A hybrid adaptive interval type-2 fuzzy control and descriptor-based observer scheme for liquid sloshing crane systems

Abstract

This paper investigates the control problem of liquid slosh suppression in overhead crane systems. In industrial applications, the liquid sloshing crane system (LSCS) transfers liquid within a container in a short time with minimal oscillation to ensure process efficiency and safety under model parameter uncertainties and unmeasurable state variables. To address these challenges, an adaptive interval type-2 fuzzy control with descriptor-based observer scheme (AIT2FC-DO) is proposed. The adaptive interval type-2 fuzzy control (AIT2FC) employs a composite sliding surface that couples actuated and unactuated states, combined with a fuzzy approximator to estimate uncertainties based on Interval Type-2 Fuzzy systems using the Nie–Tan type-reduction method to reduce computational complexity, thereby generating the control signal through adaptive update laws, while all state variables required by the AIT2FC are estimated using a descriptor-based observer (DO) constructed from a partially linearized Takagi–Sugeno descriptor model. The stability of the AIT2FC is first established, followed by the convergence of the estimation error of the DO, and finally the stability of the overall AIT2FC-DO system is proven. The ME, RMS, ITAE, and settling time indices are also provided to quantitatively evaluate the controller performance. Compared with input shaping, the proposed closed-loop control scheme demonstrates improved effectiveness and robustness for the LSCS by reducing the liquid elevation during position tracking by approximately 5 mm.

Keywords

adaptive type-2 fuzzy control composite sliding surface descriptor-based observer liquid sloshing suppression crane system

1. Introduction

The demand for liquid transportation using overhead crane systems has been steadily increasing in harsh industrial environments, where high precision is required to prevent excessive sloshing and vibrations. As a typical underactuated system with strong nonlinearities and parameter uncertainties, it poses a significant control challenge. In practice, two main objectives must be achieved simultaneously: the payload should be transferred to the target position as quickly as possible to improve productivity, while the liquid sloshing must be minimized to meet technological requirements.

Due to the significant difficulty in measuring certain system states during liquid transportation (Ibrahim, 2005) the implementation of feedback control architectures becomes challenging. As a result, input shaping (Alshaya and Alghanim, 2020) and feedforward methods are more commonly employed. For instance, in a typical industrial casting process, heavy liquid metal must be transported by overhead cranes, which is considered highly hazardous due to high temperature, humidity, and dust conditions (Terashima and Yano, 2001). In such scenarios, vibration suppression through controller design is essential to ensure human safety against the risks associated with sloshing of hot liquid metal. Furthermore, the use of liquid level sensors inside the container is practically infeasible.

Therefore, various input-shaping-based control schemes have been proposed for liquid sloshing crane systems (LSCS) without requiring feedback sensors. The optimal design of the shape and size of a rod container device, where the sloshing dynamics in each transfer direction are represented by a double pendulum model, is presented in Kaneshige et al. (2008). Optimal Command Shaping employing a cost-effective and smooth continuous command shaper based on a multisine-wave function with independent time parameters is introduced to suppress sloshing vibrations (Khorshid and Al-Fadhli, 2021). Additionally, a smooth polynomial-shaped command with adjustable duration is developed to eliminate residual vibrations in a multi-mode liquid-filled container transported while preserving robustness against parameter uncertainties (Alshaya and Almujarrab, 2021). A feedforward controller incorporating notch filter terms is utilized to suppress the sway of the payload in an autonomous mobile overhead crane system (Kaneshige et al., 2009). In Li et al. (2022), three optimal trajectory planning strategies based on quasi-convex optimization, namely minimum-time trajectory planning (MTTP), minimum-energy trajectory planning (METP), and time-energy optimal trajectory planning (TEOTP), are proposed to simultaneously suppress container swing and liquid sloshing. However, input-shaping methods inherently exhibit limited disturbance rejection capability in industrial environments, while feedforward approaches remain highly sensitive to parameter uncertainties.

Meanwhile, feedback control approaches based on closed-loop systems, which provide strong disturbance rejection and reduced sensitivity to parameter variations, have also been applied to slosh suppression for this class of systems. Conventional methods such as proportional–integral–derivative (PID) control have been employed to regulate lateral sloshing represented by a simple pendulum model Bandyopadhyay et al. (2009), while a data-driven PID tuning approach based on safe experimentation dynamics (SED) has been proposed for liquid sloshing control (Shukor et al., 2017). In addition, advanced control techniques such as H-infinity synthesis with pole clustering based on LMI region schemes have also been investigated for liquid slosh control (Tumari et al., 2019). Indeed, the H-infinity controller is highly promising, as it can be utilized for anti-disturbance control of Markov jump systems with mismatched disturbances, thereby ensuring closed-loop stability with the desired H-infinity performance (Shen et al., 2021). Robust control of container–sloshing coupled dynamical systems has been achieved through the synthesis of the variable gain super-twisting algorithm (VGSTA) (Mishra and Kurode, 2014). Additionally, a fuzzy sloshing controller has also been proposed for industrial liquid transport tasks requiring high-precision motion performance (Hussien et al., 2025). However, these aforementioned approaches generally require multiple state variables to be measured and utilized in the feedback control scheme.

Over the past decades, fuzzy logic systems have been widely applied for various purposes, including universal approximation of nonlinear and uncertain components in models, such as linear and multi-linear functions (Zeng and Singh, 1996); direct regulation of nonlinear systems through inference mechanisms based on human reasoning (Nguyen et al., 2025); online adaptation of controller parameters for PID or sliding mode controllers (Yesil et al., 2003); estimation of unmeasurable or unknown state variables (Dong et al., 2026); proactive compensation of unknown nonlinear behaviors to handle sensor and actuator attacks while ensuring stable consensus among agents (Khan et al., 2025); and mitigation of errors caused by friction or dead-zone nonlinearities (Jang, 2001).

Fuzzy systems are generally classified into three main categories: Mamdani-like systems, Takagi–Sugeno (T-S) fuzzy systems, and Mamdani fuzzy systems. First, T-S fuzzy systems, also referred to as TSK systems, are considered model-based approaches because they employ mathematical functions in the consequent part, enabling accurate representation of local system dynamics through the combination of linear time-invariant subsystems (Precup et al., 2024). Furthermore, T-S models can be integrated with advanced methods such as guaranteed-cost event-triggered anti-disturbance control for wind turbines (Gu et al., 2024) and fault detection (FD) for discrete-time T-S fuzzy systems via iterative interval estimation (Shen et al., 2023b). Second, Mamdani-like fuzzy systems have recently attracted increasing attention because they combine the interpretability of Mamdani systems with modern optimization and computational capabilities to effectively handle highly nonlinear systems with significant uncertainties (Nguyen et al., 2019). Third, interval type-2 Mamdani fuzzy systems directly regulate nonlinear systems based on human reasoning through inference mechanisms (Nguyen et al., 2025) and control underactuated mobile two-wheeled inverted pendulums (MTWIPs) subject to modeling uncertainties and external disturbances (Huang et al., 2018). Therefore, the application of fuzzy systems to controller and observer design for liquid sloshing crane systems (LSCS) is well justified, as it enables suppression of undesirable effects while facilitating state estimation from readily measurable variables.

Furthermore, the LSCS is a nonlinear underactuated system; therefore, many existing studies on nonlinear and underactuated systems can be extended to this class of systems. Notably, Ye (2018) employed saturated controllers to stabilize uncertain feedforward strongly nonlinear underactuated systems while guaranteeing asymptotic stability of the corresponding reduced system. In addition, networked nonlinear systems have been controlled using a data-driven approach with event-triggered output mechanisms to ensure uniform ultimate boundedness of the tracking error system (Shen et al., 2023a). Moreover, an extended Kalman filtering-based nonlinear model predictive control method has been developed for underactuated systems (Zhai et al., 2024). With the increasing demand for high-performance control of underactuated systems, fuzzy systems have been widely applied in both theoretical studies and practical applications (Huang et al., 2019).

Following the aforementioned motivations and limitations of existing methodologies for LSCS, this study focuses on the design of an adaptive interval type-2 fuzzy control integrated with a descriptor-based observer (AIT2FC-DO) for the underactuated LSCS. The adaptive interval type-2 fuzzy control (AIT2FC), employing Interval Type-2 fuzzy basis functions, acts as an approximator for uncertain components, while leveraging the advantages of a composite sliding surface that incorporates both actuated and unactuated states to enhance position tracking performance in the initial phase and prioritize liquid sloshing suppression in the final phase. The Descriptor-based Observer is designed to estimate the angular states of the underactuated system using only a minimal set of measurable variables, namely position and velocity, which are readily available in industrial environments. The estimation error is ensured to converge rapidly to zero, thereby guaranteeing the stability of the overall closed-loop system when integrated with the AIT2FC.

The main contributions of this study are summarized as follows:

• The proposed AIT2FC-DO methodology is capable of achieving stable control and suppressing liquid sloshing within a short time, while maintaining robustness against uncertainties in model parameters.

• The Descriptor-based Observer, constructed from a partially linearized Takagi–Sugeno descriptor model, is capable of estimating the full states of the LSCS, making it suitable for industrial environments where certain variables are not directly measurable.

• The stability of the AIT2FC is proven, the estimation state errors are shown to converge to zero, and the stability of the overall system under the integrated AIT2FC-DO framework is established.

2. System model

This section describes the dynamic equations of the liquid sloshing crane system (LSCS) based on the Lagrangian modeling technique to design a closed-loop control for suppressing liquid vibration in the container of the overhead crane, as shown in Figure 1. These equations are developed under the assumptions that the flow is time-invariant, no slipping of the trolley occurs, the displacement and velocity of the liquid surface are infinitesimal, and the container is a rigid, impermeable body. As established in Kaneshige et al. (2008), the LSCS model for three-dimensional transfer is adapted for two-dimensional transfer in the Oxy plane. The position of the entire LSCS is determined by the trolley, which is driven by force u₁ (N) along the x-axis, with a swing angle θ₁ (rad) oscillating from an equilibrium. The rigid rectangular container of size l × w (m) is filled with liquid to a level of h (m).

Figure 1.

The liquid sloshing crane system.

