{*} = - δ_{1} z_{1} + {\overset{\cdot}{y}}_{d}

(18)

where $δ_{1} > 1$ is a positive constant. Choose a Lyapunov function as $W_{1} = \frac{1}{2} z_{1}^{2} + \frac{1}{2} y_{2}^{2}$ . Taking the derivative of $W_{1}$ with respect to time, it yields that

\begin{matrix} {\overset{\cdot}{W}}_{1} = z_{1} {\overset{\cdot}{z}}_{1} + y_{2} {\overset{\cdot}{y}}_{2} \\ = z_{1} (ξ_{2} - {\overset{\cdot}{y}}_{d}) + y_{2} {\overset{\cdot}{y}}_{2} \\ = z_{1} (z_{2} + {\tilde{ξ}}_{2} + y_{2} + ϖ_{2}^{*} - {\overset{\cdot}{y}}_{d}) - y_{2} (\frac{1}{τ_{1}} y_{2} + z_{1} + {\overset{\cdot}{ϖ}}_{2}^{*}) \\ \leq - (δ_{1} - 1) z_{1}^{2} + z_{1} z_{2} - (\frac{1}{τ_{1}} - \frac{1}{2}) y_{2}^{2} + \frac{1}{4} {\tilde{ξ}}^{T} \tilde{ξ} \\ + \frac{1}{2} {| {\overset{\cdot}{ϖ}}_{2}^{*} |}^{2} \end{matrix}

(19)

where the term $z_{1} z_{2}$ will be cancelled in the next step.

Step $i$ $(i = 2, 3)$ : Similar to step 1, we consider the candidate Lyapunov function as

W_{i} = W_{i - 1} + \frac{1}{2} z_{i}^{2} + \frac{1}{2} y_{i + 1}^{2} .

(20)

The derivative of $W_{i}$ with respect to time can be expressed as

\begin{matrix} {\overset{\cdot}{W}}_{i} = {\overset{\cdot}{W}}_{i - 1} + z_{i} (z_{i + 1} + l_{i} {\tilde{ξ}}_{1} + y_{i + 1} + ϖ_{i + 1}^{*} - {\overset{\cdot}{ϖ}}_{id}) \\ - y_{i + 1} (\frac{1}{τ_{i}} y_{i + 1} + z_{i} + {\overset{\cdot}{ϖ}}_{i + 1}^{*}) . \end{matrix}

(21)

The virtual control law $ϖ_{i + 1}^{*}$ can be obtained as

ϖ_{i + 1}^{*} = - δ_{i} z_{i} - l_{i} {\tilde{ξ}}_{1} - \frac{1}{τ_{i - 1}} y_{i} - 2 z_{i - 1}

(22)

where $δ_{i} > 0$ . Combing (21) and (22), we can obtain that

\begin{matrix} {\overset{\cdot}{W}}_{i} = {\overset{\cdot}{W}}_{i - 1} - δ_{i} z_{i}^{2} + z_{i} z_{i + 1} - y_{i + 1} (\frac{1}{τ_{i}} y_{i + 1} + {\overset{\cdot}{ϖ}}_{i + 1}^{*}) \\ \leq - \sum_{j = 1}^{i} δ_{j} z_{j}^{2} - \sum_{j = 1}^{i} (\frac{1}{τ_{j}} - \frac{1}{2}) y_{j + 1}^{2} + z_{i} z_{i + 1} \\ + \sum_{j = 1}^{i} \frac{1}{2} {| {\overset{\cdot}{ϖ}}_{j + 1}^{*} |}^{2} + z_{1}^{2} + \frac{1}{4} {\tilde{ξ}}^{T} \tilde{ξ} \end{matrix}

(23)

where the term $z_{i} z_{i + 1}$ will be handled in the next step.

Step 4: The last candidate Lyapunov function $W_{4}$ is configured as

W_{4} = W_{3} + \frac{1}{2} z_{4}^{2} .

(24)

The time derivative of $W_{4}$ is computed as

{\overset{\cdot}{W}}_{4} = {\overset{\cdot}{W}}_{3} + z_{4} (b_{0} u + f + {\hat{ξ}}_{5} + l_{4} {\tilde{ξ}}_{1} - {\overset{\cdot}{ϖ}}_{4 d}) .

