Takagi-Sugeno fuzzy fault estimator design for nonlinear dynamics of autonomous ground vehicles

Abstract

In the context of cloud and edge computing for real-time fault diagnosis (FD) in autonomous ground vehicles (AGVs), this paper proposes a novel fault estimation method for AGV actuators. A two degrees-of-freedom nonlinear vehicle model is first used to characterize the AGV dynamics. Based on this model, a Takagi-Sugeno (TS) fuzzy observer is designed to estimate actuator fault signals. To handle unmeasurable premise variables, a nonlinear partitioning method reconstructs the TS fuzzy model into an N-TS fuzzy form. Unlike conventional TS fuzzy models with linear consequents, this reformulation incorporates the differential mean value theorem to explicitly address unmeasured nonlinearities–a common difficulty in TS fuzzy observer design. Using Lyapunov stability theory, the fault estimator is formulated as an optimization problem subject to strict linear matrix inequalities (LMIs), which can be solved efficiently with numerical tools. The proposed observer is validated through co-simulations in Simulink and CarSim, demonstrating its potential for deployment in cloud computing environments to enhance fault management in AGVs.

1 Introduction

The advancement and deployment of cloud and edge computing technologies have significantly enhanced the active safety framework of AGVs (Zhou et al., 2022; Li et al., 2025b,a). A critical requirement for enabling these safety-critical functions is the reliable availability of real-time on-board sensor data (Deng et al., 2023; Zhang et al., 2025). However, the economic and technical constraints associated with sensor deployment often limit both the quantity and quality of such data, raising substantial concerns about its reliability and accuracy (Nguyen et al., 2018). These limitations underscore a crucial operational challenge: ensuring the fault-free operation of AGV systems—particularly the steering system—is vital for maintaining driving safety. Therefore, effective fault diagnosis and timely fault estimation are not merely technical objectives but operational necessities for AGV deployment (Mu et al., 2021; Nguyen et al., 2021a). In this context, while the existing literature offers various approaches for system monitoring and fault detection, a notable gap remains in accurately estimating fault signals under conditions of limited measurability and system nonlinearity. This paper specifically addresses this challenge by developing a novel fault estimation methodology that overcomes the limitations of conventional approaches through an enhanced TS fuzzy framework capable of handling unmeasurable premise variables, thereby providing more robust and accurate fault diagnostics essential for the safe operation of AGVs.

Currently, accurate and timely fault estimation (FE) is critical for ensuring the operational safety of AGVs, serving as a fundamental input to fault-tolerant control systems that are essential to vehicle active safety (Zemzemi et al., 2019). While FE has drawn considerable research attention, existing methods often encounter practical limitations. For example, a fuzzy Proportional-Integral observer for TS fuzzy AGV systems was proposed in ElYoussfi and Oudghiri (2018), yet it is restricted to continuous-time models and involves a comparatively complex design procedure, limiting its applicability to real-world digital implementations. In response to these challenges, and inspired by Nguyen et al. (2021b) on handling unmeasured premise variables in observer design, this paper develops a discrete-time TS fuzzy observer to estimate actuator faults in AGV steering systems under disturbance conditions (Tanaka and Wang, 2004). The approach builds on the assumption that the second-order derivative of the fault signal is negligible, enabling system reformulation and facilitating a model-based estimator derived via direct Lyapunov stability analysis. The design is cast as a convex optimization problem in the form of LMIs, which can be efficiently solved with available numerical tools. By integrating the differential mean-value theorem (DMVT) into an N-TS fuzzy structure, the proposed estimator not only accommodates unmeasured nonlinearities but also delivers accurate fault estimates, thereby bridging the gap between theoretical design and real-time feasibility for AGV safety systems.

