A reduced-complexity Takagi–Sugeno fuzzy descriptor control approach for double-pendulum overhead crane

Abstract

This paper investigates the control problem of a highly nonlinear double-pendulum overhead crane (DPOC) system. In industrial applications, such cranes are typically modeled as a trolley coupled with a double pendulum, where strong nonlinear couplings and underactuated dynamics make controller design a challenging task. To effectively handle these nonlinearities, a reduced-complexity Takagi–Sugeno fuzzy descriptor (RC-TSFD) model is proposed, which accurately represents the nonlinear behavior of the system using a limited number of linear submodels. Based on this modeling framework, a Linear Matrix Inequality (LMI)-based state-feedback control scheme is developed to ensure closed-loop stability and disturbance attenuation according to Lyapunov stability theory. The proposed method significantly reduces the computational burden associated with the conventional Takagi–Sugeno fuzzy descriptor (TSFD) approach, while maintaining comparable control performance and robustness. Simulation studies demonstrate that the proposed control scheme achieves precise trolley movement and swing reduction, confirming both high efficiency and robustness for the DPOC system.

Keywords

Takagi–Sugeno fuzzy descriptor model double-pendulum overhead crane linear matrix inequality reduced-complexity underacuated system

Introduction

Overhead cranes play a vital role in modern manufacturing, construction, and logistics, serving as essential equipment for the efficient and safe transportation of heavy loads (Mojallizadeh et al., 2023; Yao et al., 2025; Zhang et al., 2023). The primary goal of crane operation is to achieve fast and accurate load transfer while minimizing undesired payload oscillations. Overhead crane systems are inherently nonlinear and underactuated, since the number of control inputs is less than their degrees of freedom. This results in strong coupling between the trolley movement and the load oscillation, making controller development challenging (Zhang et al., 2020; Shen et al., 2021). Therefore, considerable research effort has focused on designing control approaches that ensure accurate trolley motion while suppressing payload swing in both theory and practice.

The single-pendulum overhead crane model serves as a simplified configuration for analyzing crane dynamics and has provided the foundation for numerous control studies. The Input Shaping (IS) method, characterized by its simple design based on the system’s natural oscillation frequency, has been widely proposed to suppress payload sway (Maghsoudi et al., 2017, 2019; Mojallizadeh et al., 2023). Moreover, trajectory planning strategies have been introduced to optimize motion time and reduce oscillations while satisfying specified state and control constraints (Zhang et al., 2014; Chen et al., 2016). Additionally, techniques such as sliding mode control (SMC) (Chwa, 2017; Wang et al., 2021; Zhang et al., 2018), adaptive control (Liu and Xu, 2023; Yang et al., 2022; Huang et al., 2021) have been developed to achieve precise motion control and effective load swing suppression. Furthermore, various intelligent and hybrid control strategies, including fuzzy proportional-integral-derivative (PID) control (Sun et al., 2021; Esleman et al., 2021), neural network–based control (Yang et al., 2021), and SMC–neural network schemes (Wen et al., 2022; Le et al., 2019), have been proposed to enhance control performance and to better handle nonlinearities and system uncertainties. However, control methods derived from the single-pendulum approximation are insufficient to fully represent the complex dynamics of real-world crane systems. In reality, the single-pendulum assumption neglects factors such as the mass of the hoisting assembly and the physical connection between the cable and payload via a hook. Consequently, the oscillations of the cable and payload motion are more accurately described by a double-pendulum configuration. This additional dynamic complexity, characterized by a higher degree of underactuation and strong coupling among system states, significantly increases the challenge of control design and necessitates more sophisticated strategies to achieve both precise trolley positioning and effective swing suppression.

Recent research has increasingly concentrated on the development of effective control schemes for double-pendulum overhead crane (DPOC) systems (Sun et al., 2017; Zhao et al., 2024; Rigatos, 2024). To suppress double-pendulum oscillations, the model reference command shaping method has been proposed to generate reference trajectories for closed-loop control; however, residual vibrations remain (Jaafar et al., 2019). Considering constraints on motion time, swing angle, and obstacle avoidance, trajectory planning approaches have been employed as a foundation for designing tracking controllers (Chen et al., 2017; Zhang et al., 2021). In addition, due to the highly nonlinear and strongly coupled dynamics of DPOC systems, advanced nonlinear control strategies have been developed to achieve better performance than conventional approaches. In Guo et al. (2023), an SMC scheme combined with an extended state observer was proposed to simultaneously regulate trolley position and suppress both pendulum swings. A nonlinear coupled tracking control method based on passivity analysis was also introduced, achieving precise positioning and suppression of double-pendulum oscillations (Li et al., 2024). However, cranes often operate under external disturbances such as gusty winds and parameter uncertainties, which can degrade control performance. To address these challenges, several hybrid and improved control methods have been investigated. In Zhang et al. (2019), an enhanced coupled PD–SMC approach was proposed, where the SMC component improves robustness while the PD component stabilizes the system. Moreover, neural network–based control schemes have been employed to approximate unknown nonlinear components and uncertainties in the system dynamics, thereby improving the overall robustness of the controller (Zhang et al., 2021). Nevertheless, these methods generally require careful controller parameter tuning and involve complex implementation procedures.