As reported in Qi et al. (2023), higher sloshing dynamics may become relevant under broader or more aggressive operating conditions, for which a multi-degree-of-freedom pendulum model can provide improved representation. In this study, the container is operated under moderate and smooth acceleration; therefore, the influence of higher sloshing modes is assumed to be negligible. Consequently, the rod–container assembly is represented by the first pendulum, while the liquid vibration corresponds to the second. The nonlinear equations of the LSCS are subsequently derived into compact matrix-vector form using the Lagrangian mechanics framework as follows:

M (X) \ddot{X} + C (X, \dot{X}) \dot{X} + D \dot{X} + G (X) = R u

(1)

where

X = {[x, θ_{1}, θ_{2}]}^{T}

denotes the state vector, M(X) is the inertia matrix,

C (X, \dot{X})

represents the Coriolis and centrifugal effects, D is the damping coefficient matrix, while G(X) stands for the gravity vector. The influence of the calculated control input

u = {[u_{1}, 0,0]}^{T}

on the state variables is determined by the mapping matrix R. The specific components for each of these vectors and matrices are detailed below:

\begin{aligned} M (X) & = [\begin{matrix} M_{11} & M_{12} & M_{13} \\ M_{21} & M_{22} & M_{23} \\ M_{31} & M_{32} & M_{33} \end{matrix}], R = [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] \\ C (X) & = [\begin{matrix} 0 & C_{12} & C_{13} \\ 0 & 0 & C_{23} \\ 0 & C_{32} & 0 \end{matrix}], D = [\begin{matrix} 0 & 0 & 0 \\ 0 & ν l_{1}^{2} & 0 \\ 0 & 0 & 0 \end{matrix}] \\ G (X) & = [\begin{matrix} 0 \\ G_{2} \\ G_{3} \end{matrix}] \\ M_{11} & = m_{1} + m_{2}, \\ M_{13} & = M_{31} = m_{2} l_{2} \cos θ_{2} = m_{13} \cos θ_{2} \\ M_{12} & = M_{21} = (m_{1} + m_{2}) l_{1} \cos θ_{1} = m_{12} \cos θ_{1} \\ M_{22} & = i_{1} + (m_{1} + m_{2}) l_{1}^{2}, M_{33} = i_{2} + m_{2} l_{2}^{2} \\ M_{23} & = M_{32} = m_{2} l_{1} l_{2} \cos (θ_{1} - θ_{2}) \\ = m_{23} \cos (θ_{1} - θ_{2}) \\ C_{12} & = - (m_{1} + m_{2}) l_{1} \sin θ_{1} \dot{θ_{1}} = c_{12} \sin θ_{1} \dot{θ_{1}} \\ C_{13} & = - m_{2} l_{1} l_{2} \sin θ_{2} \dot{θ_{2}} = c_{13} \sin θ_{2} \dot{θ_{2}} \\ C_{23} & = m_{2} l_{1} l_{2} \sin (θ_{1} - θ_{2}) \dot{θ_{2}} \\ C_{32} & = - m_{2} l_{1} l_{2} \sin (θ_{1} - θ_{2}) \dot{θ_{1}} \\ = c_{32} \sin (θ_{1} - θ_{2}) \dot{θ_{1}} \\ G_{2} & = (m_{1} + m_{2}) g l_{1} \sin θ_{1} = g_{2} \sin θ_{1} \\ G_{3} & = m_{2} g l_{2} \sin θ_{2} = g_{3} \sin θ_{2} \end{aligned}

(2)

where m₁ and m₂ denote the masses of the rod–container system and the internal liquid, respectively, l₁ represents the distance to the equilibrium point of the rod–container system, l₂ is the equivalent length of the sloshing pendulum while θ₂ represents the swing angle of the sloshing model, i₁ and i₂ signify the moments of inertia for the rod–container system and the liquid relative to their respective equilibrium points, g is the gravitational acceleration. Finally,

\dot{X} = {[\dot{x}, {\dot{θ}}_{1}, {\dot{θ}}_{2}]}^{T}

and

\ddot{X} = {[\ddot{x}, {\ddot{θ}}_{1}, {\ddot{θ}}_{2}]}^{T}

represent the velocity and acceleration vectors of the state variables.

The surface elevation of the liquid at the container’s edge, as shown in Figure 1, is calculated based on the swing angle of the sloshing model θ₂ and is formulated as follows:

ς (t) = \frac{l}{2} \tan (θ_{2} (t) - θ_{1} (t))

(3)

3. Control methodology

As previously mentioned, the LSCS is an underactuated system characterized by strong nonlinearities and uncertainties. Additionally, the difficulty in measuring specific state variables using sensors poses a significant obstacle to conventional closed-loop feedback methods.

To overcome these challenges, an AIT2FC-DO scheme is proposed as illustrated in Figure 2, which integrates an adaptive interval type-2 fuzzy controller (AIT2FC) with a descriptor-based observer (DO) to achieve high-precision control performance. Within this framework, the AIT2FC employs a composite sliding surface to couple actuated and unactuated states, while an Interval Type-2 fuzzy approximator based on the Nie–Tan type-reduction method efficiently estimates uncertainties with reduced computational complexity via adaptive laws. Meanwhile, a descriptor-based observer constructed from a T-S descriptor model reconstructs the required state variables. The red lines represent measurable variables, the blue lines indicate the estimated variables, and the black lines denote the desired points and the composite sliding surface.

Figure 2.

Control structure.

The theoretical proof is completed by establishing the stability of the AIT2FC, followed by the convergence of the observer, and finally the Lyapunov-based stability of the overall integrated AIT2FC-DO system.

3.1. Actuated–underactuated decomposition

According to equations (1) and (2), the LSCS is defined as a single-input multi-output (SIMO) underactuated system. In this setup, the position x serves as the actuated state, whereas the angles θ₁ and θ₂ represent the underactuated variables. These angles reflect the coupled sloshing dynamics, which are influenced indirectly through internal system interactions involving the trolley acceleration (Yang et al., 2021).

Assumption 1

Given the predefined system parameters m₁, m₂, l₁, l₂, i₁, i₂, ν and the bounded state variables $θ_{1}, θ_{2}, {\dot{θ}}_{1}, {\dot{θ}}_{2}$ , there exist positive bounding constants $\underset{̲}{m}, \bar{m}, \bar{c}, \bar{d}$ , and $\bar{g}$ such that:

\begin{aligned} \underset{̲}{m} I & \leq M (X) \leq \bar{m} I \\ {\bar{m}}^{- 1} I & \leq M^{- 1} (X) \leq {\underset{̲}{m}}^{- 1} I \end{aligned}

(4)

\begin{aligned} ‖ C (X, \dot{X}) \dot{X} ‖ \leq \bar{c} ‖ \dot{X} ‖^{2} \\ ‖ D \dot{X} ‖ \leq \bar{d} ‖ \dot{X} ‖, ‖ G (X) ‖ & \leq \bar{g} ‖ X ‖ \end{aligned}

(5)

where I is the identity matrix. For the numerical experiment, the prescribed operating bounds are selected according to the admissible motion range and actuator capability of the considered system. The closed-loop responses are verified to remain within these bounds, over which M(X) is positive definite and invertible, thereby ensuring the validity of the state-space representation and the associated LMI-based controller synthesis.

Under Assumption 1, equation (1) can be reformulated to facilitate further analysis as below:

\begin{aligned} \ddot{X} = & M^{- 1} (X) R u - M^{- 1} (X) C (X, \dot{X}) \dot{X} \\ - M^{- 1} (X) D \dot{X} - M^{- 1} (X) G (X) \\ = & N (X) u - ζ (X, \dot{X}) \end{aligned}

(6)

where

N (X) = M^{- 1} (X) = {[n_{1}^{T}, n_{2}^{T}, n_{3}^{T}]}^{T}

is also a symmetric and positive-definite matrix. Meanwhile, the vector

ζ (X, \dot{X})

combines all known or partially measurable dynamic components, including the Coriolis, centrifugal, friction, and gravitational terms, and is defined as follows:

\begin{aligned} ζ (X, \dot{X}) & = M^{- 1} (X) [C (X, \dot{X}) + D \dot{X} + G (X)] \\ = {[ζ_{1}, ζ_{2}, ζ_{3}]}^{T} \end{aligned}

(7)

Based on the kinematic equation (1) of the LSCS, it is evident that the actuated state x is strongly coupled with the two underactuated states θ₁ and θ₂ via the horizontal acceleration $\ddot{x}$ . Consequently, equation (6) can be rearranged to explicitly reflect this relationship as follows:

[\begin{matrix} \ddot{x} \\ {\ddot{q}}_{x} \end{matrix}] = [\begin{matrix} n_{1} u - ζ_{1} \\ N_{x} u - ζ_{x} \end{matrix}]

(8)

where

N_{x} = {[n_{2}^{T}, n_{3}^{T}]}^{T} \in R^{2 \times 3}

and

ζ_{x} = {[ζ_{2}, ζ_{3}]}^{T} \in R^{2 \times 1}

3.2. Adaptive interval type-2 fuzzy control with composite surface

The composite surface of the proposed method is formulated using the tracking errors between the actual state variables $X = {[x, θ_{1}, θ_{2}]}^{T}$ and the desired states $X_{d} = {[x_{d}, θ_{1 d}, θ_{2 d}]}^{T}$ , defined as follows:

\begin{aligned} e & = X - X_{d} = {[e_{1}, e_{2}, e_{3}]}^{T} \\ \dot{e} & = {[{\dot{e}}_{1}, {\dot{e}}_{2}, {\dot{e}}_{3}]}^{T} \\ \ddot{e} & = {[{\ddot{e}}_{1}, {\ddot{e}}_{2}, {\ddot{e}}_{3}]}^{T} \end{aligned}

(9)

Then, the normal sliding surfaces are defined as the auxiliary variables:

\begin{aligned} s_{1} & = {\dot{e}}_{1} + λ_{1} e_{1} \\ s_{2} & = {\dot{e}}_{2} + λ_{2} e_{2} \\ s_{3} & = {\dot{e}}_{3} + λ_{3} e_{3} \end{aligned}

(10)

where λ_i (i = 1, 2, 3) are positive control parameters.

The corresponding unactuated auxiliary variables are then integrated with the actuated state to construct the composite surface S, as follows:

S = s_{1} + α_{2} s_{2} + α_{3} s_{3}

(11)

where α_j (j = 2, 3) represent positive adjustable parameters. Both λ_i and α_j are selected to guarantee the asymptotic stability of the closed-loop equilibrium. Furthermore, the time derivative of the composite surface S is denoted as:

\begin{aligned} \dot{S} & = {\dot{s}}_{1} + α_{2} {\dot{s}}_{2} + α_{3} {\dot{s}}_{3} \\ = {\ddot{e}}_{1} + λ_{1} {\dot{e}}_{1} + α_{2} {\ddot{e}}_{2} + α_{3} {\ddot{e}}_{3} + α_{2} λ_{2} {\dot{e}}_{2} + α_{3} λ_{3} {\dot{e}}_{3} \\ = {\ddot{e}}_{1} + Γ {\ddot{e}}_{x} + χ \end{aligned}

(12)

where Γ = [α₂, α₃],

{\ddot{e}}_{x} = {[{\ddot{e}}_{2}, {\ddot{e}}_{3}]}^{T}

, and

χ = λ_{1} {\dot{e}}_{1} + α_{2} λ_{2} {\dot{e}}_{2} + α_{3} λ_{3} {\dot{e}}_{3}

By substituting equations (7) and (8) into equation (12), the time derivative of the composite surface S is represented as:

\begin{aligned} \dot{S} & = (n_{1} u - ζ_{1}) + χ + Γ (N_{x} u - ζ_{x}) \\ = (n_{1} + Γ N_{x}) u - (ζ_{1} + Γ ζ_{x} - χ) \\ = Φ u_{1} - (ζ_{1} + Γ ζ_{x} - χ) \\ = (\bar{Φ} + Δ Φ) u_{1} - (ζ_{1} + Γ ζ_{x} - χ) \\ = \bar{Φ} u_{1} - (ζ_{1} + Γ ζ_{x} - χ - Δ Φ u_{1}) \\ = \bar{Φ} u_{1} + f (X, \dot{X}, u_{1}) \end{aligned}