Building upon the foundation of TS fuzzy models for nonlinear vehicle dynamics, the primary limitation of existing methods lies in their reliance on measurable premise variables, which restricts their applicability to systems with partial state observability. To address this, we propose a reformulation of the AGV system into an N-TS fuzzy framework, which systematically integrates unmeasurable premise variables into a localized nonlinear consequent using the DMVT. This approach offers several distinct advantages over conventional TS fuzzy observers:

• (i) The proposed N-TS fuzzy framework provides notable advances over conventional TS approaches. Unlike existing methods that rely on fully measurable premise variables–restricting their use to systems with complete state observability–the N-TS formulation explicitly accommodates unmeasurable nonlinearities through a localized consequent structure. This enhances its applicability to AGV systems with partial or uncertain sensing. Moreover, by embedding finite-frequency characteristics of fault signals and disturbances–grounded in established physical insights–the observer design is optimized for realistic operational bandwidths (Liu et al., 2021; Wang et al., 2021), improving estimation accuracy while reducing tuning conservatism. These contributions collectively support a more adaptable and reliable fault estimation strategy for AGVs.

• (ii) The observer gains are derived through LMIs, ensuring both asymptotic stability of estimation errors and computational tractability for real-time implementation.

• (iii) The proposed fault estimation scheme is rigorously validated via co-simulations in Simulink and CarSim, demonstrating effective performance under varied driving scenarios, and showcasing its practical relevance for AGV safety-critical applications.

The structure of this paper is delineated as follows: Section 2 presents the derivation of a tailored N-TS fuzzy model for AGVs. It also outlines the preliminary preparations for the design of the fault estimation algorithm, including the observer and the differential mean value theorem. Useful Technicality such as lemma, theorem and LMI-based conditions are formulated in Section 3 for FE observer design. The visual representations of the simulation are detailed in Section 4, whereas Section 5 encompasses the concluding remarks.

2 Vehicle modeling

This section first describes the vehicle modeling utilized in observer-based fault estimation methodologies, then elaborates on the challenges in designing a fault estimation observer for AGV systems, and finally provides a collection of pertinent technical resources.

2.1 Nonlinear vehicle model

2.1.1 Vehicle dynamics model

This study adopts a 2-DOF nonlinear lateral vehicle model (Kang et al., 2023; Nguyen et al., 2024), which is depicted in Figure 1. Its dynamic model is given by:

where v_x[m/s] denotes the longitudinal velocity, v_y[m/s] signifies the lateral velocity, r[rad/s] represents the yaw rate of the vehicle, and T_eng[Nm] indicates the torque input affecting the longitudinal dynamics (Deng et al., 2023; Zhang et al., 2025). The lateral wind force is expressed as F_w[N]. F_yf and F_yr correspond to the front tire forces and rear tire cornering forces, respectively, with other vehicle nomenclature provided in Table 1.

Figure 1

For our analysis, we assume typical driving conditions while adhering to the small angle approximation (Kang et al., 2025; Deng et al., 2023; Zhang et al., 2025). Furthermore, the lateral forces generated by the tires are directly related to the slip angles observed at each axle (Kang et al., 2025). Hence, the cornering forces at the front tires F_yf and at the rear tires F_yr can be approximated by:

where δ is the front wheel steering angle. By analyzing Equations 1, 2, we can derive the governing nonlinear dynamics that characterize the behavior of the vehicle in the following manner:

In this scenario, the state vector of the vehicle is articulated as , whereas the control input is represented by . Additionally, the external disturbance is defined as w = F_w. The state-space matrices that pertain to the nonlinear vehicle model described in Equation 3 are delineated as follows:

with

Furthermore, we acknowledge that although both the vehicle's longitudinal velocity v_x and the yaw rate r can be accurately measured, the lateral velocity v_y remains inaccessible due to financial limitations. As a result, the nonlinear model represented in Equation 3 can be restructured into the following descriptor form:

where

This research investigates the frequency and implications of actuator failures in AGVs. In pursuit of this objective, we analyze the failures associated with both the steering angle and Longitudinal torque, as detailed in Zhang and Wang (2017). The steering angle of the front-wheel steering, denoted as δ, together with the engine torque, represented by T, can be expressed mathematically in the following formulation:

In this framework, the parameters δ_d and T_d signify the desired steering angles for the front wheels as well as the associated engine torque, respectively. The variable δ_f represents the steering angle that is influenced by system faults, while T_f denotes the perturbation in engine torque. The coefficients b_δ and b_T are established based on the gain values of the actuators. We analyze model Equation 4 in conjunction with the fault model detailed in Equation 5, as follows:

where

2.1.2 Mathematical formulation of fault models

To rigorously characterize actuator faults, we consider two representative types:

(1) Bias injection fault: a constant or slowly varying bias is superimposed on the nominal control input:

where is the bias vector, H(·) is the Heaviside step function, and t_f denotes the fault onset time. In continuous time, the fault signal vector in Equation 6 becomes:

(2) Actuator scaling (loss-of-effectiveness) fault: The actuator's effectiveness degrades over time, modeled as a time-varying scaling factor:

where ⊙ denotes element-wise multiplication. A linear degradation model is adopted:

with degradation rate β > 0. The corresponding fault signal is then:

These two explicit fault models cover common real-world actuator abnormalities and are used in the subsequent observer design and simulations.

Remark 1. In order to better represent the practical value of the designed observer, we considered the actual situation of the vehicle driving. We consider that the external disturbance during vehicle driving is mainly the obstructive effect of wind„ and, this signal is a low frequency signal (Liu et al., 2021), we assume it is a constant and is w = 800[N]. In this work, in designing the fault estimator considering the faults of both steering angle and longitudinal torque, and considering the fault estimation performance when there is a disturbance signal in the actuator.

2.2 Fuzzy modeling for AGVs

In order to enable real-time execution, it is imperative to reconfigure the continuous model represented by Equation 6 into a discrete framework. For this transformation, we utilize the classical Euler method, defined by the equation , where s = 0.01 signifies the sampling interval. As a result, the model Equation 6 is reformulated as follows:

where

and I is the identity matrix of suitable dimension, corresponding to A(x_k).

In view of the fundamental physical constraints commonly present in typical driving scenarios (Nguyen et al., 2024), we can articulate a concise and comprehensive representation of the vehicle's state as follows:

Taking into account the parameters v_x = 5 [m/s], [m/s], v_y = −1.5 [m/s], [m/s], r = −0.55 [rad/s], and [rad/s], it is evident that the vehicle system described by Equation 12 exhibits three distinct nonlinear characteristics, which are referred to as premise variables: namely, [], , and v_y. Therefore, we define the vector of premise variables as . Utilizing the sector nonlinearity method as detailed in Tanaka and Wang (2004), one can systematically develop the standard eight-rule TS fuzzy model that accurately represents the nonlinear dynamics of the vehicle as articulated in:

The membership functions (MFs) represented as h_i(ϖ), where i is an element of the index set , are defined by the criteria h_i(ϖ) ≥ 0 and the normalization condition . The local state-space matrices A_vi, associated with each index , are formulated by substituting the premise variables v_x, , and v_y with their corresponding upper and lower bounds within the framework of A(x_k). For conciseness, additional information regarding the local matrices and membership functions is deliberately omitted in this section. As of now, the conventional TS fuzzy representation has been widely utilized in the assessment of vehicle dynamics (Zhang et al., 2016). However, this traditional TS fuzzy framework introduces both theoretical and practical difficulties in the development of vehicle observers, primarily owing to the presence of unmeasured premise variables within the MFs Nguyen et al. (2021a). To overcome this notable limitation, we draw upon concepts from N-TS fuzzy modeling (Coutinho et al., 2020) and reformulate the vehicle system Equation 12 in the following manner:

where

and the system matrix A_i, for , , . The forthcoming section delineates the specifications pertaining to the local matrices alongside their associated moment functions, which are represented as h_i(ξ)A_i:

the MFs, satisfy the following convex sum property:

Remark 2. The TS fuzzy models, which are derived from the sector nonlinearity approach, result in a polytypic embedding characterized by a vertex count of 2ⁿ, where n represents the quantity of premise variables. As such, an increase in the number of premise variables leads not only to heightened computational requirements but also to an escalation in the structural complexity of the observational frameworks. Current methodologies for observer design in the context of AGVs rely on the Lipschitz condition that is intrinsic to conventional TS fuzzy modeling (Pan et al., 2020), culminating in a TS fuzzy model that is governed by eight fuzzy rules, as depicted in Equation 14. By reconfiguring the nonlinear descriptor system illustrated in Equation 14 into the format presented in Equation 15, it is feasible to attain a more accurate fuzzy representation that employs only four fuzzy rules. This reformulation serves to mitigate both the computational burden associated with observer design and the structural complexity of the observers necessary for real-time applications. Additionally, a salient feature of the N-TS fuzzy model delineated in Equation 15 is the evaluation of the nonlinear consequent g(ξ_k) alongside the segregation of all measured premise variables within the unmeasured nonlinear consequent ϕ(x_k), thereby establishing a robust framework conducive to the design of fuzzy observers for AGVs.