It is observed that few studies have simultaneously addressed the need for an accurate modeling framework that captures the nonlinear dynamics of the DPOC system and a control strategy that maintains implementation simplicity while ensuring precise positioning, robustness to uncertainties, and effective disturbance rejection. To address these challenges, this study proposes a control strategy based on the Takagi–Sugeno fuzzy descriptor (TSFD) approach. The TSFD model is well known for its capability to accurately represent nonlinear systems as a convex combination of multiple linear submodels (Huang et al., 2023; Pham et al., 2024; Nguyen et al., 2021). The state-feedback gain matrices can be obtained by formulating the design problem as a set of Linear Matrix Inequality (LMI) based on Lyapunov stability theory, which can then be efficiently solved using convex optimization tools (Boyd et al., 1994). Moreover, during the construction of the LMI conditions, additional performance criteria such as input and output constraints and disturbance rejection can be readily incorporated to enhance system performance (Nguyen et al., 2023; Pourasghar et al., 2025; Sun et al., 2020). Recent studies have shown that Takagi–Sugeno fuzzy modeling combined with Lyapunov-based LMI synthesis offers a systematic convex framework to guarantee stability and robust performance for nonlinear systems (Dong and Nguyen, 2025a, 2025b).

However, the conventional TSFD approach suffers from the exponential growth of fuzzy rules relative to the count of premise variables (Nguyen et al., 2024). As the number of premise variables increases, reflecting higher system nonlinearity, the exponential growth of fuzzy rules imposes a heavy computational burden on numerical solvers, thereby limiting the practical implementation of TSFD-based controllers. Several approaches have been proposed to reduce the number of fuzzy rules. The Eigenvalue Decomposition (ED) method employs orthogonal transformation, where the most significant rules are retained based on the largest eigenvalues, while the remaining ones are considered redundant and eliminated (Yen and Wang, 1999). Alternatively, the Sparse Fuzzy Inference System (FIS) approach formulates system identification as a nonlinear sparsity regularization problem by assigning a distinct weight vector to the fuzzy rules. Rule reduction is then achieved by imposing sparsity constraints, which force the weights of redundant or insignificant rules to converge to zero (Lughofer and Kindermann, 2010). Although these methods effectively reduce the rule base, they often lead to a degradation in the system’s modeling accuracy. The study in Lu and Bai (2019) proposed a novel rule reduction framework that utilizes a rule fusion mechanism to merge fuzzy rules and a space projection technique to capture nonlinear dynamics, thereby maintaining high modeling accuracy. However, this method is currently restricted to system identification and modeling tasks, and its extension to feedback control design remains an open problem. In a different approach, a systematic design procedure using a generalized Takagi–Sugeno fuzzy form was introduced in Taniguchi et al. (2002). This approach facilitates a rapid reduction in the number of fuzzy rules depending on the specific nonlinear terms chosen for replacement, allowing for flexible model simplification. Nevertheless, a limitation is that this simplification introduces modeling errors that must be treated as parameter uncertainties, necessitating the formulation of complex LMIs to ensure robust stability. Inspired by the work in Dehak et al. (2022), this study adopts a modeling scheme of reduced complexity designed for nonlinear descriptor systems. Consequently, a reduced-complexity Takagi–Sugeno fuzzy descriptor (RC-TSFD) model is utilized to exactly describe the nonlinear dynamic behavior of the DPOC system. Instead of relying on conventional product-inference mechanism, the proposed approach exploits the affine structure of the system matrices to decouple the premise variables. This significantly reduces the number of fuzzy rules from an exponential 2^r to a linear 2r (where r is the number of premise variables), thereby enhancing both scalability and computational tractability. Based on the reduced rule set, a state-feedback controller is developed, and the corresponding LMI-based stability conditions are formulated using the Lyapunov stability criterion to ensure asymptotic stability. Furthermore, by explicitly accounting for external disturbances and model uncertainties, the H∞ performance is incorporated into the Lyapunov framework, ensuring system stability while enhancing disturbance attenuation capability.

Based on the above motivation, this study offers the following main contributions:

• A reduced-complexity Takagi–Sugeno fuzzy descriptor (RC-TSFD) model is developed to accurately represent the nonlinear dynamics of DPOC system. The proposed formulation effectively reduces the fuzzy rule base from 2^r to 2r rules, thereby enhancing model scalability and computational feasibility for complex nonlinear systems.

• A control law is formulated using the RC-TSFD model, and the corresponding LMI-based conditions are established through Lyapunov stability analysis to guarantee asymptotic stability of the closed-loop DPOC system.

• H∞ performance is incorporated into the LMI framework to enhance system robustness and disturbance attenuation, enabling the proposed controller to maintain stable and reliable operation under external disturbances and parameter uncertainties.

Notations: For a matrix R, R⁻¹ denotes its inverse, and R^T represents its transpose. The expression R ≻ 0 (R ≺ 0) indicates that R is a symmetric positive (negative) definite matrix. The operator He(R) = R + R^T. The operator diag(⋅) constructs a diagonal matrix from its arguments. The symbol $I_{n}$ denotes the index set of natural numbers {1, 2, …, n}. The symbol ∗ indicates the transposed entries within a symmetric matrix.

Problem formulation

This section first presents the general nonlinear model. Then, the fuzzy modeling approach and the reduced-complexity fuzzy model are introduced as the foundation for the controller design developed in the subsequent section.