(25)

Ultimately, the real control input $u$ can be proposed as

u = \frac{1}{b_{0}} (- δ_{4} z_{4} - f - {\hat{ξ}}_{5} - l_{4} {\tilde{ξ}}_{1} - \frac{1}{τ_{3}} y_{4} - 2 z_{3}) .

(26)

Substituting (26) into (25), we generate the result that

\begin{matrix} {\overset{\cdot}{W}}_{4} \leq - \sum_{j = 1}^{4} δ_{j} z_{j}^{2} - \sum_{j = 2}^{4} (\frac{1}{τ_{j - 1}} - \frac{1}{2}) y_{j}^{2} \\ + \frac{1}{2} \sum_{j = 2}^{4} | {\overset{\cdot}{ϖ}}_{j} |^{2} + z_{1}^{2} + \frac{1}{4} {\tilde{ξ}}^{T} \tilde{ξ} . \end{matrix}

(27)

Note that ${\overset{\cdot}{ϖ}}_{j}^{*}$ $(j = 2, 3, 4)$ are the continuous functions and bounded in a compact set, which implies that there exists constants $Δ_{j} \geq | {\overset{\cdot}{ϖ}}_{j}^{*} | \geq 0$ . And we can define $Λ = \frac{1}{2} \sum_{j = 2}^{4} Δ_{j} \geq 0$ . Thus the ultimate result of ${\overset{\cdot}{W}}_{4}$ can be rewritten as

\begin{matrix} {\overset{\cdot}{W}}_{4} \leq - (δ_{1} - 1) z_{1}^{2} - \sum_{j = 2}^{4} δ_{j} z_{j}^{2} - \sum_{j = 2}^{4} (\frac{1}{τ_{j - 1}} - \frac{1}{2}) y_{j}^{2} \\ + \frac{1}{4} {\tilde{ξ}}^{T} \tilde{ξ} + Λ . \end{matrix}

(28)

Theorem 1. Assume that Assumptions 1–2 are satisfied for system (8) and the following inequality holds

PA + A^{T} P + PE E^{T} P + \frac{1}{4} I \leq - Q

(29)

where $P$ and $Q$ are the positive definite matrices. Under the proposed controller (26), we have that

(a) all signals of the closed-loop system are uniformly ultimately bounded (UUB);

(b) system output state $y$ tracks the reference signal $y_{d}$ .

Proof. Consider the candidate Lyapunov function $W = W_{\tilde{ξ}} + W_{4}$ where $W_{\tilde{ξ}} = {\tilde{ξ}}^{T} P \tilde{ξ}$ . Hence,

\begin{matrix} {\overset{\cdot}{W}}_{\tilde{ξ}} = {\overset{\cdot}{\tilde{ξ}}}^{T} P \tilde{ξ} + {\tilde{ξ}}^{T} P \overset{\cdot}{\tilde{ξ}} \\ = {\tilde{ξ}}^{T} (A^{T} P + PA) \tilde{ξ} + 2 {\tilde{ξ}}^{T} PEh (t) \\ \leq {\tilde{ξ}}^{T} (A^{T} P + PA) \tilde{ξ} + {({\tilde{ξ}}^{T} PE)}^{2} + h (t)^{2} \\ \leq {\tilde{ξ}}^{T} (A^{T} P + PA + PE E^{T} P) \tilde{ξ} + L^{2} . \end{matrix}

(30)

By combing the results in (28) and (30), one has

\begin{matrix} \overset{\cdot}{W} \leq {\tilde{ξ}}^{T} (A^{T} P + PA + PE E^{T} P + \frac{1}{4} I) \tilde{ξ} + Λ + L^{2} \\ - (δ_{1} - 1) z_{1}^{2} - \sum_{j = 2}^{4} δ_{j} z_{j}^{2} - \sum_{j = 2}^{4} (\frac{1}{τ_{j - 1}} - \frac{1}{2}) y_{j}^{2} \\ \leq - {\tilde{ξ}}^{T} Q \tilde{ξ} - (δ_{1} - 1) z_{1}^{2} - \sum_{j = 2}^{4} δ_{j} z_{j}^{2} \\ - \sum_{j = 2}^{4} (\frac{1}{τ_{j - 1}} - \frac{1}{2}) y_{j}^{2} + Λ + L^{2} \\ \leq - Θ_{1} W + Θ_{2} \end{matrix}