2.3 Fault estimation problem formulation

Since the frequency of fault signals is limited in practical applications, using finite frequency technology for fault estimation can reduce its conservativeness (Zhang and Wang, 2017). We assume that the actuator fault signal f_k dynamics is represented as a triple integrator (Wang et al., 2022), namely,

By analyzing Equations 14, (17), one can derive the following comprehensive N-TS fuzzy system,

The system matrices in Equation 18 are give by:

where . and the MFs-dependent matrices are give by: . It is important to acknowledge that the MFs adhere to the convex sum property as delineated in Equation 16.

For the purposes of estimating faults, we will examine the following configuration of the observer:

In this context, z_k denotes the state variable observed by the observer, while represents the estimated value of . The matrices N(h), L(h), M, and E, which are contingent upon the membership functions, must be meticulously designed to fulfill specific criteria,

Considering the state estimation error defined as , one can deduce from Equations 19, 20 that

Consequently, in light of the prevailing circumstances,

The dynamics of estimation error can be articulated based on the formulations presented in Equation 21 through Equation 23.

The presence of the mismatched term Δ_ϕ in Equation 24 presents a significant obstacle in the design of observers, as highlighted in Pan et al. (2020) and Pan et al. (2024). In order to adeptly address this term and ensure the convergence of the asymptotic estimation error, it is necessary to reformulate the expression for as a function of the estimation error e_k, as demonstrated in the subsequent Lemma 1.

2.4 Technical lemmas

Lemma 1.Zemouche et al. (2008) Let be a differentiable function defined on the convex hull co(a, b), where a and b are elements of . Under these conditions, it follows that there exist constant vectors c_i belonging to co(a, b) such that c_i is neither equal to a nor to b for every index i within the index set .

By utilizing Lemma 1 in relation to the function , it can be deduced that there exist elements ν_i belonging to the convex hull of the points and , for indices i drawn from the set ,

where ρ_ij is defined as for every pair . Given that x resides within the set S_x, as outlined in Equation 13, the parameter ρ is constrained to a bounded convex region denoted as S_ϕ, characterized by its vertices, which are specified as follows:

Based on the formulations presented in Equations 24, 26, we can reformulate the dynamics of the residual as follows:

where and

The formulation of the observer design problem can be articulated as follows.

Problem 1. In the context of nonlinear vehicle dynamics represented in N-TS form, achieving an effective estimation of faults for AGVs necessitates that the state estimate , the fault signal f_k, and its corresponding estimation converge asymptotically. Additionally, it is crucial to ascertain the matrices N(h), L(h), M, and E associated with the observer described in Equation 19. By employing arguments grounded in Lyapunov's theory, this study delineates a feasible and systematic approach to address the aforementioned observer design challenge.

3 Fault estimation observer design for vehicle system

In this section, we shall initially revisit the following lemma and theorem, which will serve as foundational components in the formulation of the principal outcomes presented in this manuscript.

3.1 Useful technicality

The subsequent lemma plays a pivotal role in facilitating the convexification process associated with the design of observers.

Lemma 2.Rao and Mitra (1971) Let us examine the matrix W ∈ ℝ^m×n where m ≥ n and the matrix Z ∈ ℝ^k×n. The matrix X, defined as X = ZW^† + Y(I − WW^†), where Y is an arbitrary matrix in ℝ^k×n, serves as a solution to the equation XW = Z under the prerequisite that the relation ZW^†W = Z is satisfied.

The subsequent theorem establishes the foundational theoretical framework necessary for the realization of our objective in designing a fault estimation observer.