(13)

where

Φ = \bar{Φ} + Δ Φ = n_{1} + Γ N_{x}

consists of two parts: the nominal constant

\bar{Φ}

and the unknown component ΔΦ. These terms relate to the nominal values and the uncertainties of the matrix

M (X, \dot{X})

in equation (1) and its inverse matrix N(X). Subsequently, all unknown elements and system uncertainties are grouped into a single function

f (X, \dot{X}, u_{1})

To mitigate the impact of $f (X, \dot{X}, u_{1})$ , a universal approximator is developed using interval type-2 fuzzy logic and its inference rules, following the structure in Cao et al. (2019). Consequently, the function $f (X, \dot{X}, u_{1})$ can be written as:

f (X, \dot{X}, u_{1}) = f (χ) = φ^{* T} ω (χ) + ε

(14)

where

χ = {[X^{T}, {\dot{X}}^{T}, u_{1}]}^{T} \in R^{7}

represents the input vector of the type-2 fuzzifier. To reduce the computational load, the Nie–Tan type-reduction method (Nie and Tan, 2008) is used instead of the Karnik-Mendel (KM) algorithm. This technique is applied to each component ω_j(χ) of the fuzzy basis function vector

ω (χ) = {[ω_{1} (χ), \dots, ω_{9} (χ)]}^{T} \in R^{9}

as follows:

ω_{j} (χ) = \frac{\prod_{i = 1}^{7} [{\underset{̲}{μ}}_{i}^{j} (χ_{i}) + {\bar{μ}}_{i}^{j} (χ_{i})]}{\sum_{j = 1}^{9} \{\prod_{i = 1}^{7} [{\underset{̲}{μ}}_{i}^{j} (χ_{i}) + {\bar{μ}}_{i}^{j} (χ_{i})]\}}

(15)

where j = 1, 2, …, 9 indicates the j-th fuzzy rule, while

{\underset{̲}{μ}}_{i}^{j} (χ_{i})

and

{\bar{μ}}_{i}^{j} (χ_{i})

denote the lower and upper membership functions, respectively, for the i-th input variable.

It should be noted that, in (14), $φ^{*} = [φ_{1}^{*}, \dots, φ_{j}^{*}] \in R^{9}$ denotes the ideal optimal constant weight vector associated with the neural-network approximation and is introduced only for stability analysis. Therefore, it is not required to be known or identified by the online adaptive strategy. Moreover, φ∗ is assumed to be bounded, with existing upper bounds ${\bar{φ}}_{j}^{*}$ satisfying ${\bar{φ}}_{j}^{*} \geq | φ_{j}^{*} | \geq 0$ , while the approximation error ɛ is also bounded by a positive constant ɛ∗ such that ɛ∗ ≥|ɛ|.

Finally, an adaptive control law is constructed to guarantee stability and tracking performance, defined as:

\begin{aligned} u_{1} (t) = & {\bar{Φ}}^{- 1} [- K S - {\hat{φ}}^{T} ω (χ) sign (S) \\ - \hat{ε} sign (S) - K_{s} sign (S)] \end{aligned}

(16)

where

\hat{φ} = [{\hat{φ}}_{1}, \dots, {\hat{φ}}_{9}]

and

\hat{ε}

denote the estimated values of the ideal parameters φ∗ and ɛ, respectively. The term sign(S) refers to the standard sign function. Furthermore, K is a positive scalar constant, while K_s is a positive gain constant satisfying K_s > ϖ, here ϖ is a parameter specified in the stability analysis.

The corresponding adaptation laws for $\hat{φ}$ and $\hat{ε}$ are then defined as follows:

\begin{aligned} Ξ & = Λ ω S \\ \dot{\hat{φ}} & = proj (Ξ) \\ {\dot{\hat{φ}}}_{j} & = proj (Ξ_{j}) \\ = \{\begin{cases} 0, & {\hat{φ}}_{j} = {\underset{̲}{φ}}_{j} and Ξ_{j} < 0 \\ Ξ_{j}, & else \end{cases} \\ \dot{\hat{ε}} & = ψ S sign (S) = ψ | S | \end{aligned}

(17)

where

{\underset{̲}{φ}}_{j} \in R (j = 1, \dots, 9)

represents the predefined lower bound of

\hat{φ}

. Additionally, the positive parameters Λ and ψ determine the learning rates of the approximation. Furthermore, to guarantee system stability, the initial condition

{\hat{φ}}_{j} (0)

must be chosen to satisfy

{\hat{φ}}_{j} (0) \geq {\underset{̲}{φ}}_{j}

(Lemma 1).

3.3. T-S descriptor modeling

Before designing the T-S descriptor-based observer for the LSCS, equation (1) must be transformed into a nonlinear descriptor state-space form (Guelton et al., 2008). By defining the state vector as $q = {[x, θ_{1}, θ_{2}, \dot{x}, {\dot{θ}}_{1}, {\dot{θ}}_{2}]}^{T} = {[X^{T}, {\dot{X}}^{T}]}^{T}$ , the generic nonlinear descriptor state-space model of the LSCS is formulated as follows:

\{\begin{cases} E (q) \dot{q} = A (q) q + B u \\ Y = C q \end{cases}

(18)

where

\begin{aligned} E (q) & = [\begin{matrix} I_{3 \times 3} & 0_{3 \times 3} \\ 0_{3 \times 3} & M (X) \end{matrix}] \\ A (q) & = [\begin{matrix} 0_{3 \times 3} & I_{3 \times 3} \\ - [\begin{matrix} 0 & 0 & 0 \\ 0 & \frac{G_{2}}{θ_{1}} & 0 \\ 0 & 0 & \frac{G_{3}}{θ_{2}} \end{matrix}] & - [\begin{matrix} 0 & C_{12} & C_{13} \\ 0 & ν l_{1}^{2} & C_{23} \\ 0 & C_{32} & 0 \end{matrix}] \end{matrix}] \\ B & = [\begin{matrix} 0_{3 \times 3} \\ R \end{matrix}], C = [\begin{matrix} I 0_{3 \times 3} \end{matrix}] \end{aligned}

(19)

In descriptor form, the T-S fuzzy representation produces multiple LTI models that are smoothly combined using nonlinear functions:

\{\begin{cases} \sum_{k = 1}^{l} υ_{k} (z (t)) E_{k} \dot{q} = \sum_{m = 1}^{r} η_{e} (z (t)) (A_{m} q + B_{m} u (t)) \\ y = \sum_{m = 1}^{r} η_{e} (z (t)) C_{m} q \end{cases}

(20)

where l and r represent the number of fuzzy sets for the left-hand and right-hand sides of the descriptor formulation, respectively. The premise vector z(t) is constructed from the state variables within vector p, which are associated with the nonlinear components of the system. The functions υ_k(z(t)) > 0 (k = 1, …, l) and η_m(z(t)) > 0 (m = 1, …, r) ensure the convex sum properties:

\sum_{k = 1}^{l} υ_{k} (z (t)) = 1

and

\sum_{e = 1}^{r} η_{m} (z (t)) = 1

. Finally, the matrices E_k, A_m, B_m, and C_m correspond to the l × r LTI subsystems that constitute the T-S descriptor model.

In the T-S descriptor framework, the number of LTI subsystems depends on the nonlinear terms considered. As the number of nonlinear terms increases, the resulting analysis becomes more conservative. For the generic nonlinear descriptor state-space model of the LSCS, Assumption 2 is introduced to mitigate this conservatism by neglecting negligible nonlinear terms.

Assumption 2

Within the LSCS system, the swing angles θ₁ and θ₂ are assumed to remain sufficiently small, aligning with the control objective to minimize their magnitudes. Consequently, the associated nonlinear terms are linearized under the constraints $| θ_{1} | \leq {\bar{θ}}_{1}$ and $| θ_{2} | \leq {\bar{θ}}_{2}$ as follows:

\begin{aligned} \cos (θ_{1} - θ_{2}) ≃ 1, \sin (θ_{1} - θ_{2}) & ≃ 0, \\ \frac{\sin θ_{1}}{θ_{1}} ≃ 1, \frac{\sin θ_{2}}{θ_{2}} ≃ 1 \end{aligned}

(21)

where

{\bar{θ}}_{1} > 0

and

{\bar{θ}}_{2} > 0

denote the bounding radius that defines the valid regions for the linearizations.

Subsequently, since only the matrices E(q) and A(q) contain nonlinear terms, which are selected as the premise variables, defined as below:

z_{1} = \cos (θ_{1}) \in [{\underset{̲}{z}}_{1}, {\bar{z}}_{1}]

(22)

z_{2} = \cos (θ_{2}) \in [{\underset{̲}{z}}_{2}, {\bar{z}}_{2}]

(23)

z_{3} = \sin (θ_{1}) {\dot{θ}}_{1} \in [{\underset{̲}{z}}_{3}, {\bar{z}}_{3}]

(24)

z_{4} = \sin (θ_{2}) {\dot{θ}}_{2} \in [{\underset{̲}{z}}_{4}, {\bar{z}}_{4}]

(25)

where

{\underset{̲}{z}}_{p}

and

{\bar{z}}_{p} (p = 1,2,3,4)

represent the minimum and maximum values of the corresponding premise variable z_p, respectively. These bounds are derived from the bounded regions of the state variables:

| θ_{1} | \leq {\bar{θ}}_{1}

| θ_{2} | \leq {\bar{θ}}_{2}

| {\dot{θ}}_{1} | \leq {\dot{\bar{θ}}}_{1}

, and

| {\dot{θ}}_{2} | \leq {\dot{\bar{θ}}}_{2}

, which is provided in the model parameters. Accordingly, the associated membership functions are defined as follows:

\begin{aligned} υ_{1} = \frac{({\bar{z}}_{1} - z_{1}) ({\bar{z}}_{2} - z_{2})}{({\bar{z}}_{1} - {\underset{̲}{z}}_{1}) ({\bar{z}}_{2} - {\underset{̲}{z}}_{2})}, υ_{2} = \frac{({\bar{z}}_{1} - z_{1}) (z_{2} - {\underset{̲}{z}}_{2})}{({\bar{z}}_{1} - {\underset{̲}{z}}_{1}) ({\bar{z}}_{2} - {\underset{̲}{z}}_{2})} \\ υ_{3} = \frac{(z_{1} - {\underset{̲}{z}}_{1}) ({\bar{z}}_{2} - z_{2})}{({\bar{z}}_{1} - {\underset{̲}{z}}_{1}) ({\bar{z}}_{2} - {\underset{̲}{z}}_{2})}, υ_{4} = \frac{(z_{1} - {\underset{̲}{z}}_{1}) (z_{2} - {\underset{̲}{z}}_{2})}{({\bar{z}}_{1} - {\underset{̲}{z}}_{1}) ({\bar{z}}_{2} - {\underset{̲}{z}}_{2})} \\ η_{1} = \frac{({\bar{z}}_{3} - z_{3}) ({\bar{z}}_{4} - z_{4})}{({\bar{z}}_{3} - {\underset{̲}{z}}_{3}) ({\bar{z}}_{4} - {\underset{̲}{z}}_{4})}, η_{2} = \frac{({\bar{z}}_{3} - z_{3}) (z_{4} - {\underset{̲}{z}}_{4})}{({\bar{z}}_{3} - {\underset{̲}{z}}_{3}) ({\bar{z}}_{4} - {\underset{̲}{z}}_{4})} \\ η_{3} = \frac{(z_{3} - {\underset{̲}{z}}_{3}) ({\bar{z}}_{4} - z_{4})}{({\bar{z}}_{3} - {\underset{̲}{z}}_{3}) ({\bar{z}}_{4} - {\underset{̲}{z}}_{4})}, η_{4} = \frac{(z_{3} - {\underset{̲}{z}}_{3}) (z_{4} - {\underset{̲}{z}}_{4})}{({\bar{z}}_{3} - {\underset{̲}{z}}_{3}) ({\bar{z}}_{4} - {\underset{̲}{z}}_{4})} \end{aligned}