Description of Takagi–Sugeno fuzzy descriptor model

The dynamic behavior of the overhead crane can be formulated as a general nonlinear descriptor system described by:

\begin{aligned} M (x) \dot{x} & = f (x, u, d) \\ y & = g (x, u) \end{aligned}

(1)

where

x \in R^{n_{x}}

is the state vector,

u \in R^{n_{u}}

is the control input vector,

d \in R^{n_{d}}

is the external disturbance, and

M (x) \in R^{n_{x} \times n_{x}}

is a state-dependent matrix.

f (x, u, d) : R^{n_{x}} \times R^{n_{u}} \times R^{n_{d}} \to R^{n_{x}}

and

g (x, u) : R^{n_{x}} \times R^{n_{u}} \to R^{n_{y}}

are differentiable with respect to state x.

Remark 1

The dynamic equations of the crane system are derived from the Lagrange equation (Yao et al., 2024); thus, M(x) is regular. While explicit inversion of M(x) yields a standard state-space model, it leads to a highly complex structure (Dong and Nguyen, 2025c). Therefore, based on the TSFD model (Nguyen et al., 2024), system (1) can be described in descriptor form to avoid complexity and reduce the number of fuzzy rules.

The definition $z (t) = {[z_{1} (t), \dots, z_{r} (t)]}^{⊤}$ is introduced as the premise variable of system (1). Owing to the physical limitation of the state x(t), z(t) is subject to bounds. Let x_min and x_max denote the lower and upper bound vectors of the state x(t), respectively. Since z(t) is dependent on x(t), the corresponding range z(t) ∈ [z_min, z_max] for the premise variables can be easily calculated Dong et al. (2025). Firstly, consider a class of nonlinear descriptor systems (1), which can be reformulated in the TSFD model form as follows:

\begin{aligned} E (z) \dot{x} (t) & = A (z) x (t) + B (z) u (t) + D (z) d (t) \\ y & = C (z) x (t) \end{aligned}

(2)

Based on the sector nonlinearity formulation Sala and Arino(2009), the nonlinear descriptor system (2) can be expressed by 2^r linear submodels. After using fuzzy inference, the TSFD model is obtained as follows:

\begin{aligned} \sum_{i = 1}^{2^{r}} w_{i} (z) E_{i} \dot{x} (t) & = \sum_{i = 1}^{2^{r}} w_{i} (z) (A_{i} x (t) + B_{i} u (t) + D_{i} d (t)) \\ y & = \sum_{i = 1}^{2^{r}} w_{i} (z) C_{i} x (t) \end{aligned}

(3)

where E_i, A_i, B_i, D_i, C_i are constant matrices associated with the i^th rule. The membership functions w_i(z) are defined as

w_{i} (z) = \prod_{j = 1}^{r} ζ_{i j} (z_{j} (t)) / \sum_{i = 1}^{2^{r}} \prod_{j = 1}^{r} ζ_{i j} (z_{j} (t))

, where

ζ_{i j} (z_{j} (t))

is membership grade of z_j(t) with respect to the corresponding fuzzy sets. Note that membership function w_j(z) satisfy the convex sum property as follow:

\sum_{i = 1}^{2^{r}} w_{i} (z) = 1

, 0 ≤ w_i(z) ≤ 1.

Remark 2

The fuzzy modeling approach provides a flexible solution for addressing nonlinear problems. However, the application of the sector nonlinearity approach results in 2^r fuzzy rules, which grow exponentially with the number of premise variables. Consequently, for large and complex models, this becomes a drawback that imposes a significant computational burden on the subsequent LMI-based controller design. To overcome this limitation, a reduced-complexity modeling approach is adopted in this study and applied to the DPOC system.

Reduced-complexity model

The matrices E(z), A(z), B(z), D(z), C(z) of nonlinear system (2) affinely depend on z(t), which can be decomposed as follow (Cox et al., 2018):

X (z) = X_{0} + \sum_{j = 1}^{r} z_{j} X_{j}

(4)

where

X \in \{A, B, C, D, E\}

and X₀ denotes the constant system matrix, and X_j represents the constant coefficient matrix associated with the j-th premise variable z_j,

j \in I_{r}

. This decomposition effectively isolates the system nonlinearities, facilitating the construction of a reduced-complexity model. Premise variable z_j can be express through using sector nonlinearity approach:

z_{j} (t) = υ_{j}^{0} (z_{j}) z_{jmin} + υ_{j}^{1} (z_{j}) z_{jmax}

(5)

where

υ_{j}^{0} (z_{j}) = z_{jmax} - z_{j} / z_{jmax} - z_{jmin}

and

υ_{j}^{1} (z_{j}) + υ_{j}^{0} (z_{j}) = 1

with

j \in I_{r}

. By replacing (5) in equation (4), the following expression is obtained:

\begin{align} X (z) & = X_{0} + \sum_{j = 1}^{r} (υ_{j}^{0} (z_{j}) z_{jmin} + υ_{j}^{1} (z_{j}) z_{jmax}) X_{j} \\ = \sum_{j = 1}^{r} \frac{1}{r} (υ_{j}^{0} (z_{j}) + υ_{j}^{1} (z_{j})) X_{0} \\ + \sum_{j = 1}^{r} (υ_{j}^{0} (z_{j}) z_{jmin} + υ_{j}^{1} (z_{j}) z_{jmax}) X_{j} \\ = \sum_{j = 1}^{r} \frac{1}{r} υ_{j}^{0} (z_{j}) (X_{0} + r z_{jmin} X_{j}) \\ + \frac{1}{r} υ_{j}^{1} (z_{j}) (X_{0} + r z_{jmax} X_{j}) \end{align}