(31)

where $Θ_{1} = min {\frac{λ_{min} (Q)}{λ_{max} (P)}, 2 (δ_{1} - 1), 2 δ_{j}, \frac{2}{τ_{j - 1}} - 1}$ and $Θ_{2} = Λ + L^{2}$ , $j = 2, 3, 4$ . Solving the above inequality yields that

0 \leq W (t) \leq \frac{Θ_{2}}{Θ_{1}} + (W (0) - \frac{Θ_{2}}{Θ_{1}}) e^{- Θ_{1} t}

(32)

which implies that $W (t)$ is eventually bounded by $\frac{Θ_{2}}{Θ_{1}}$ . Based on (32), we obtain that $z_{1}$ , $z_{2}$ , $z_{3}$ , $z_{4}$ , $y_{2}$ , $y_{3}$ , $y_{4}$ , and $\tilde{ξ}$ which are uniformly bounded. Since $z_{1}$ is bounded, it readily follows that $y = ξ_{1}$ is bounded. Furthermore, we can conclude that ${\hat{ξ}}_{1}, \dots, {\hat{ξ}}_{5}$ , $ξ_{2}, \dots, ξ_{5}$ , $ϖ_{2}^{*}$ , $ϖ_{3}^{*}$ , $ϖ_{4}^{*}$ , and $u$ are bounded. Thus, all signals of the closed-loop system are UUB. Moreover, we can obtain that the tracking error $y - y_{d}$ converges to a small neighborhood of zero.

Remark 2. This approach considers disturbance $D$ as an extended state, then an ESO (10) is triumphantly introduced to estimate both the system states and the external disturbance. Furthermore, by incorporating the one-order filter technique at each design step, we can avoid the need for repetitive differentiation of virtual control laws.

Sliding mode controller with FTDO design and analysis

In this section, based on a specifically designed finite-time disturbance observer, a sliding mode control approach is proposed to ensure the finite-time convergence of system states. Moreover, this method not only accurately estimates the external perturbation, but also rapidly improves the system control performance. For the coming controller development, some new variables are introduced as

x_{1} = ξ_{1} - P_{d}, x_{2} = ξ_{2}, x_{3} = ξ_{3}, x_{4} = ξ_{4} .

So the dynamic equation (8) can be rewritten as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2}, \\ {\overset{\cdot}{x}}_{2} = x_{3}, \\ {\overset{\cdot}{x}}_{3} = x_{4}, \\ {\overset{\cdot}{x}}_{4} = b_{0} u + f + D . \end{matrix}

(33)

For system (33), the control objective is equivalently transformed into $lim_{t \to T} {[\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} \end{matrix}]}^{T} = {[\begin{matrix} 0 & 0 & 0 & 0 \end{matrix}]}^{T}$ .

Disturbance observer design

To enhance the system robustness and achieve the swift and precise disturbance estimation, it is recommended to use the FTDO. As motivated by Borkar and Patil,³⁹ an FTDO is introduced as

\begin{matrix} {\overset{\cdot}{\hat{x}}}_{4} = b_{0} u + f + v, \\ v = - λ_{0} L^{\frac{1}{2}} {({\hat{x}}_{4} - x_{4})}^{[\frac{1}{2}]} + \hat{D}, \\ \overset{\cdot}{\hat{D}} = - λ_{1} L sign (\hat{D} - v) \end{matrix}

(34)

where ${\hat{x}}_{4}$ is the estimate of state $x_{4}$ ; $\hat{D}$ is the estimate of lumped disturbance $D$ ; $L$ is the observer gain, and $λ_{0}$ , $λ_{1}$ are the positive constants to be selected. The error estimation errors of observer can be defined as

{\begin{matrix} {\tilde{x}}_{4} = {\hat{x}}_{4} - x_{4}, \\ \tilde{D} = \hat{D} - D . \end{matrix}