(26)

Thus, the matrices E_k and A_m for the LTI subsystems are formulated as:

\begin{aligned} E_{1} & = [\begin{matrix} I & 0 \\ 0 & [\begin{matrix} M_{11} & m_{12} {\underset{̲}{z}}_{1} & m_{13} {\underset{̲}{z}}_{2} \\ m_{12} {\underset{̲}{z}}_{1} & M_{22} & m_{23} \\ m_{13} {\underset{̲}{z}}_{2} & m_{23} & M_{33} \end{matrix}] \end{matrix}] \\ E_{2} & = [\begin{matrix} I & 0 \\ 0 & [\begin{matrix} M_{11} & m_{12} {\underset{̲}{z}}_{1} & m_{13} {\bar{z}}_{2} \\ m_{12} {\underset{̲}{z}}_{1} & M_{22} & m_{23} \\ m_{13} {\bar{z}}_{2} & m_{23} & M_{33} \end{matrix}] \end{matrix}] \\ E_{3} & = [\begin{matrix} I & 0 \\ 0 & [\begin{matrix} M_{11} & m_{12} {\bar{z}}_{1} & m_{13} {\underset{̲}{z}}_{2} \\ m_{12} {\bar{z}}_{1} & M_{22} & m_{23} \\ m_{13} {\underset{̲}{z}}_{2} & m_{23} & M_{33} \end{matrix}] \end{matrix}] \\ E_{4} & = [\begin{matrix} I & 0 \\ 0 & [\begin{matrix} M_{11} & m_{12} {\bar{z}}_{1} & m_{13} {\bar{z}}_{2} \\ m_{12} {\bar{z}}_{1} & M_{22} & m_{23} \\ m_{13} {\bar{z}}_{2} & m_{23} & M_{33} \end{matrix}] \end{matrix}] \\ A_{1} & = [\begin{matrix} 0 & I \\ - [\begin{matrix} 0 & 0 & 0 \\ 0 & g_{2} & 0 \\ 0 & 0 & g_{3} \end{matrix}] & - [\begin{matrix} 0 & c_{12} {\underset{̲}{z}}_{3} & c_{13} {\underset{̲}{z}}_{4} \\ 0 & ν l_{1}^{2} & 0 \\ 0 & 0 & 0 \end{matrix}] \end{matrix}] \\ A_{2} & = [\begin{matrix} 0 & I \\ - [\begin{matrix} 0 & 0 & 0 \\ 0 & g_{2} & 0 \\ 0 & 0 & g_{3} \end{matrix}] & - [\begin{matrix} 0 & c_{12} {\underset{̲}{z}}_{3} & c_{13} {\bar{z}}_{4} \\ 0 & ν l_{1}^{2} & 0 \\ 0 & 0 & 0 \end{matrix}] \end{matrix}] \\ A_{3} & = [\begin{matrix} 0 & I \\ - [\begin{matrix} 0 & 0 & 0 \\ 0 & g_{2} & 0 \\ 0 & 0 & g_{3} \end{matrix}] & - [\begin{matrix} 0 & c_{12} {\bar{z}}_{3} & c_{13} {\underset{̲}{z}}_{4} \\ 0 & ν l_{1}^{2} & 0 \\ 0 & 0 & 0 \end{matrix}] \end{matrix}] \\ A_{4} & = [\begin{matrix} 0 & I \\ - [\begin{matrix} 0 & 0 & 0 \\ 0 & g_{2} & 0 \\ 0 & 0 & g_{3} \end{matrix}] & - [\begin{matrix} 0 & c_{12} {\bar{z}}_{3} & c_{13} {\bar{z}}_{4} \\ 0 & ν l_{1}^{2} & 0 \\ 0 & 0 & 0 \end{matrix}] \end{matrix}] \end{aligned}

(27)

Finally, the LSCS can be represented by the following simplified T-S fuzzy descriptor model:

\{\begin{cases} \sum_{k = 1}^{4} υ_{k} (z (t)) E_{k} \dot{q} = \sum_{e = 1}^{4} η_{e} (z (t)) (A_{m} q + B u (t)) \\ y = C q \end{cases}

(28)

3.4. Fuzzy T-S descriptor-based observer

By utilizing equation (28), a nonlinear descriptor-based observer for the LSCS is proposed as follows:

\{\begin{cases} \begin{aligned} \sum_{k = 1}^{4} υ_{k} (z (t)) E_{k} \dot{\hat{q}} = \sum_{m = 1}^{4} η_{m} (\hat{z} (t)) [A_{m} \hat{q} + B u (t)] \\ + \sum_{k = 1}^{4} \sum_{m = 1}^{4} υ_{k} (z (t)) η_{m} (z (t)) L_{m k} (y - \hat{y}) \end{aligned} \\ \hat{y} = C \hat{q} \end{cases}

(29)

where

\hat{q}

and

\hat{y}

denote the estimated state and output vectors, respectively. Compared with the standard T-S fuzzy observer, the proposed descriptor-based observer preserves the descriptor structure of the original nonlinear model and is implementable using the estimated premise variables. Moreover, according to Guerra et al. (2015), the descriptor-observer formulation can reduce the conservatism of the resulting LMI conditions by providing greater flexibility in handling the derivative terms arising in the Lyapunov stability analysis. This formulation also offers a convenient basis for further extensions through the direct use of Finsler’s Lemma. Furthermore, L_mk represents the observer gain matrices, which are determined by solving linear matrix inequality (LMI) to guarantee convergence time, based on the proof of estimation error convergence for the T-S observer.

Remark 1

Notably, in the convergence analysis, the existence of the mismatched term $\hat{z}$ leads to several challenges, especially because these terms contain unmeasured states that cannot be directly compensated by the measured states. Although many approaches have been proposed to handle this problem, such as the use of Lipschitz conditions (Nguyen et al., 2021) and the mean value theorem (Xie et al., 2019), the approach based on Lipschitz constants often increases conservatism, while the mean value theorem usually increases the complexity due to the need to consider the bounds of Jacobian matrices. This may lead to a significant computational burden when the number of fuzzy rules is large. Fortunately, for the LSCS, since the initial conditions are set to zero, the mismatched term can be avoided, thereby reducing the complexity. Therefore, in the remainder of this work, during the observer design process, the term z(t) can be replaced by $\hat{z} (t)$ .

4. Stability and convergence analysis

This section presents a step-by-step stability analysis of the proposed control scheme. First, the stability of the adaptive interval type-2 fuzzy control (AIT2FC) is established by demonstrating the convergence of the composite surface by Lyapunov techniques. Next, the convergence of the estimation error for the descriptor-based observer (DO) is proven. Finally, the overall closed-loop system is shown to be asymptotically stable under the combined AIT2FC-DO framework.

4.1. Stability of the AIT2FC

Lemma 1

If the initial parameters in equation (17) are selected such that ${\hat{φ}}_{j} (0) > {\underset{̲}{φ}}_{j}$ , the following two conclusions hold true:

{\hat{φ}}_{j} \geq {\underset{̲}{φ}}_{j}

(30)

ϑ = - | S | {\hat{φ}}^{T} ω (χ) + S φ^{* T} ω (χ) - {\tilde{φ}}^{T} Λ^{- 1} \dot{\hat{φ}} \leq 0

(31)

Proof: Appendix A.

To establish the stability of the proposed AIT2FC controller and prove that the composite surface and tracking errors converge to zero within a finite time T, a candidate Lyapunov function is selected as follows:

V_{1} = \frac{1}{2} S^{2} + \frac{1}{2} {\tilde{φ}}^{T} Λ^{- 1} \tilde{φ} + \frac{1}{2} ψ^{- 1} {\tilde{ε}}^{2}

(32)

where

\tilde{φ} = φ^{*} - \hat{φ}

and

\tilde{ε} = ε^{*} - \hat{ε}

denote the estimation errors of φ∗ and ɛ∗, respectively. Since the ideal parameters are constant, their time derivatives are given by

\dot{\tilde{φ}} = - \dot{\hat{φ}}

and

\dot{\tilde{ε}} = - \dot{\hat{ε}}

. The derivative of V₁ with respect to time is expressed as follows:

{\dot{V}}_{1} = S \dot{S} - {\tilde{φ}}^{T} Λ^{- 1} \dot{\hat{φ}} - ψ^{- 1} \tilde{ε} \dot{\hat{ε}}

(33)

Combining equations (13), (14), and (16), the time derivative of the composite surface $\dot{S}$ is transformed as follows:

\begin{aligned} \dot{S} = & \bar{Φ} u_{1} + f (X, \dot{X}, u_{1}) = \bar{Φ} u_{1} + φ^{* T} ω (χ) + ε \\ = & - K S - {\hat{φ}}^{T} ω (χ) sign (S) - \hat{ε} sign (S) \\ - K_{s} sign (S) + φ^{* T} ω (χ) + ε \end{aligned}

(34)

By substituting equation (34) into equation (33), the time derivative of V₁ can be rewritten as follows:

\begin{aligned} {\dot{V}}_{1} = & S (\bar{Φ} u_{1} + f (X, \dot{X}, u_{1})) - {\tilde{φ}}^{T} Λ^{- 1} \dot{\hat{φ}} - ψ^{- 1} \tilde{ε} \dot{\hat{ε}} \\ = & - K S^{2} - | S | {\hat{φ}}^{T} ω (χ) - \hat{ε} | S | - K_{s} | S | \\ + S φ^{* T} ω (χ) + ε S - {\tilde{φ}}^{T} Λ^{- 1} \dot{\hat{φ}} - ψ^{- 1} \tilde{ε} \dot{\hat{ε}} \\ = & - K S^{2} - K_{s} | S | - \hat{ε} | S | + ε S - \tilde{ε} | S | \\ - | S | {\hat{φ}}^{T} ω (χ) + S φ^{* T} ω (χ) + {\tilde{φ}}^{T} Λ^{- 1} \dot{\hat{φ}} \\ = & - K S^{2} - K_{s} | S | + ε S - ε^{*} | S | \\ - | S | {\hat{φ}}^{T} ω (χ) + S φ^{* T} ω (χ) + {\tilde{φ}}^{T} Λ^{- 1} \dot{\hat{φ}} \\ \leq & - K S^{2} - K_{s} | S | + ε | S | - ε^{*} | S | + ϑ \end{aligned}

(35)

By using equation (31) of the Lemma 1, the inequalities −KS² ≤, − K_s|S| ≤ 0, and the fact that ɛ|S| − ɛ∗|S| ≤ 0 (since |ɛ| ≤ ɛ∗), the following inequality is proven:

\begin{aligned} {\dot{V}}_{1} & \leq - K S^{2} \\ \leq 0 \end{aligned}

(36)

Therefore, the estimation error $\tilde{ε}$ of the nonlinear function approximated by the type-2 fuzzy system has an upper bound. In other words, there exists a constant ϖ such that $\tilde{ε} \leq ϖ$ .