(6)

In the generalized form, the premise variable-dependent matrices are expressed as:

\begin{align} X (z) & = \sum_{j = 1}^{r} (υ_{2 j - 1} (z_{j}) X_{2 j - 1} + υ_{2 j} (z_{j}) X_{2 j}) \\ = \sum_{i = 1}^{2 r} υ_{i} (z) X_{i}, i \in I_{2 r} \end{align}

(7)

with

υ_{2 j - 1} (z_{j}) = 1 / r υ_{j}^{0} (z_{j})

υ_{2 j} = 1 / r υ_{j}^{1} (z_{j})

, and X_2j−1 = X₀ + rz_jminX_j,

X_{2 j} = X_{0} + r z_{jmax} X_{j}, j \in I_{r}

The nonlinear system (2) use new weight function in the form:

\begin{aligned} \sum_{i = 1}^{2 r} υ_{i} (z) E_{i} \dot{x} (t) & = \sum_{i = 1}^{2 r} υ_{i} (z) (A_{i} x (t) + B_{i} u (t) + D_{i} d (t)) \\ y & = \sum_{i = 1}^{2 r} υ_{i} (z) C_{i} x (t) \end{aligned}

(8)

where the function υ_i(z) is calculated from equation (7), which satisfies properties:

\begin{aligned} \sum_{i = 1}^{2 r} υ_{i} (z) & = 1, 0 \leq υ_{i} (z) \leq 1 \\ υ_{2 p - 1} (z) + υ_{2 p} (z) & = \frac{1}{r}, p \in I_{r} \end{aligned}

(9)

Equation (8) describes a nonlinear system that is exactly reformulated into 2r linear subsystems. Compared with the fuzzy model (3), this reduced model decreases the number of rules from 2^r to 2r.

To facilitate the later presentation and proofs, the following concise notation is used:

Γ (υ) = \sum_{i = 1}^{2 r} υ_{i} (z) Γ_{i}, Γ \in \{E; A; B; D; C\}

(10)

with

υ = {[υ_{1} (z), \dots, υ_{2 r} (z)]}^{⊤}

Remark 3

It is worth noting that, derived from the affine structure of the system matrices representation in equation (4), the RC-TSFD model (8) is an algebraic reformulation of the original TSFD model (2) Dehak et al. (2022). This implies that the reduced-complexity model not only reduces the fuzzy rule base but also fully preserves the exact nonlinear dynamics of the system.

LMI-based control design

In this section, the reduced fuzzy model, as given in equation (8), is utilized to represent the nonlinear dynamic system. The controller is developed via LMIs based on the Lyapunov stability theory, which guarantees closed-loop stability and simultaneously reduces the influence of disturbances. An LMI-based procedure for the following RC-TSFD control design is presented as follows:

Let us define extend vector $\bar{x} (t) = {[x {(t)}^{⊤} \dot{x} {(t)}^{⊤}]}^{⊤}$ . Then, the reduced-complexity model (8) can be reformulated as:

\begin{aligned} \bar{E} \dot{\bar{x}} (t) & = \bar{A} (υ) \bar{x} (t) + \bar{B} (υ) u (t) + \bar{D} (υ) d (t) \\ y & = \bar{C} (υ) \bar{x} (t) \end{aligned}

(11)

where

\bar{E} = [\begin{matrix} I & 0 \\ 0 & 0 \end{matrix}], \bar{A} (υ) = [\begin{matrix} 0 & I \\ A (υ) & - E (υ) \end{matrix}],

\bar{B} (υ) = [\begin{matrix} 0 \\ B (υ) \end{matrix}], \bar{D} (υ) = [\begin{matrix} 0 \\ D (υ) \end{matrix}], \bar{C} (υ) = [\begin{matrix} C (υ) & 0 \end{matrix}]

The nonlinear feedback control law utilized to design the controller for system (8) takes the following form:

u (t) = - \sum_{i = 1}^{2 r} υ_{i} (z) K_{i} x (t) = - K (υ) x (t)

(12)

with

K_{i} \in R^{n_{u} \times n_{x}}

denoting the control gain, which will be obtained by solving the LMI conditions to be constructed later. By substituting the control law (12) into system (11), the closed-loop system is represented as:

\bar{E} \dot{\bar{x}} (t) = (\bar{A} (υ) - \bar{B} (υ) \bar{K} (υ)) \bar{x} (t) + \bar{D} (υ) d (t)

(13)

where

\bar{K} (υ) = [K (υ) 0] \in R^{n_{u} \times 2 n_{x}}

In the following, the control problem in this paper is formulated:

Problem 1

Consider nonlinear system (1), which is reformulated in the form (8). Under control law (12), determine gain K_i such that the closed-loop system (13) satisfies the following properties:

(i) For d(t) = 0, ∀t ≥ 0, the system (13) is exponentially stable with a decay rate α > 0.

(ii) H∞ performance: Given a positive scalar γ and assuming zero initial condition, with t_f > 0, we have:

\int_{0}^{t_{f}} y {(t)}^{⊤} y (t) \leq γ^{2} \int_{0}^{t_{f}} d {(t)}^{⊤} d (t)

(14)

The following relaxation condition is useful for constructing the LMI conditions.