(35)

Remark 3. According to Assumption 2, the first derivative of lumped disturbance is limited. Selecting the observer gain $L$ which satisfies Assumption 2 ensures the estimation errors converge to zero within a finite time. Therefore, the estimation errors $({\tilde{x}}_{4} and \tilde{D})$ converge to zero, allowing us to express $x_{4} = {\hat{x}}_{4}$ , $D = \hat{D}$ after a finite time $T_{\tilde{D} = 0}$ , which satisfies that when $t \geq T_{\tilde{D} = 0}$ , $\tilde{D} = 0$ .

Remark 4. The observer gains should be selected to ensure that the observer time is shorter than the controller time ( $T_{\tilde{D} = 0} \leq T$ , $T$ will be introduced later). This ensures that when the system states reach the origin, there is no impact on performance from lumped disturbances or disturbance estimation errors.

Control law design and stability analysis

A sliding mode manifold⁴⁰ for dynamics (33) is designed as

\begin{matrix} σ = b_{0} u + \hat{D} + f + c_{4} x_{4}^{[α_{4}]} + c_{3} x_{3}^{[α_{3}]} \\ + c_{2} x_{2}^{[α_{2}]} + c_{1} x_{1}^{[α_{1}]} \end{matrix}

(36)

where $c_{i}$ and $α_{i}$ $(i = 1, 2, 3, 4)$ are the positive constants. The coefficients $c_{i}$ should be selected such that the following polynomial

P (s) = s^{4} + c_{4} s^{3} + c_{3} s^{2} + c_{2} s + c_{1}

(37)

is Hurwitz. The fractional powers $α_{i}$ are determined using the following recursion relations

α_{i - 1} = \frac{α_{i} α_{i + 1}}{2 α_{i + 1} - α_{i}}, i = 2, 3, 4 .

(38)

where $α_{5} = 1$ , $α_{4} = α_{c} \in (1 - ε, 1)$ with $ε \in (0, 1)$ , and $α_{c}$ is the pretuned control exponent.

Theorem 2. For system (33) with the designed nonlinear sliding mode surface (36), if the control law is designed as⁴⁰

u = - b_{0}^{- 1} (u_{n} + u_{eq}),

(39)

{\overset{\cdot}{u}}_{n} = κ sign (σ),

(40)

u_{eq} = \hat{D} + f + c_{4} x_{4}^{[α_{4}]} + c_{3} x_{3}^{[α_{3}]} + c_{2} x_{2}^{[α_{2}]} + c_{1} x_{1}^{[α_{1}]}

(41)

where $κ > 0$ is a switching gain, then under the designed disturbance observer (34), the designed controller can guarantee that all the closed-loop system states are bounded and finite-time convergent, that is

lim_{t \to T} {[\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} \end{matrix}]}^{T} = {[\begin{matrix} 0 & 0 & 0 & 0 \end{matrix}]}^{T} .

Proof. Using the control law specified in (39) and applying it to the sliding dynamic results in

σ = - u_{n} .

(42)

Taking the time derivative of (42) yields that

\overset{\cdot}{σ} = - κ sign (σ) .

(43)

Now, a candidate Lyapunov function is considered as

V_{σ} = \frac{1}{2} σ^{2}

(44)

whose derivative $V_{σ}$ with respect to time satisfies that

{\overset{\cdot}{V}}_{σ} = σ \overset{\cdot}{σ} = - κ | σ | = - \sqrt{2} κ V_{σ}^{\frac{1}{2}} .

(45)

Hence, $V_{σ}$ and $σ$ converge to zero within a finite time $T_{σ = 0}$ , which can be expressed as

T_{σ = 0} \leq \frac{\sqrt{2}}{κ} V_{σ}^{\frac{1}{2}} (0) .

(46)

Next, another candidate Lyapunov function $V$ defined in terms of system states $x_{i}$ $(i = 1, 2, 3, 4)$ and sliding variable $σ$ is given as

V = \frac{1}{2} (σ^{2} + \sum_{i = 1}^{4} x_{i}^{2}) .