Moreover, to prove that the composite surface S converges to zero within a finite time, the Lyapunov function V₂ is constructed as:

V_{2} = \frac{1}{2} S^{2}

(37)

Taking the time derivative of V₂ and substituting equation (34), the following equation is yielded:

\begin{aligned} {\dot{V}}_{2} = & S \dot{S} \\ = & - K S^{2} - | S | {\hat{φ}}^{T} ω (χ) - \hat{ε} | S | - K_{s} | S | \\ + S φ^{* T} ω (χ) + ε S \\ \leq & - K S^{2} - | S | {\hat{φ}}^{T} ω (χ) - \hat{ε} | S | - K_{s} | S | \\ + | S | φ^{* T} ω (χ) + ε^{*} | S | \\ \leq & - K S^{2} + (ε^{*} - \hat{ε}) | S | - K_{s} | S | \\ - | S | ({\hat{φ}}^{T} - φ^{* T}) ω (χ) \end{aligned}

(38)

Based on equation (30), the condition ${\hat{φ}}_{j} \geq {\underset{̲}{φ}}_{j} \geq φ_{j}^{*}$ is guaranteed. Furthermore, by noting that $ε^{*} - \hat{ε} = \tilde{ε}$ , $\tilde{ε} \leq ϖ$ the expression for ${\dot{V}}_{2}$ is transformed as below:

\begin{aligned} {\dot{V}}_{2} \leq & \tilde{ε} | S | - K_{s} | S | \leq ϖ | S | - K_{s} | S | \\ \leq & - (K_{s} - ϖ) | S | \leq - ρ | S | \end{aligned}

(39)

where K_s is a parameter in the control law (16) that ensures ρ = K_s − ϖ > 0. Then, from equation (37), the composite surface S can be rewritten in terms of the Lyapunov function V₂ as:

\begin{aligned} | S (t) | & = \sqrt{2 V_{2} (t)} \\ ρ | S (t) | & \leq - ρ \sqrt{2 V_{2} (t)} \\ {\dot{V}}_{2} (t) & \leq - ρ \sqrt{2 V_{2} (t)} \\ \frac{d V_{2}}{\sqrt{V_{2}}} & \leq - ρ \sqrt{t} d t \end{aligned}

(40)

Integrating both sides of the differential inequality in equation (40), the following results are yielded:

\begin{aligned} 2 \sqrt{V_{2} (t)} - 2 \sqrt{V_{2} (0)} \leq - ρ \sqrt{2} t \\ t \leq \frac{\sqrt{2 V_{2} (0)} - \sqrt{2 V_{2} (t)}}{ρ} \end{aligned}

(41)

Consequently, V₂(T) = 0 is achieved within a finite time T. In other words, the composite surface converges to zero after the convergence time T, which is proven by the following expression:

\begin{aligned} T & \leq \frac{\sqrt{2 V_{2} (0)} - \sqrt{2 V_{2} (T)}}{ρ} \\ T & \leq \frac{\sqrt{2 V_{2} (0)}}{ρ} \end{aligned}

(42)

Therefore, after the finite time T, since S = 0 for all t ≥ T, equation (11) can be rewritten as follows:

\begin{aligned} 0 & = {\dot{e}}_{1} + λ_{1} e_{1} + α_{2} ({\dot{e}}_{2} + λ_{2} e_{2}) + α_{3} ({\dot{e}}_{3} + λ_{3} e_{3}) \\ {\dot{e}}_{1} & = - λ_{1} e_{1} - α_{2} {\dot{e}}_{2} - α_{3} {\dot{e}}_{3} - α_{2} λ_{2} e_{2} - α_{3} λ_{3} e_{3} \end{aligned}

(43)

By introducing a new state vector $ϵ = {[e_{2}, {\dot{e}}_{2}, e_{3}, {\dot{e}}_{3}, e_{1}]}^{T} \in R^{5}$ to denote the position errors and their respective time derivatives, and combining this definition with equation (8) while adding and subtracting the terms ϵ₃ and ϵ₅, the error system is rewritten as follows:

\begin{aligned} {\dot{ϵ}}_{1} = & ϵ_{2} \\ {\dot{ϵ}}_{2} = & {\ddot{e}}_{2} = ϵ_{3} + (n_{2} u_{1} - ζ_{2} - ϵ_{3}) \\ {\dot{ϵ}}_{3} = & ϵ_{4} \\ {\dot{ϵ}}_{4} = & {\ddot{e}}_{3} = ϵ_{5} + (n_{3} u_{1} - ζ_{3} - ϵ_{5}) \\ {\dot{ϵ}}_{5} = & {\dot{e}}_{1} = - λ_{1} e_{1} - α_{2} {\dot{e}}_{2} - α_{3} {\dot{e}}_{3} \\ - α_{2} λ_{2} e_{2} - α_{3} λ_{3} e_{3} \end{aligned}

(44)

Subsequently, equation (44) can be rewritten in matrix-vector form as follows:

\dot{ϵ} = A_{ϵ} ϵ + δ_{ϵ}

(45)

where

\begin{aligned} A_{ϵ} & = [\begin{matrix} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ - α_{2} λ_{2} & - α_{2} & - α_{3} λ_{3} & - α_{3} & - λ_{1} \end{matrix}] \\ δ_{ϵ} & = [\begin{matrix} 0 \\ n_{2} u_{1} - ζ_{2} - ϵ_{3} \\ 0 \\ n_{3} u_{1} - ζ_{3} - ϵ_{5} \\ 0 \end{matrix}] \end{aligned}

(46)

Furthermore, since the matrix A_ϵ is in a canonical form, the controller parameters λ₁, λ₂, λ₃, α₂, and α₃ can be designed to be strictly positive such that the characteristic polynomial of A_ϵ satisfies the Hurwitz criterion. Consequently, according to the Lyapunov equation, there exist two positive definite matrices P_ϵ and Q_ϵ such that: P_ϵA + A^TP_ϵ = −Q_ϵ. Then, the following Lyapunov function is selected:

V_{3} = ϵ^{T} P_{ϵ} ϵ

(47)

The derivative respect to time of V₃ can be yielded as:

\begin{aligned} {\dot{V}}_{3} & = ϵ^{T} (P_{ϵ} A_{ϵ} + A_{ϵ}^{T} P_{ϵ}) ϵ + 2 ϵ^{T} P_{ϵ} δ_{ϵ} \\ = ϵ^{T} (Q_{ϵ}) ϵ + 2 ϵ^{T} P_{ϵ} δ_{ϵ} \\ \leq - σ ‖ ϵ ‖^{2} + 2 ‖ P_{ϵ} ‖ ‖ ϵ ‖ ‖ δ_{ϵ} ‖ \end{aligned}

(48)

where σ denotes the minimum eigenvalue of Q_ϵ. Furthermore, when S = 0 for all t ≥ T, the sign function becomes sign(S) = 0, which results in u₁ = 0. At this point, δ_ϵ contains only the terms ζ₂, ζ₃, ϵ₃, and ϵ₅; thus, it can be evaluated using the Euclidean norm as follows:

‖ δ_{ϵ} ‖ \leq ‖ ζ ‖ + ‖ ϵ ‖

(49)

Under Assumption 1 and based on the components of ζ defined in equation (7), the following evaluation is established:

‖ ζ ‖ \leq {\underset{̲}{m}}^{- 1} (\bar{c} ‖ \dot{X} ‖^{2} + \bar{d} ‖ \dot{X} ‖ + \bar{g} ‖ X ‖)

(50)

From equations (9) and (43), the following inequality is also derived:

\begin{aligned} ‖ X ‖ & \leq ‖ ϵ ‖ \\ | {\dot{e}}_{1} | & \leq ϱ ‖ ϵ ‖ \\ ‖ \dot{X} ‖ & \leq ‖ ϵ ‖ + | {\dot{e}}_{1} | \\ \leq ‖ ϵ ‖ + ϱ ‖ ϵ ‖ \leq (1 + ϱ) ‖ ϵ ‖ \end{aligned}

(51)

where

ϱ = \sqrt{{(α_{2} λ_{2})}^{2} + {(α_{3} λ_{3})}^{2} + α_{2}^{2} + α_{3}^{2} + λ_{1}^{2}}

By substituting equations (49), (50), and (51) into equation (48), the time derivative of V₃ is evaluated as follows:

\begin{aligned} {\dot{V}}_{3} \leq & - σ ‖ ϵ ‖^{2} + 2 ‖ P_{ϵ} ‖ ‖ ϵ ‖ (‖ ζ ‖ + ‖ ϵ ‖) \\ \leq & - σ ‖ ϵ ‖^{2} + 2 ‖ P_{ϵ} ‖ ‖ ϵ ‖ ({\underset{̲}{m}}^{- 1} [\bar{c} ‖ \dot{X} ‖^{2} + \bar{d} ‖ \dot{X} ‖ \\ + \bar{g} ‖ X ‖] + ‖ ϵ ‖) \\ \leq & - σ ‖ ϵ ‖^{2} + 2 ‖ P_{ϵ} ‖ ‖ ϵ ‖ ({\underset{̲}{m}}^{- 1} [\bar{c} {((1 + ϱ) ‖ ϵ ‖)}^{2} \\ + \bar{d} ((1 + ϱ) ‖ ϵ ‖) + \bar{g} ‖ ϵ ‖] + ‖ ϵ ‖) \\ \leq & - σ ‖ ϵ ‖^{2} + 2 ‖ P_{ϵ} ‖ {\underset{̲}{m}}^{- 1} \bar{c} {(1 + ϱ)}^{2} ‖ ϵ ‖^{3} \\ + 2 ‖ P_{ϵ} ‖ [{\underset{̲}{m}}^{- 1} \bar{d} (1 + ϱ) + {\underset{̲}{m}}^{- 1} \bar{g} + 1] ‖ ϵ ‖^{2} \\ \leq & ‖ ϵ ‖ (2 ‖ P_{ϵ} ‖ {\underset{̲}{m}}^{- 1} \bar{c} {(1 + ϱ)}^{2} ‖ ϵ ‖^{2} - σ ‖ ϵ ‖ \\ + 2 ‖ P_{ϵ} ‖ [{\underset{̲}{m}}^{- 1} \bar{d} (1 + ϱ) + {\underset{̲}{m}}^{- 1} \bar{g} + 1] ‖ ϵ ‖) \\ \leq & ‖ ϵ ‖ (Ψ_{1} ‖ ϵ ‖^{2} - Ψ_{2} ‖ ϵ ‖) \end{aligned}

(52)

where

\begin{aligned} Ψ_{1} & = 2 ‖ P_{ϵ} ‖ {\underset{̲}{m}}^{- 1} \bar{c} {(1 + ϱ)}^{2} \\ Ψ_{2} & = σ - 2 ‖ P_{ϵ} ‖ [{\underset{̲}{m}}^{- 1} d (1 + ϱ) + {\underset{̲}{m}}^{- 1} \bar{g} + 1] \end{aligned}

Remark 2

Note that f(‖ϵ‖) = Ψ₁‖ϵ‖² − Ψ₂‖ϵ‖ represents a parabolic function with respect to ‖ϵ‖ > 0. The condition Ψ₁ > 0 is satisfied by appropriately designing the control gain acting on ϱ such that $σ > 2 ‖ P_{ϵ} ‖ [{\underset{̲}{m}}^{- 1} d (1 + ϱ) + {\underset{̲}{m}}^{- 1} \bar{g} + 1]$ . Thus, the inequality f(‖ϵ‖) < 0 holds if and only if ‖ϵ‖ ∈ (0, Ψ₂/Ψ₁).