Lemma 1

Consider the following inequality:

Ψ (h) = \sum_{i = 1}^{2 r} \sum_{i = 1}^{2 r} υ_{i} (z) υ_{j} (z) Ψ_{i j} ≺ 0

(15)

with Ψ_ij representing symmetric matrices with appropriate dimensions. The inequality (15) holds if (Tuan et al., 2001):

\begin{aligned} Ψ_{i i} ≺ 0 \\ Ψ_{i j} + Ψ_{j i} ≺ 0, i, j \in I_{2 r}, i < j \end{aligned}

According to Dehak et al. (2022), the transformation of the nonlinear model into the reduced-complexity form (8) introduces conservatism in controller design due to modeling overbounding. The lemma presented below relaxes the LMI conditions to reduce conservatism, thereby improving the feasibility of the solution search.

Lemma 2

Consider a family of matrices $\{W_{i}, i \in I_{2 r}\}$ with appropriate dimensions satisfying W_2p = W_2p−1 for $p \in I_{r}$ and $W = 1 / r \sum_{p = 1}^{r} W_{2 p}$ . The following equality holds:

X (υ) = \sum_{i = 1}^{2 r} υ_{i} (z) (X_{i} + W_{i} - W)

(16)

where X ∈ {E, A, B, D, C}.

Proof: The summation term in (16) can be expanded as follows:

\begin{align} \sum_{i = 1}^{2 r} υ_{i} (z) (X_{i} + W_{i} - W) \\ = X (υ) + \sum_{i = 1}^{2 r} υ_{i} (z) W_{i} - W \\ = X (z) + \sum_{p = 1}^{r} (υ_{2 p} (z) W_{2 p} + υ_{2 p - 1} (z) W_{2 p - 1}) - W \\ = X (υ) + \sum_{p = 1}^{r} (υ_{2 p} (z) + υ_{2 p - 1} (z)) W_{2 p} - W \end{align}

(17)

By utilizing property (9), which states that υ_2p(z) + υ_2p−1(z) = 1/r, and the definition $W = 1 / r \sum_{p = 1}^{r} W_{2 p}$ (with $p \in I_{r}$ ), it follows that $\sum_{i = 1}^{2 r} υ_{i} (z) W_{i} - W = 0$ . This completes the proof of Lemma 2.

Remark 4

The introduction of slack matrices W_i into the system representation (16) provides additional degrees of freedom for the LMI formulation. Consequently, this structural relaxation helps mitigate the conservatism in the LMI-based design, thereby enhancing the feasibility of the optimization problem.

Subsequently, an LMI-based approach is proposed for the controller design of system (1). The derived conditions are sufficient to guarantee the solution of Problem 1.

Theorem 1

Consider the closed-loop system (13). If there exists a positive definite matrix P₁, matrices P₃, P₄, symmetric matrices W_i, Z_j and a positive scalar α such that the following optimization problem is feasible:

\begin{matrix} minimize & γ^{2} \\ subject to & \{\begin{cases} Ψ_{i j} ≺ 0 \\ Ψ_{i j} + Ψ_{j i} ≺ 0, i, j \in I_{2 r}, i < j \end{cases} \end{matrix}

(18)

where

\begin{array}{l} Ψ_{i j} & = Π_{i j} + W_{i} + Z_{j} - (W + Z), \end{array}

\begin{array}{l} W_{2 p} & = W_{2 p - 1}, W = \frac{1}{r} \sum_{p = 1}^{r} W_{2 p}, \end{array}

\begin{array}{l} Z_{2 p} & = Z_{2 p - 1}, Z = \frac{1}{r} \sum_{p = 1}^{r} Z_{2 p}, p \in I_{r} \end{array}

\begin{array}{l} Π_{i j} & = [\begin{matrix} P_{3}^{⊤} + P_{3} + 2 α P_{1} & * & * & * \\ Θ_{i j} & - E_{i} P_{4} - P_{4}^{⊤} E_{i}^{⊤} & * & * \\ 0 & D_{i}^{⊤} & - γ_{1}^{2} I & * \\ C_{i} P_{1} & 0 & 0 & - I \end{matrix}], \end{array}

\begin{array}{l} Θ_{i j} & = A_{i} P_{1} - B_{i} Y_{j} - E_{i} P_{3} + P_{4}^{⊤} \end{array}

then the properties stated in Problem 1 are satisfied with the control gains

K_{j} = Y_{j} P_{1}^{- 1}

Proof: First, a candidate Lyapunov function is given as follows

V (\bar{x}) = {\bar{x}}^{⊤} \bar{E} P^{- 1} \bar{x}

(19)

with the matrix

P = [\begin{matrix} P_{1} & 0 \\ P_{3} & P_{4} \end{matrix}]

and P₁ ≻ 0, P₄ is invertible

The following two conditions serve to ensure the properties in Problem 1:

\begin{align} \dot{V} (\bar{x}) < - 2 α V (\bar{x}) \end{align}

(20)

\begin{align} y^{⊤} y - γ^{2} d^{⊤} d \leq 0 \end{align}

(21)

It is clear from (20) that $V (\bar{x}) < V (\bar{x} (0)) e^{- 2 α t}$ . Hence the closed-loop system is exponentially asymptotically stable with decay rate α. The following condition allows the formulation of the LMI according to Theorem 1:

\dot{V} (\bar{x}) + 2 α V (\bar{x}) + y^{⊤} y - γ^{2} d^{⊤} d < 0

(22)

where

\dot{V} (\bar{x}) = {(\bar{E} \dot{\bar{x}})}^{⊤} P^{- 1} \bar{x} + {\bar{x}}^{⊤} {(P^{- 1})}^{⊤} \bar{E} \dot{\bar{x}}