(47)

Differentiating $V$ along system dynamic (33) and sliding surface dynamic (43), we have

\begin{matrix} \overset{\cdot}{V} = - κ | σ | + \sum_{i = 1}^{3} x_{i} x_{i + 1} + x_{4} {\overset{\cdot}{x}}_{4} \\ \leq \sum_{i = 1}^{3} \frac{x_{i}^{2} + x_{i + 1}^{2}}{2} + x_{4} {\overset{\cdot}{x}}_{4} \\ \leq 2 V + x_{4} {\overset{\cdot}{x}}_{4} . \end{matrix}

(48)

With estimation error (35) in mind, combining (33) and (36), one has

\begin{array}{l} {\dot{x}}_{4} = b_{0} u + f + [D - \hat{D}] + \hat{D} \\ = σ - c_{4} x_{4}^{[α_{4}]} - c_{3} x_{3}^{[α_{3}]} - c_{2} x_{2}^{[α_{2}]} - c_{1} x_{1}^{[α_{1}]} - \tilde{D} \\ \leq | σ | + c_{4} | x_{4} |^{α_{4}} + c_{3} | x_{3} |^{α_{3}} + c_{2} | x_{2} |^{α_{2}} + c_{1} | x_{1} |^{α_{1}} + | \tilde{D} | . \end{array}

(49)

Note that for parameters $α_{i}$ $(i = 1, 2, 3, 4)$ satisfying $0 < α_{i} < 1$ , the following inequalities hold

| x_{i} |^{α_{i}} < 1 + α_{i} | x_{i} | < 1 + | x_{i} | .

(50)

Substituting (50) into (49) leads to

{\overset{\cdot}{x}}_{4} < | σ | + \sum_{i = 1}^{4} (c_{i} + c_{i} | x_{i} |) + | \tilde{D} |

(51)

which is substituted into (48) to obtain

\begin{matrix} \overset{\cdot}{V} \leq 2 V + | x_{4} σ | + | x_{4} | \sum_{i = 1}^{4} (c_{i} + c_{i} | x_{i} |) + | x_{4} \tilde{D} | \\ \leq 2 V + \frac{x_{4}^{2} + σ^{2}}{2} + \sum_{i = 1}^{4} (c_{i} \frac{x_{4}^{2} + x_{i}^{2}}{2}) \\ + (\sum_{i = 1}^{4} c_{i} + | \tilde{D} |) \frac{x_{4}^{2} + 1}{2} \\ \leq (2 + 1 + 3 \sum_{i = 1}^{4} c_{i} + | \tilde{D} |) V + \frac{1}{2} (\sum_{i = 1}^{4} c_{i} + | \tilde{D} |) . \end{matrix}

The above inequality can be written in the following compact form

\overset{\cdot}{V} \leq Ψ_{1} V + Ψ_{2}

(52)

with

Ψ_{1} = 3 + 3 \sum_{i = 1}^{4} c_{i} + | \tilde{D} | \leq Ψ_{1 \max},

Ψ_{2} = \frac{1}{2} (\sum_{i = 1}^{4} c_{i} + | \tilde{D} |) \leq Ψ_{2 \max} .

Since $c_{i}$ are the positive bounded constants and the estimation error $\tilde{D}$ is bounded and will converge to zero within a finite time, both terms $Ψ_{1}$ and $Ψ_{2}$ are positive and bounded. Therefore, we conclude that $V$ and system states $x_{i}$ $(i = 1, 2, 3, 4)$ will not flee to infinity within a finite time.

Since the sliding surface $σ$ converges to zero within a finite time, as shown in (46), there exists a bounded time $T_{σ = 0}$ such that the system dynamic (33) remains in the sliding surface manifold $σ = 0$ when $t \geq T_{σ = 0}$ . Recalling that

{\overset{\cdot}{x}}_{4} = - \sum_{i = 1}^{4} c_{i} x_{i}^{[α_{i}]} - \tilde{D}

(53)

and defining

T = max {T_{σ = 0}, T_{\tilde{D} = 0}},

(54)

it can be claimed that when $t \geq T$ , there holds

0 = {\overset{\cdot}{x}}_{4} + \sum_{i = 1}^{4} c_{i} x_{i}^{[α_{i}]} .