Consequently, by initializing the system such that ‖ϵ(0)‖ ∈ (0, Ψ₂/Ψ₁), we obtain ${\dot{V}}_{3} < 0$ . This ensures that ‖ϵ‖ monotonically decreases over time until it reaches zero (‖ϵ‖ = 0), yielding the following result:

\begin{aligned} \lim_{t \to \infty} ϵ (t) & = 0 \\ \lim_{t \to \infty} e_{1} (t) & = e_{2} (t) = e_{3} (t) = {\dot{e}}_{1} (t) = {\dot{e}}_{2} (t) = {\dot{e}}_{3} (t) = 0 \end{aligned}

(53)

Hence, the control system using AIT2FC is stabilized at the equilibrium point.

4.2. Convergence of the estimation error

Sufficient convergence conditions for the T-S fuzzy descriptor-based observer can be derived by expressing equation (29) in an augmented state-space representation:

\{\begin{cases} \begin{aligned} E^{*} {\dot{\hat{q}}}^{*} = & \sum_{k = 1}^{4} \sum_{m = 1}^{4} υ_{k} (\hat{z} (t)) η_{m} (\hat{z} (t)) A_{m k}^{*} {\dot{\hat{q}}}^{*} \\ + \sum_{k = 1}^{4} \sum_{m = 1}^{4} υ_{k} (z (t)) η_{m} (\hat{z} (t)) L_{m k} (y - \hat{y}) \\ + \sum_{m = 1}^{4} η_{m} ((\hat{z} (t)) B_{m}^{*} u \end{aligned} \\ \hat{y} = C^{*} \hat{q} \end{cases}

(54)

where

\begin{aligned} E^{*} & = [\begin{matrix} I_{3 \times 3} & 0 \\ 0 & 0 \end{matrix}], A_{m k}^{*} = [\begin{matrix} 0 & I_{3 \times 3} \\ A_{m} & - E_{k} \end{matrix}], B_{m}^{*} = [\begin{matrix} 0 \\ B_{m} \end{matrix}] \\ q^{*} & = {[q^{T}, {\dot{q}}^{T}]}^{T}, C^{*} = [\begin{matrix} C & 0 \end{matrix}] \end{aligned}

The augmented estimation error is defined as ${\tilde{q}}^{*} = {[{(q - \hat{q})}^{T}, {(\dot{q} - \dot{\hat{q}})}^{T}]}^{T}$ with its derivative can be presented as below:

\begin{aligned} E^{*} \dot{\tilde{q}} = & \sum_{k = 1}^{4} \sum_{m = 1}^{4} \sum_{n = 1}^{4} υ_{k} (\hat{z} (t)) η_{m} (\hat{z} (t)) η_{n} (\hat{z} (t)) \\ \times [A_{m k}^{*} - L_{n k}^{*} C] {\tilde{q}}^{*} \end{aligned}

(55)

Therefore, the convergence of the state estimation error is ensured by finding a feasible set of gains L_mk that satisfies the LMI conditions in Theorem 1.

Theorem 1

Considering the Takagi–Sugeno (T-S) fuzzy descriptor state-space representation in equation (55), the convergence of the estimation error is ensured if and only if there exist a symmetric matrix $P_{1} = P_{1}^{T}$ , nonsingular matrices P₃ and P₄, and matrices L_mk (for k, m, n ∈ {1, 2, 3, 4} and n > m) such that:

\begin{aligned} ϒ_{m m}^{k} < 0 \\ ϒ_{m n}^{k} + ϒ_{n m}^{k} < 0 \end{aligned}

(56)

where

\begin{aligned} ϒ_{m n}^{k} & = [\begin{matrix} ϒ_{1} & ϒ_{2} \\ ϒ_{3} & ϒ_{4} \end{matrix}] \\ ϒ_{1} & = A_{m}^{T} P_{3} + P_{3}^{T} - C^{T} L_{n k}^{T} P_{3} - P_{3}^{T} L_{n k} C \\ ϒ_{2} & = ϒ_{3} = P_{1} - E_{k}^{T} P_{3} + P_{4}^{T} A_{m} - P_{4}^{T} L_{n k} C \\ ϒ_{4} & = - E_{k}^{T} P_{4} - P_{4}^{T} E_{k} \end{aligned}

Proof: Appendix B.

4.3. Stability analysis of the close-loop system

Given that the estimated states $\hat{X}$ and $\dot{\hat{X}}$ is obtained from $\hat{q}$ to be applied in the AIT2FC, the time derivative of the composite surface, $\dot{S}$ , in equation (13) is transformed into:

\dot{S} = \bar{Φ} u_{1} (\hat{X}) + f (X, \dot{X}, u_{1})

(57)

where

u_{1} (\hat{X})

is the control signal calculated according to equation (16) by replacing X and

\dot{X}

with

\hat{X}

and

\dot{\hat{X}}

\begin{aligned} ‖ u_{1} (\hat{X}) - u_{1} (X) ‖ & \leq L_{u} ‖ X - \hat{X} ‖ \\ S (\bar{Φ} u_{1} (\hat{X}) - \bar{Φ} u_{1} (X)) & \leq L | S | ‖ \tilde{X} ‖ \end{aligned}

(58)

where

L = \bar{Φ} L_{u}, \tilde{X} = \hat{X} - X

Based on the estimated states, the candidate Lyapunov function $V_{1}^{*}$ for the AIT2FC controller is constructed as follows:

V_{1}^{*} = \frac{1}{2} S^{2} + \frac{1}{2} {\tilde{φ}}^{T} Λ^{- 1} \tilde{φ} + \frac{1}{2} ψ^{- 1} {\tilde{ε}}^{2}

(59)

When $V_{1}^{*}$ is differentiated with respect to time and combined with equations (57), (33), and (35), the expression is transformed as follows:

\begin{aligned} {\dot{V}}_{1}^{*} = & S (\bar{Φ} u_{1} (\hat{X}) + f (X, \dot{X}, u_{1})) \\ - {\tilde{φ}}^{T} Λ^{- 1} \dot{\hat{φ}} - ψ^{- 1} \tilde{ε} \dot{\hat{ε}} \\ = & S (\bar{Φ} u_{1} (\hat{X}) + f (X, \dot{X}, u_{1}) + \bar{Φ} u_{1} (X) + \bar{Φ} u_{1} (X)) \\ - {\tilde{φ}}^{T} Λ^{- 1} \dot{\hat{φ}} - ψ^{- 1} \tilde{ε} \dot{\hat{ε}} \\ = & [S (\bar{Φ} u_{1} (X) + f (X, \dot{X}, u_{1})) - {\tilde{φ}}^{T} Λ^{- 1} \dot{\hat{φ}} - ψ^{- 1} \tilde{ε} \dot{\hat{ε}}] \\ + S (\bar{Φ} u_{1} (\hat{X}) - \bar{Φ} u_{1} (X)) \\ = & {\dot{V}}_{1} + S (\bar{Φ} u_{1} (\hat{X}) - \bar{Φ} u_{1} (X)) \\ \leq & - K | S |^{2} + L | S | ‖ \tilde{X} ‖ \end{aligned}

(60)

For the closed-loop control system, the overall candidate Lyapunov function is selected as:

V = V_{1}^{*} + γ V_{4}^{*}

(61)

By differentiating this function with respect to time and incorporating equations (60) and (74), the following inequality is derived:

\begin{aligned} \dot{V} & \leq - K | S |^{2} + L | S | ‖ \tilde{X} ‖ - γ β ‖ \tilde{X} ‖^{2} \\ \leq {[\begin{matrix} | S | \\ ‖ \tilde{X} ‖ \end{matrix}]}^{T} [\begin{matrix} K & - \frac{L}{2} \\ - \frac{L}{2} & γ β \end{matrix}] [\begin{matrix} | S | \\ ‖ \tilde{X} ‖ \end{matrix}] \end{aligned}

(62)

where

\begin{aligned} Q = [\begin{matrix} K & - \frac{L}{2} \\ - \frac{L}{2} & γ β \end{matrix}] > 0 & ⟺ K γ β - \frac{L^{2}}{4} > 0 \\ ⟺ γ > \frac{L^{2}}{4 β K} \end{aligned}

(63)

Provided that the control parameters K, β, and L are bounded and strictly determined, there always exists a scalar γ that satisfies the condition in equation (63). As a result, the time derivative of the composite candidate Lyapunov function is negative semi-definite (i.e., $\dot{V} \leq 0$ ), thereby guaranteeing the stability of the closed-loop system.

5. Performance evaluation via simulation

In this section, the proposed control methodology for the LSCS is evaluated through simulation results obtained under model parameter uncertainties and compared with the input shaping.

Initially, the parameters of the LSCS employed in this study are given as: m₁ = 1.334 kg, m₂ = 1.152 kg, l₁ = 0.3903 m, l₂ = 0.2044 m, i₁ = 0.2742 kgm², i₂ = 0.001 kgm², l = 0.24 m, h = 0.03 m g = 9.8 m/s², and ν = 0.065 Ns/m. Furthermore, the bounded ranges of the position-related variables and the control signal u₁ are also specified to construct the parameters for the membership functions of the fuzzy approximator as follows: |x| ≤ 1 m, $| \dot{x} | \leq 2 m/s$ , and |u₁| ≤ 2 N. Besides, the feasible ranges of the unactuated state variables corresponding to the angles θ₁ and θ₂ are bounded by: |θ₁| ≤ π/15, |θ₂| ≤ π/15, $| {\dot{θ}}_{1} | \leq π / 3$ , and $| {\dot{θ}}_{2} | \leq π / 3$ .

For the adaptive interval type-2 fuzzy control (AIT2FC) scheme, the parameters of the composite surface S (11), the adaptive laws (17), and control signal (16) are selected as follows: λ₁ = 0.8, λ₂ = 0.5, λ₃ = 2, α₂ = 0.1, α₃ = 0.01, ψ = 5, K = 0.1, K_s = 0.05, and Λ is a 7 × 7 diagonal matrix with all diagonal elements equal to 50.

To overcome the chattering phenomenon commonly encountered in methods related to conventional SMC, all sign(S) functions in (16) have been replaced by tanh(5S).

Remark 3

The control parameters λ₁, λ₂, λ₃, α₂, and α₃ in (11) are selected similarly to those in conventional SMC, where λ₃ is assigned the largest value to prioritize the stabilization of θ₂. In (16), K and K_s are the main parameters affecting the convergence time of x and can be tuned to achieve a trade-off between convergence speed and the deviations of θ₁ and θ₂. The parameters ψ and Λ in (17), associated with the fuzzy approximation, are selected through trial and error to effectively compensate for the effects of unknown parameters and system nonlinearities.