\bar{E} P^{- 1} = {(P^{- 1})}^{⊤} \bar{E} = [\begin{matrix} {(P_{1})}^{- 1} & 0 \\ 0 & 0 \end{matrix}]

By substituting the closed-loop system (13) and introducing $ζ = {[{(P^{- 1} \bar{x})}^{⊤} d^{⊤}]}^{⊤}$ , inequality (22) is reformulated as follows:

\dot{V} (\bar{x}) + 2 α V (\bar{x}) + y^{⊤} y - γ^{2} d^{⊤} d = ζ^{⊤} Π (υ) ζ < 0

(23)

with

$Π (υ) = He ([\begin{matrix} ϕ (υ) + 1 / 2 {(\bar{C} (υ) P)}^{⊤} \bar{C} (υ) P & 0 \\ \bar{D} {(υ)}^{⊤} & - γ^{2} / 2 I \end{matrix}])$ , and $ϕ (υ) = (\bar{A} (υ) - \bar{B} (υ) \bar{K} (υ)) P + α \bar{E} P$ .

By applying the Schur complement (Boyd et al., 1994) and considering ζ ≠ 0, inequality (23) can be rewritten as:

He ([\begin{matrix} ϕ (υ) & 0 & 0 \\ \bar{D} {(υ)}^{⊤} & - \frac{γ^{2}}{2} I & 0 \\ \bar{C} (υ) P & 0 & - \frac{I}{2} \end{matrix}]) ≺ 0

(24)

Utilizing the slack matrix property established in Lemma 2 and the definition of Ψ(υ) yields Ψ(υ) = Π(υ) ≺ 0. Finally, substituting the matrices $\bar{E}$ , $\bar{A} (υ)$ , $\bar{B} (υ)$ , $\bar{K} (υ)$ , $\bar{C} (υ)$ , $\bar{D} (υ)$ from the closed-loop system (13) into LMI (24) and utilizing Lemma 1 establishes Theorem 1, guaranteeing the satisfaction of the properties defined in Problem 1.

It is noting that when the external disturbance is absent, the control problem reduces to a stabilization problem, as stated in the following corollary.

Corollary 1

Consider the closed-loop system (13). The system (13) is exponentially stable with a decay rate α > 0 if there exists a positive definite matrix P₁, matrices P₃, P₄, symmetric matrices W_i, Z_j, and a positive scalar α such that the following optimization problem is feasible:

\{\begin{cases} Ψ_{i j} ≺ 0 \\ Ψ_{i j} + Ψ_{j i} ≺ 0, i, j \in I_{2 r}, i < j \end{cases}

(25)

where

\begin{array}{l} Ψ_{i j} & = Π_{i j} + W_{i} + Z_{j} - (W + Z), \\ W_{2 p} & = W_{2 p - 1}, W = \frac{1}{r} \sum_{p = 1}^{r} W_{2 p}, \\ Z_{2 p} & = Z_{2 p - 1}, Z = \frac{1}{r} \sum_{p = 1}^{r} Z_{2 p}, p \in I_{r} \\ Π_{i j} & = [\begin{matrix} P_{3}^{⊤} + P_{3} + 2 α P_{1} & * \\ A_{i} P_{1} - B_{i} Y_{j} - E_{i} P_{3} + P_{4}^{⊤} & - E_{i} P_{4} - P_{4}^{⊤} E_{i}^{⊤} \end{matrix}] \end{array}

then the control gains can be derived as:

K_{j} = Y_{j} P_{1}^{- 1}

Proof: In the absence of external disturbance (d(t) = 0), condition (22) in Theorem 1 can be reduced to:

\dot{V} (\bar{x}) + 2 α V (\bar{x}) < 0

(26)

Similar to Theorem 1, with the Lyapunov function candidate $V (\bar{x}) = {\bar{x}}^{⊤} \bar{E} P^{- 1} \bar{x}$ , it follows from (26) that:

\begin{align} {(\bar{E} \dot{\bar{x}})}^{⊤} P^{- 1} \bar{x} + {\bar{x}}^{⊤} {(P^{- 1})}^{⊤} \bar{E} \dot{\bar{x}} + 2 α {\bar{x}}^{⊤} \bar{E} P^{- 1} \bar{x} \\ = {(P^{- 1} \bar{x})}^{⊤} Π (υ) (P^{- 1} \bar{x}) < 0 \end{align}

(27)

where

Π (υ) = He ((\bar{A} (υ) - \bar{B} (υ) \bar{K} (υ)) P + α \bar{E} P)

. The proof of Corollary 1 is completed by substituting the system matrices

\bar{E}

\bar{A} (υ)

\bar{B} (υ)

, and

\bar{K} (υ)

from (13) and invoking the properties presented in Lemma 1 and Lemma 2.

Simulations

In this section, simulations are conducted to evaluate the proposed control algorithm for the DPOC system. The RC-TSFD model is employed for system modeling and controller design. In addition to the two controllers derived from Theorem 1 and Corollary 1, a conventional TSFD-based controller and adaptive hierarchical sliding mode control (AHSMC) (Ouyang et al., 2019) is also implemented for comparison, highlighting the control performance of the proposed approach.