(55)

Based on the conclusion that state variables of the closed-loop system are convergent within a finite time, in the sense that

lim_{t \to T} {[\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} \end{matrix}]}^{T} = {[\begin{matrix} 0 & 0 & 0 & 0 \end{matrix}]}^{T}

(56)

that is

lim_{t \to T} {[\begin{matrix} x & \overset{\cdot}{x} & θ & \overset{\cdot}{θ} \end{matrix}]}^{T} = {[\begin{matrix} P_{d} & 0 & 0 & 0 \end{matrix}]}^{T}

(57)

Hence, the proof of this theorem is completed.

Remark 5. The proposed FTDO can demonstrate accurate estimation of the unknown disturbance within a finite time. Moreover, the sliding mode controller implemented with FTDO guarantees the finite-time convergence of the system.

Simulation results

In this section, we provide numerical results that validate the superior functioning of the two proposed approaches. The simulation outcomes demonstrate the remarkable resistance to disturbances of the two controllers when compared to current methods. Furthermore, the system parameters are set as $M = 24 kg$ , $m = 12 kg$ , $l = 2 m$ , and $g = 9.8 m / s^{2}$ . During the control process, the following disturbance is imposed on the crane system: $d (t) = 5 \sin (0.4 π t)$ , which is generated by use of the Simulink block.

Comparative results of the first method

In this subsection, a comparison is made between the first proposed control technique and existing control methods to demonstrate superior performance. First, we have selected the following reference trajectory as the desired input

P_{1 d} = {\begin{matrix} \begin{matrix} (P_{d} - x_{0}) (\frac{t}{t_{f}} - \frac{1}{2 π} \sin (\frac{2 π t}{t_{f}})) + α_{0}, t \in [0, t_{f}) \\ P_{d}, t \in [t_{f}, + \infty) \end{matrix} \end{matrix}

where $x_{0}$ represents the initial position, $P_{d}$ represents the final target position and $t_{f}$ represents the reaching time, where $x_{0} = x (0) = 0 m$ and $t_{f} = 5 s$ . The representation of the Enhanced Damping-Based Control (EDC) method in Wu and He⁴¹ is

\begin{matrix} F = (M + m \sin θ^{2}) [- k_{d} (k_{p} \overset{\cdot}{x} - λ k_{p} \sin θ - k_{E} \overset{\cdot}{θ} \cos θ) \\ + k_{E} \sin θ - k_{p} (x - λ \int_{0}^{t} \sin θ (τ) d τ - P_{d})] \\ - m \sin θ (g \cos θ + l {\overset{\cdot}{θ}}^{2}) \end{matrix}

wherein $k_{p}$ , $k_{d}$ , $k_{E}$ , $λ$ are positive control gains and turned sufficiently to be chosen as $k_{p} = 0.5$ , $k_{d} = 3$ , $k_{E} = 2$ , $λ = 2$ . As described in Zhang et al.,⁴² the expression of the PSCDC method for partially saturated coupled-dissipation control is

\begin{array}{l} F = - k_{p} (M + m \sin^{2} θ) \tanh (x - l \int_{0}^{t} \sin θ (τ) d τ - P_{d} - \sin θ) \\ - \frac{k_{d}}{M + m \sin^{2} θ} (\dot{x} - l \sin θ - \frac{1}{l} \cos θ \dot{θ}) \\ - m \sin θ (g \cos θ + l {\dot{θ}}^{2}) - (M + m \sin^{2} θ) l \cos θ \dot{θ} \end{array}

wherein $k_{p} = 3$ , $k_{d} = 2500$ .