The lower and upper membership functions of the type-2 fuzzy basis function in (15) are defined as:

\begin{aligned} {\underset{̲}{μ}}_{j}^{j} & = 0.6 e^{- \frac{(X_{i} - σ_{i j})}{1}} \\ {\bar{μ}}_{j}^{j} & = e^{- \frac{(X_{i} - σ_{i j})}{6}} \end{aligned}

where i = 1, 2; j = 1, …, 9; and σ_ij is related to the i-th component of χ by the following formula:

σ_{i j} = \min (χ_{i}) + (j - 1) \frac{\max (χ_{i}) - \min (χ_{i})}{9 - 1}

where χ_i represents one of the variables

x, θ_{1}, θ_{2}, \dot{x}, {\dot{θ}}_{1}, {\dot{θ}}_{2}

, or u₁, whose bounded ranges have been previously defined.

For the descriptor-based observer (DO), the minimum and maximum values of the premise variables z₁, z₂, z₃, and z₄ are calculated based on the feasible ranges of the unactuated state variables provided in the physical parameters section. Additionally, the 6 × 6 gain matrices L_mk in equation (29) are computed by solving the LMI conditions in equation (56), in which P₄ = 0.1P₃ is imposed to retain the linear formulation. For presentation, only the nonzero first and fourth columns of each 6 × 6 gain matrix L_mk are arranged in the following reduced form:

\begin{matrix} L_{11} = & L_{12} = [\begin{matrix} 3.1095 & 0.9971 \\ 0.6969 & 52.6296 \\ 0.1929 & 46.7684 \\ 2.1212 & 133.1256 \\ 0.0864 & - 30.5980 \\ 0.6020 & 51.0740 \end{matrix}] [\begin{matrix} 3.1095 & 0.9969 \\ 0.6974 & 38.3317 \\ 0.1929 & 42.4827 \\ 2.1228 & 100.9669 \\ 0.0865 & - 28.7143 \\ 0.6024 & 38.1585 \end{matrix}] \end{matrix}

\begin{matrix} L_{13} = & L_{14} = [\begin{matrix} 3.1095 & 0.9971 \\ 0.6957 & 58.3588 \\ 0.1929 & 46.7061 \\ 2.1177 & 143.2193 \\ 0.0861 & - 36.4243 \\ 0.6010 & 56.3544 \end{matrix}] [\begin{matrix} 3.1095 & 0.9971 \\ 0.6961 & 63.8653 \\ 0.1929 & 42.2265 \\ 2.1191 & 163.2219 \\ 0.0861 & - 25.2110 \\ 0.6014 & 59.1410 \end{matrix}] \end{matrix}

\begin{matrix} L_{21} = & L_{22} = [\begin{matrix} 3.1095 & 0.9973 \\ 0.6964 & 92.1599 \\ 0.1933 & 26.5752 \\ 2.1200 & 231.1259 \\ 0.0862 & - 21.1327 \\ 0.6016 & 81.1997 \end{matrix}] [\begin{matrix} 3.1095 & 0.9970 \\ 0.6974 & 68.4234 \\ 0.1933 & 24.1439 \\ 2.1232 & 175.4348 \\ 0.0866 & - 22.0473 \\ 0.6024 & 60.8220 \end{matrix}] \end{matrix}

\begin{matrix} L_{23} = & L_{24} = [\begin{matrix} 3.1095 & 0.9975 \\ 0.6952 & 112.3255 \\ 0.1933 & 22.1009 \\ 2.1167 & 278.6785 \\ 0.0860 & - 21.4876 \\ 0.6006 & 97.5971 \end{matrix}] [\begin{matrix} 3.1095 & 0.9973 \\ 0.6957 & 99.7670 \\ 0.1933 & 22.0819 \\ 2.1184 & 252.3443 \\ 0.0861 & - 16.4357 \\ 0.6011 & 86.2850 \end{matrix}] \end{matrix}

\begin{matrix} L_{31} = & L_{32} = [\begin{matrix} 3.1095 & 0.9969 \\ 0.6967 & 52.3434 \\ 0.1932 & 24.6937 \\ 2.1202 & 156.6417 \\ 0.0856 & 15.0946 \\ 0.6019 & 42.8652 \end{matrix}] [\begin{matrix} 3.1095 & 0.9968 \\ 0.6976 & 41.1945 \\ 0.1933 & 23.4366 \\ 2.1231 & 129.2295 \\ 0.0858 & 12.0563 \\ 0.6027 & 33.5304 \end{matrix}] \end{matrix}

\begin{matrix} L_{33} = & L_{34} = [\begin{matrix} 3.1095 & 0.9968 \\ 0.6960 & 47.3466 \\ 0.1933 & 18.6464 \\ 2.1183 & 144.6459 \\ 0.0853 & 17.7006 \\ 0.6015 & 37.3009 \end{matrix}] [\begin{matrix} 3.1095 & 0.9970 \\ 0.6959 & 62.2625 \\ 0.1933 & 20.7770 \\ 2.1181 & 182.8291 \\ 0.0854 & 18.9286 \\ 0.6014 & 50.0524 \end{matrix}] \end{matrix}

\begin{matrix} L_{41} = & L_{42} = [\begin{matrix} 3.1095 & 0.9972 \\ 0.6963 & 93.1792 \\ 0.1936 & 5.1272 \\ 2.1194 & 257.3205 \\ 0.0855 & 23.4964 \\ 0.6017 & 74.2972 \end{matrix}] [\begin{matrix} 3.1095 & 0.9969 \\ 0.6973 & 71.7233 \\ 0.1937 & 5.3431 \\ 2.1226 & 204.6459 \\ 0.0857 & 18.4752 \\ 0.6025 & 56.6354 \end{matrix}] \end{matrix}

\begin{matrix} L_{43} = & L_{44} = [\begin{matrix} 3.1095 & 0.9973 \\ 0.6955 & 108.1105 \\ 0.1937 & - 5.0377 \\ 2.1172 & 296.0111 \\ 0.0852 & 30.6018 \\ 0.6011 & 84.6845 \end{matrix}] [\begin{matrix} 3.1095 & 0.9972 \\ 0.6955 & 99.5264 \\ 0.1937 & 0.7553 \\ 2.1171 & 274.7739 \\ 0.0852 & 27.0466 \\ 0.6010 & 78.4477 \end{matrix}] \end{matrix}

Then, the simulations are specifically designed under three distinct cases to demonstrate the robustness and effectiveness of the proposed approach:

• Case 1: The LSCS is controlled by the AIT2FC-DO under nominal conditions. This case aims to evaluate the DO capability to estimate all system states using only the measurable variables x and $\dot{x}$ , while assessing the settling time and slosh suppression performance during trolley motion.

• Case 2: The liquid mass in the container m₂ and the container length l are increased by (a) +50% and (b) +100% relative to their nominal values. This case is designed to evaluate the robustness and adaptability of the proposed method against parameter uncertainties.

• Case 3: The proposed closed-loop control scheme is compared with the input shaping ${ZVD}_{ω_{1}}$ and ${ZVD}_{ω_{n 1}}$ introduced in Khorshid and Al-Fadhli (2021).

In Case 1, the trolley position x, moving from an initial position of 0 m to a reference position of 0.4 m, and the liquid surface elevation ς, are depicted in Figures 3 and 4. Under the AIT2FC-DO scheme, the trolley rapidly reaches the reference position without overshoot, with a settling time of 7.7 s, as presented in Table 1. According to Table 1, during the trolley motion, liquid sloshing occurs with an ME of 2.085 mm (approximately 7% compared to the liquid depth h) and a RMSE value of 0.0072 mm. As observed in Figure 4, the sloshing amplitude gradually decreases over time and steadily converges to zero before 10 s. Even the accumulated ITAE in Case 1 remains very small at only 0.0727 mm. These results demonstrate that the proposed methodology can accurately and rapidly drive the LSCS to the reference position while effectively suppressing liquid surface oscillations.

Figure 3.

Case 1-2-3: The tracking position x(t) of the trolley.

Figure 4.

Case 1-2: The swing angle θ₁(t) of the rod–container system and the liquid surface elevation ς(t).

Table 1.

Maximum error (ME), root mean square error (RMSE), integral of time-weighted absolute error (ITAE), and settling time (T_s) for all cases.

		ME	RMSE	ITAE	T _s
Case 1	θ₁ (rad)	0.0339	0.0072	0.0727	6.8 s
	ς (mm)	2.0850	0.3545	3.8979	7.7 s
Case 2(a)	θ₁ (rad)	0.0278	0.0063	0.0773	5.1 s
	ς (mm)	2.2198	0.3564	2.3758	5.6 s
Case 2(b)	θ₁ (rad)	0.0232	0.0058	0.0833	4.8 s
	ς (mm)	2.2608	0.3565	1.9513	5.0 s
${ZVD}_{ω_{1}}$	θ₁ (rad)	0.2024	0.1314	13.603	*
	ς (mm)	11.638	5.7335	537.17	*
${ZVD}_{ω_{n 1}}$	θ₁ (rad)	0.0772	0.0394	3.8017	*
	ς (mm)	7.3430	3.9188	398.00	*

Furthermore, the remaining state variables of the LSCS and their corresponding estimations in Case 1 are presented in Figures 5 and 6 to verify the control capability of the AIT2FC and the state estimation performance of the DO. Specifically, the rod swing angle from equilibrium θ₁ and the sloshing-pendulum angle θ₂ are maintained within the range of −0.03 rad to 0.02 rad, where the ME of θ₁ is 0.0339 rad, as reported in Table 1, indicating relatively small oscillations. Based on Figure 5, both θ₁ and θ₂ rapidly converge to zero immediately after 6.8 s, and the oscillations are effectively suppressed. In Figure 6, by replacing sign(S) with tanh(5S) in (16), the control signal becomes smooth without exhibiting chattering phenomena. Moreover, all LSCS states involved in the control signal synthesis are accurately estimated with negligible and rapidly converging estimation errors. Consequently, the AIT2FC maintains effective performance when DO-estimated states are employed, demonstrating the effectiveness of the AIT2FC-DO scheme in scenarios where certain state variables are not directly measurable (Figure 7).

Figure 5.

Case 1: Estimation of the angle and angular velocity for θ₁ and θ₂ during trolley position regulation.

Figure 6.

Case 1: Estimation of the trolley position, trolley velocity, and control signal u₁ during the regulation process.

Figure 7.

Cases 1-3: Comparison of the proposed method, ${ZVD}_{ω_{1}}$ , and ${ZVD}_{ω_{n 1}}$ in terms of the performance of θ₁ and ς.

In Case 2, the liquid mass m₂ is varied according to scenarios (a) and (b), directly affecting the mathematical model, while the container length l influences the liquid elevation through equation (3). The corresponding results are shown in Figures 3 and 4. From Figure 3, Cases 2(a) and 2(b), represented by the blue and yellow-orange curves, exhibit overshoots of approximately 1.43% and 2.95%, respectively. However, the settling time in Case 2 decreases by approximately 2 s compared to Case 1, as evidenced in Table 1, with T_s = 5.6 s for Case 2(a) and T_s = 5.0 s for Case 2(b). Nevertheless, the overall stability performance of the trolley position is well maintained under the AIT2FC-DO scheme.