First, the dynamic equations of the DPOC system, illustrated in Figure 1, are given in Chen et al. (2017) as follows:

\begin{array}{l} (M + m_{h} + m_{p}) \ddot{x} + (m_{h} + m_{p}) l_{1} (\cos θ_{1} \ddot{θ_{1}} - {\dot{θ_{1}}}^{2} \sin θ_{1}) \end{array}

\begin{align} + m_{p} l_{2} \ddot{θ_{2}} \cos θ_{2} - m_{p} l_{2} {\dot{θ_{2}}}^{2} \sin θ_{2} = F + d \end{align}

(28)

\begin{array}{l} (m_{h} + m_{p}) l_{1} \cos θ_{1} \ddot{x} + (m_{h} + m_{p}) g l_{1} s i n θ_{1} \end{array}

\begin{array}{l} + (m_{h} + m_{p}) l_{1}^{2} \ddot{θ_{1}} + m_{p} l_{1} l_{2} \cos (θ_{1} - θ_{2}) \ddot{θ_{2}} \end{array}

\begin{align} + m_{p} l_{1} l_{2} \sin (θ_{1} - θ_{2}) \dot{θ_{1}} \dot{θ_{2}} = 0 \end{align}

(29)

\begin{array}{l} m_{p} l_{2} \cos θ_{2} \ddot{x} + m_{p} l_{1} l_{2} \cos (θ_{1} - θ_{2}) \ddot{θ_{1}} + m_{p} l_{2}^{2} \ddot{θ_{2}} \end{array}

\begin{align} - m_{p} l_{1} l_{2} \sin (θ_{1} - θ_{2}) {\dot{θ_{1}}}^{2} + m_{p} g l_{2} \sin θ_{2} = 0 \end{align}

(30)

where M, m_h, m_p denote masses of the trolley, the hook, and the payload, respectively. The parameters l₁ and l₂ represent the cable length and the distance from the hook center to the payload center, respectively. The angles θ₁ and θ₂ denote the swing angles of the hook and the payload, respectively. F is the control force applied to the trolley, d denotes the external disturbance, and g denotes the gravitational acceleration.

The following premise variables are defined:

\begin{aligned} z_{1} & = (m_{h} + m_{p}) l_{1} \cos θ_{1}, & z_{2} & = m_{p} l_{2} \cos θ_{2} \\ z_{3} & = m_{p} l_{1} l_{2} \cos (θ_{1} - θ_{2}), & z_{4} & = (m_{h} + m_{p}) l_{1} \sin θ_{1} {\dot{θ}}_{1} \\ z_{5} & = m_{p} l_{2} \sin θ_{2} {\dot{θ}}_{2}, & z_{7} & = m_{p} l_{1} l_{2} \sin (θ_{1} - θ_{2}) {\dot{θ}}_{1} \\ z_{6} & = - m_{p} l_{1} l_{2} \sin (θ_{1} - θ_{2}) {\dot{θ}}_{2} \end{aligned}

(31)

Equations (28)–(30) can be reformulated into the nonlinear descriptor system form, consistent with (2), where the state vector is defined as $x^{*} = {[x, θ_{1}, θ_{2}, \dot{x}, {\dot{θ}}_{1}, {\dot{θ}}_{2}]}^{⊤}$ and the control input u = F, yielding:

\begin{aligned} E (z) {\dot{x}}^{*} & = A (z) x^{*} + B u + D d \\ y & = C x^{*} \end{aligned}

(32)

E (z) = [\begin{matrix} I & 0 \\ 0 & E_{22} (z) \end{matrix}]

A (z) = [\begin{matrix} 0 & I \\ A_{21} (z) & A_{22} (z) \end{matrix}]

E_{22} (z) = [\begin{matrix} M + m_{h} + m_{p} & z_{1} & z_{2} \\ z_{1} & (m_{h} + m_{p}) l_{1}^{2} & z_{3} \\ z_{2} & z_{3} & m_{p} l_{2}^{2} \end{matrix}]

A_{21} (z) = [\begin{matrix} 0 & 0 & 0 \\ 0 & - (m_{l} + m_{p}) g l_{1} & z_{6} \\ 0 & z_{7} & - m_{p} g l_{2} \end{matrix}]

A_{22} (z) = [\begin{matrix} 0 & z_{4} & z_{5} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

B = D = {[\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \end{matrix}]}^{⊤}

C = diag(1, 1, 1, 1, 1, 1)

The ranges of the state variables of the DPOC system are given as:

$\begin{matrix} | θ_{1} | \leq π / 12 rad, & | {\dot{θ}}_{1} | \leq π / 4 rad/s \\ | θ_{2} | \leq π / 12 rad, & | {\dot{θ}}_{2} | \leq π / 4 rad/s \end{matrix}$

The parameter values of the DPOC system used in this study are as follows: M = 8 kg, m_h = 2 kg, m_p = 0.85 kg, l₁ = 0.6 m, l₂ = 0.2 m, and g = 9.8 m/s². Based on the DPOC system parameters and the operating ranges of the state variables, the minimum and maximum values of the premise variables can be determined. By employing the RC-TSFD modeling approach, the DPOC system can be represented with only 2 × 7 = 14 fuzzy rules, compared to 2⁷ = 128 in the conventional TSFD model as presented in (3). This reduction not only decreases the computational complexity of solving the LMI problem but also reduces the real-time computational load required to implement the Takagi–Sugeno fuzzy-based control law, thereby facilitating the application of the proposed framework to large-scale nonlinear systems.