The physical parameters are chosen as $P_{d} = 2 m$ , $δ_{1} = 1$ , $δ_{2} = 3$ , $δ_{3} = 10$ , $δ_{4} = 20$ , $τ_{1} = 0.1$ , $τ_{2} = 0.13$ , $τ_{3} = 0.13$ , $ω = 100$ , $l_{1} = 5 ω$ , $l_{2} = 10 ω^{2}$ , $l_{3} = 20 ω^{3}$ , $l_{4} = 10 ω^{4}$ , and $l_{5} = 10 ω^{5}$ . The initial conditions are set as ${[\begin{matrix} ξ_{1} (0) & ξ_{2} (0) & ξ_{3} (0) & ξ_{4} (0) \end{matrix}]}^{T} = {[\begin{matrix} 0 & 0 & 0 & 0 \end{matrix}]}^{T}$ , ${[\begin{matrix} {\hat{ξ}}_{1} (0) & {\hat{ξ}}_{2} (0) & {\hat{ξ}}_{3} (0) & {\hat{ξ}}_{4} (0) \end{matrix}]}^{T} = {[\begin{matrix} 0 & 0 & 0 & 0 \end{matrix}]}^{T}$ , $θ (0) = 0$ .

The simulation results in Figures 2 and 3 provide a comprehensive evaluation of the first control method’s performance. As shown in Figure 2, we can see that the state $ξ_{i}$ can be estimated well by the designed ESO. As depicted in Figure 3(a) to (c), the first method can make $x$ track $P_{d}$ quickly without residual pendulum angle, and its control input is bounded, which indicates that the steady-state performance of this proposed method is superior to the other. In addition, Figure 3(d) demonstrates that the control method exhibits good robustness by effectively handling the disturbance to ensure stable and accurate control performance. Overall, these findings provide strong evidence supporting the performance and applicability of the proposed approach.

Comparative results of the second method

In this subsection, we do some optimization based on the DOB method in Wu et al.³⁵ and show the effectiveness and robustness of the second method. The associated control variables are configured as $P_{d} = 5 m$ , $λ_{0} = 3$ , $λ_{1} = 2$ , $L = 8$ , $κ = 0.001$ , $α_{4} = 0.9$ , $α_{3} = 0.8182$ , $α_{2} = 0.75$ , $α_{1} = 0.6923$ , $c_{1} = 1$ , $c_{2} = 3$ , $c_{3} = 3.6$ , and $c_{4} = 2.5$ . The initial conditions are set as ${[\begin{matrix} x_{1} (0) & x_{2} (0) & x_{3} (0) & x_{4} (0) \end{matrix}]}^{T} = {[\begin{matrix} 0 & 0 & 0 & 0 \end{matrix}]}^{T}$ , ${[\begin{matrix} {\hat{x}}_{4} (0) & \hat{D} (0) \end{matrix}]}^{T} = {[\begin{matrix} 0 & 0 \end{matrix}]}^{T}$ , $x (0) = 0$ , $θ (0) = 0$ .

Simulation results are presented in Figure 4, which provides valuable insights into the performance of the second proposed method. After a careful look at Figure 4, we can see that the trolley can follow the reference trajectory precisely and the swing angle is also suppressed into a small range of 3 degrees, which is less than the DOB method. Additionally, no residual cargo swing is seen after the trolley stops. The control input $F$ is bounded. Furthermore, $\hat{D}$ can track $D$ exactly. Based on these results, it can be inferred that the proposed method exhibits good anti-disturbance capabilities, efficient and satisfactory control performance.

Conclusion

This paper presents two distinct methods for mitigating sway motions and enhancing stability for overhead cranes. The first approach involves the development of an extended state observer that treats disturbances as part of the state. When combined with backstepping control and filter techniques, this approach effectively reduces sway and enhances stability. The second method proposes a sliding mode controller with a finite-time disturbance observer, which ensures the finite-time convergence of the closed-loop system and enables accurate estimation of the disturbance within a finite time. Furthermore, the anti-disturbance capability of both methods is supported by theoretical analysis. Finally, simulation results demonstrate that the proposed control methods exhibit superior effectiveness and robustness compared to existing control methods.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partly supported by the National Natural Science Foundation of China (62173207), the China Postdoctoral Science Special Foundation (2023T160334), the Youth Innovation Team Project of Colleges and Universities in Shandong Province (2022KJ176), and the Graduate Teaching Case Base Project of Shandong Province (SDYAL20109). The National Key Research and Development Program of China (2021YFE0193900).

ORCID iD

Zhongcai Zhang

Data availability statement

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

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