Specifically, the liquid elevation slightly increases with the variations in m₂ and l, reaching an ME of 2.2198 mm for Case 2(a) and 2.2608 mm for Case 2(b), compared to 2.085 mm in the nominal case, as shown in Table 1. Even when the model parameters are increased by +50% and +100%, the oscillation amplitude remains sufficiently small, demonstrating the capability of the AIT2FC-DO controller to handle parameter uncertainties. Moreover, increasing m₂ improves the stability of the swing angle θ₁, with the RMSE significantly reduced to 0.0063 rad and 0.0058 rad for Cases 2(a) and 2(b), respectively. However, based on the ITAE criterion, the damping performance slightly deteriorates since the ITAE values in Case 2 are consistently higher than those in Case 1. Conversely, the sloshing vibration exhibits reduced chattering and shorter settling times T_s for both θ₁ and ς. These results indicate that the proposed control scheme maintains accurate positioning performance and strong robustness against parameter uncertainties.

The proposed methodology is compared with the input shaping ${ZVD}_{ω_{1}}$ and ${ZVD}_{ω_{n 1}}$ in Khorshid and Al-Fadhli (2021). As shown in Figure 3, the trolley position under the input shaping reaches the reference position within 2 s. However, as illustrated in Figure 4, although the input shaping effectively reduce oscillations, the suppression of residual oscillations after the trolley reaches the desired position mainly relies on the inherent damping characteristics of the internal dynamics. In contrast, as reported in Table 1, the ME of the swing angle θ₁ is 0.2024 rad for ${ZVD}_{ω_{1}}$ and 0.0772 rad for ${ZVD}_{ω_{n 1}}$ , while the ME of the liquid surface elevation is 11.638 mm for ${ZVD}_{ω_{1}}$ and 7.343 mm for ${ZVD}_{ω_{n 1}}$ , which are significantly larger than those obtained using the proposed control scheme. The comparison with input shaping validates the effectiveness and robustness of the proposed closed-loop control scheme for the LSCS, as it achieves an approximately 5 mm lower liquid elevation during position tracking. Nevertheless, the AIT2FC-DO achieves this improved oscillation suppression at the expense of a settling time for x that is nearly twice as long as those of the input shaping.

Based on the performance analysis of the LSCS in Cases 1 and 2, the AIT2FC-DO exhibits strong adaptability, robustness, and effectiveness in driving the trolley position to the reference point with no or minimal overshoot, while reducing oscillation amplitude and rapidly suppressing oscillations once the desired position is reached, using only two assumed measurable variables from the system model under both nominal conditions and parameter uncertainties.

6. Conclusion

The LSCS presents a significant challenge due to its underactuated nature and strong dynamic coupling under the presence of unmeasured variables and model parameter uncertainties, thereby requiring a robust and effective control strategy. In this study, the proposed AIT2FC-DO addresses both issues related to unmeasured variables and model parameter uncertainties. By incorporating an Interval Type-2 fuzzy approximator based on the Nie–Tan type-reduction method, the uncertain components are effectively handled through adaptive update laws, while the composite sliding surface is designed to deal with the underactuated nature and strong dynamic coupling, ensuring the stability of the LSCS. The problem of unmeasured state variables is resolved by the Descriptor-based Observer, where the estimation errors are small and converge rapidly to zero. Furthermore, the stability of the overall system under the integrated AIT2FC-DO framework is guaranteed based on Lyapunov stability analysis. Simulation results demonstrate that the AIT2FC-DO not only ensures system stability under nominal conditions but also successfully stabilizes the system in the presence of uncertainties related to liquid mass and container length, confirming the effectiveness and robustness of the proposed methodology. In comparison with the input shaping, the time required to reach the desired position is longer; however, the liquid surface elevation is driven to zero immediately after the desired position is reached when the AIT2FC-DO scheme is employed. Future work will focus on the experimental implementation of the proposed approach on a real LSCS testbed to further validate its practical applicability.

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.

Appendix A

As mentioned previously, the optimal constant φ_j in equation (14) is typically bounded by an upper bound ${\bar{φ}}_{j}^{*}$ and a predefined lower estimate of this bound, denoted by ${\underset{̲}{φ}}_{j}$ , such that ${\underset{̲}{φ}}_{j} > {\bar{φ}}_{j}^{*} \geq φ_{j}^{*}$ . With the initial condition ${\hat{φ}}_{j} (0) > {\underset{̲}{φ}}_{j}$ , it is ensured that ${\hat{φ}}_{j} \geq {\underset{̲}{φ}}_{j}$ . This constraint is enforced by the projection function in the update laws (17):

Case 1: When ${\hat{φ}}_{j} = {\underset{̲}{φ}}_{j}$ and Ξ_j < 0, ${\dot{\hat{φ}}}_{j}$ is set to zero, which prevents any further decrease of ${\hat{φ}}_{j}$ and ensures that it does not fall below ${\underset{̲}{φ}}_{j}$ .

Case 2: Otherwise, ${\hat{φ}}_{j}$ is allowed to vary freely above the lower bound ${\underset{̲}{φ}}_{j}$ , and whenever this constraint is violated, the dynamics revert to Case 1. Therefore, equation (30) in Lemma 1 is always satisfied.

To prove equation (31), the term ϑ is decomposed as: (64)

ϑ = \sum_{j = 1}^{9} ϑ_{j}

where

\begin{aligned} ϑ_{j} & = - | S | {\hat{φ}}_{j} ω_{j} (χ) + S φ_{j}^{*} ω_{j} (χ) - {\tilde{φ}}_{j} Λ_{j}^{- 1} {\dot{\hat{φ}}}_{j} \\ = - | S | {\hat{φ}}_{j} ω_{j} (χ) + S φ_{j}^{*} ω_{j} (χ) - {\tilde{φ}}_{j} Λ_{j}^{- 1} proj (Ξ_{j}) \end{aligned}

Next, since the control parameter Λ_j is chosen to be strictly positive and the fuzzy basis function satisfies ω_j(χ) ≥ 0, it follows from equation (17) that the sign of the composite surface S can be inferred from Ξ_j.

Case 1: ${\hat{φ}}_{j} = {\underset{̲}{φ}}_{j}$ ; Ξ_j < 0, hence S < 0, |S| = −S and proj(Ξ_j) = 0.

(65)

\begin{aligned} ϑ_{j} & = S {\hat{φ}}_{j} ω_{j} (χ) + S φ_{j}^{*} ω_{j} (χ) \\ = S ({\hat{φ}}_{j} + φ_{j}^{*}) ω_{j} (χ) < 0 \end{aligned}

Case 2: ${\hat{φ}}_{j} = {\underset{̲}{φ}}_{j}$ ; Ξ_j ≥ 0 implies S ≥ 0 and |S| = S; ${\tilde{φ}}_{j} = φ_{j}^{*} - {\hat{φ}}_{j}$ .

(66)

\begin{aligned} ϑ_{j} & = - S {\hat{φ}}_{j} ω_{j} (χ) + S φ_{j}^{*} ω_{j} (χ) - (φ_{j}^{*} - {\hat{φ}}_{j}) ω_{j} (χ) S \\ = 0 \end{aligned}

Case 3: ${\hat{φ}}_{j} > {\underset{̲}{φ}}_{j}$ ; Ξ_j < 0, hence S < 0, |S| = −S and proj(Ξ_j) = 0; ${\tilde{φ}}_{j} = φ_{j}^{*} - {\hat{φ}}_{j}$ .

(67)

\begin{aligned} ϑ_{j} & = S {\hat{φ}}_{j} ω_{j} (χ) + S φ_{j}^{*} ω_{j} (χ) - (φ_{j}^{-} {\hat{φ}}_{j}) ω_{j} (χ) S \\ = 2 S {\hat{φ}}_{j} ω_{j} (χ) < 0 \end{aligned}

Case 4: ${\hat{φ}}_{j} > {\underset{̲}{φ}}_{j}$ ; Ξ_j > 0 implies S > 0 and |S| = S; ${\tilde{φ}}_{j} = φ_{j}^{*} - {\hat{φ}}_{j}$ .

(68)

\begin{aligned} ϑ_{j} & = - S {\hat{φ}}_{j} ω_{j} (χ) + S φ_{j}^{*} ω_{j} (χ) - (φ_{j}^{-} {\hat{φ}}_{j}) ω_{j} (χ) S \\ = 0 \end{aligned}

From the above four cases, it can be concluded that ϑ_j ≤ 0 for j = 1, …, 9. Consequently, ϑ ≤ 0, and equation (31) in Lemma 1 is proven.

Appendix B

Equation (55) is stable if and only if the candidate Lyapunov function $V_{4}^{*} = {\tilde{q}}^{* T} E^{* T} P {\tilde{q}}^{*}$ satisfies the following two conditions:

The Lyapunov function must be positive definite: (69)

E^{* T} P = P E^{*} > 0

where

P = [\begin{matrix} P_{1} & P_{2} \\ P_{3} & P_{4} \end{matrix}]

P_{1} = P_{1}^{T} > 0

and P₂ = 0.

The time derivative of the Lyapunov function must be negative definite $({\dot{V}}_{4}^{*} < 0)$ : (70)

{\dot{V}}_{4}^{*} = {\dot{\tilde{q}}}^{* T} E^{* T} P {\tilde{q}}^{*} + {\tilde{q}}^{* T} E^{* T} P {\dot{\tilde{q}}}^{*} < 0

Combining equations (55) and (69), equation (70) can be rewritten as: (71)

\begin{aligned} {\tilde{x}}^{* T} (\sum_{k = 1}^{4} \sum_{m = 1}^{4} \sum_{n = 1}^{4} υ_{k} (\hat{z} (t)) η_{m} (\hat{z} (t)) η_{n} (\hat{z} (t)) \\ \times ({[A_{m k}^{*} - K_{n k}^{*} C_{i}^{*}]}^{T} E^{* T} P + P E^{*} [A_{m k}^{*} - K_{n k}^{*} C_{i}^{*}])) {\tilde{x}}^{*} < 0 \end{aligned}

Equation (71) can be simplified into the LMI conditions given in equation (56) as follows: (72)

\begin{aligned} ϒ = \sum_{k = 1}^{4} \sum_{m = 1}^{4} \sum_{n = 1}^{4} υ_{k} (\hat{z} (t)) η_{m} (\hat{z} (t)) η_{n} (\hat{z} (t)) ϒ_{m n}^{k} < 0 \end{aligned}

Therefore, by solving the LMI conditions in equation (56), the stability of the observer is guaranteed. Moreover, from equations (72), (71), and (70), we obtain: (73)

{\dot{V}}_{4}^{*} = {\dot{\tilde{q}}}^{* T} ϒ {\dot{\tilde{q}}}^{*}

(74)

\begin{aligned} {\dot{V}}_{4}^{*} & \leq β ‖ {\dot{\tilde{q}}}^{*} ‖^{2} \\ \leq β ‖ \tilde{X} ‖^{2} \end{aligned}

where β is the minimum eigenvalue of ϒ. Given that

{\dot{\tilde{q}}}^{=} [{\tilde{q}}^{T}, {\dot{\tilde{q}}}^{T}]

and

{\tilde{q}}^{T} = [{\tilde{X}}^{T}, {\dot{\tilde{X}}}^{T}]

, it follows that

- ‖ {\tilde{q}}^{*} ‖ < - ‖ \tilde{X} ‖

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