In the following, simulation studies are carried out under three cases to validate the effectiveness and robustness of the proposed control design for the DPOC system:

Case 1

Comparison between the exact TSFD model and the proposed RC-TSFD model. The objective of this case is to compare the control performance of the RC-TSFD approach with that of the exact TSFD model. The controller gains are obtained by solving the LMIs in Corollary 1 under conditions d(t) = 0.

Case 2

Evaluation of the proposed controller performance under external disturbances. The RC-TSFD model is used, and the controller gains are computed from the LMI conditions in Theorem 1. Additionally, a comparative study with the advanced AHSMC approach introduced in Ouyang et al. (2019) is conducted to demonstrate the effectiveness of the proposed method.

Case 3

Investigation of controller robustness under parameter variations. To evaluate robustness, two parameter variation cases are considered with respect to the nominal values. In the first variation, denoted as Case 3(a), m_p = 1 kg, l₁ = 0.8 m, and l₂ = 0.1 m. In the second variation, denoted as Case 3(b), m_p = 0.6 kg, l₁ = 0.5 m, and l₂ = 0.3 m.

The simulation results for Case 1 are illustrated in Figure 2, where the reference trolley position is set to 1 m and the decay rate coefficient is α = 0.4. The responses of system states under the RC-TSFD-based controller and the TSFD-based controller are shown as solid blue and dashed red lines, respectively. Both control schemes ensure stable operation of the DPOC system without overshoot. The RC-TSFD-based controller achieves a settling time of 6.98 s, slightly longer than 6.43 s for the TSFD-based controller. However, the RC-TSFD-based controller exhibits smaller swing angles while maintaining precise trolley positioning. Specifically, for the TSFD-based design, the maximum swing angles are approximately 3.2° for θ₁ and 4° for θ₂, whereas under the RC-TSFD-based controller they are reduced to about 2.1° for both angles. These results demonstrate that the RC-TSFD-based approach not only maintains control accuracy comparable to the conventional TSFD model but also offers the advantage of reduced fuzzy rule complexity.

For Case 2, two types of external disturbances are applied, denoted as Case 2 (a) and Case 2 (b), corresponding to d(t) = 30 sin(2t) for t ≥ 0 and a pulse disturbance of d = 40 N acting from t = 10 s to t = 11 s, respectively. The simulation results corresponding to Case 2 are shown in Figures 3 and 4, where the trolley reference position is set to 1 m and the decay rate coefficient is α = 0.4. In terms of transient response, the comparative analysis reveals a distinct trade-off between convergence speed and payload stabilization. The AHSMC approach achieves the fastest settling time of approximately 3 s, which is expected given its aggressive trajectory tracking design. However, this rapid response is accompanied by increased payload sway; specifically, the sway angles under AHSMC are substantially larger—approximately 3° and 5° higher than those of the Theorem 1-based and Corollary 1-based controllers, respectively. Meanwhile, the proposed Theorem 1-based controller achieves a settling time of 4.4 s, showing an improvement over the 6.98 s observed for Corollary 1, while maintaining significantly lower sway amplitudes compared to the AHSMC. Regarding disturbance rejection capability, the differences in performance are evident. The Theorem 1-based controller demonstrates effective attenuation of external perturbations. The response of the Corollary 1-based controller is comparable, though characterized by a slightly prolonged recovery time and minor residual oscillations. In contrast, the AHSMC strategy shows reduced effectiveness under disturbance, exhibiting larger position deviations and slower recovery. Notably, AHSMC shows limited sway suppression capability in this scenario, resulting in sway angles reaching up to 5°. These results confirm that the controller derived from Theorem 1 provides improved disturbance attenuation under the tested conditions.

The simulation results corresponding to Case 3 are presented in Figure 5. It can be observed that when the system parameters including m_p, l₁, and l₂ are varied, the DPOC system still reaches the desired trolley position of 1 m with nearly identical settling times. In addition, slight differences in the swing angles are observed among the nominal, Case 3(a), and Case 3(b) conditions; however, these variations remain small. These results indicate that the RC-TSFD-based controller maintains accurate positioning performance and exhibits robustness against parameter uncertainties.

Conclusion

Accurate control of the double-pendulum overhead crane (DPOC) system presents a significant challenge due to its underactuated nature and strong dynamic coupling, requiring a robust and effective control strategy. In this study, a reduced-complexity Takagi–Sugeno fuzzy descriptor (RC-TSFD) model was developed to accurately represent the nonlinear dynamics of the DPOC system. By reducing the number of fuzzy rules from 2^r to 2r, the proposed model effectively handles high nonlinearity while mitigating the computational complexity associated with LMI-based conditions. Furthermore, by incorporating the H∞ performance criterion into the Lyapunov stability theory, the obtained controller not only ensures stability but also achieves strong disturbance rejection capability. Simulation results from the three test cases conducted demonstrate that the proposed control scheme achieves performance comparable to the exact model, while exhibiting enhanced robustness against external disturbances and parameter uncertainties. Notwithstanding these merits, an inherent constraint of the current framework is its reliance on full state availability, which is a prerequisite for the real-time computation of the premise variables. To mitigate this issue, subsequent studies will explore observer-based designs to compensate for unmeasured states and handle unmeasurable premise variables. Additionally, future work will focus on the experimental implementation of the proposed algorithms on a physical DPOC testbed to verify their practical effectiveness.

Footnotes

ORCID iD

Thi-Van-Anh Nguyen

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2025.39.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article

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