Theorem 1. Examine the observer configuration described in Equation 19 under the stipulations outlined in Equations 20, 22. The stability of the estimation error dynamics characterized by Equation 27 is ensured if there exist positive definite matrices P(h) > 0, N(h) that are dependent on membership functions, alongside matrices R, M, and E, satisfying the requisite conditions,

for h, h₊ ∈ Ω, ρ ∈ S_ϕ, and in block matrices, the notation ⋆ represents the components inferred from symmetry.

proof 1. To analyze the error system represented by Equation 27, we perform pre- and post-multiplication of Equation 29 by the matrix and its conjugate transpose. This operation yields a derived expression that facilitates further examination of the underlying dynamics.

In the context of analyzing asymptotic stability, we employ a candidate fuzzy Lyapunov function as a primary tool for our investigation,

with P(h) > 0. The variation of the fuzzy Lyapunov function Equation 31 along the trajectory of system Equation 27 is defined as follows:

From Equations 30, 32, it can be deduced that the variation in the Lyapunov function, denoted as ΔV, is less than zero for all instances where e_k ≠ 0. This observation effectively completes the proof.

3.2 LMI-based fault estimation observer design

The subsequent theorem presents a numerically feasible solution for the design of the finite element observer as articulated in Problem 1.

Theorem 2. Examine the observer framework delineated in Equation 19 under the stipulations set forth in Equations 20, 22. The stability of the estimation error system characterized by Equation 27 is guaranteed if there exist matrix functions dependent on h, specifically P(h) > 0 and N(h), along with matrices R, M, and E, such that the required conditions are met,

for h, h₊ ∈ Ω, ρ ∈ S_ϕ, and

proof 2. From Equations 20, 42, we can obtain

Due to the complexities associated with the nonlinear matrix equality presented in equation (18) and the corresponding nonlinear matrix inequality outlined in equation (20), along with their inherent dependency on the matrix function (MF), it is not feasible to directly apply Theorem 2 for the design of the finite-time (FE) observer. In light of this observation, we propose leveraging Lemma 2, with the following assumptions: matrix possesses full row rank, matrix exhibits full column rank, and the rank of product is equivalent to the rank of , we can derive the necessary results.

To facilitate the calculation, we can define,

From Equations 37, 41, we can obtain Equation 29 Note let us define,

Consequently, upon analyzing Equations 22, 23, 42, one can deduce that

From Equations 42, 43, we can obtain

The algorithm for the fault estimation observer design of autonomous vehicles is summarized in Figure 2.

Figure 2

This concludes the proof.

4 Experiments and analysis

This section presents illustrative results to validate the efficacy of the proposed TS fuzzy fault estimation observer in simultaneously estimating AGV dynamics and actuator fault signals. The simulations are conducted in a co-simulation framework integrating MATLAB/Simulink and CarSim. A high-fidelity AGV model is developed in CarSim, with key parameters summarized in Table 1, including a rear drive-toque ratio of [4:1], maximum power of 150[kW], and operation on dry asphalt road with a friction coefficient μ = 1.0. The proposed observer is implemented in Simulink, and its parameters are obtained by solving the LMIs in Theorem 2 using the SDPT3 solver. Its pseudocode is presented in Algorithm 1.

Algorithm 1

Two distinct experimental scenarios with different actuator fault types are designed to evaluate observer performance. The simulation results from CarSim are shown in Figures 3–6. Figures 3, 5 illustrate the vehicle's trajectory and dynamic states, while Figures 4, 6 present the fault estimation performance for longitudinal torque and steering angle faults, respectively. These results collectively demonstrate the observer's capability to accurately reconstruct both system states and fault signals under realistic driving conditions.

Figure 3

Figure 4

Figure 5

Figure 6

4.1 Scenario 1: driving with a random leftward vehicle trajectory

In this test scenario, the 2-DOF vehicle executes a sequence of maneuvers, specifically proceeding straight, making a right turn, and then continuing straight before executing another right turn, as illustrated in Figure 3a. The initial conditions of the observer are set to [5, −1, 0]. The AGV system experiences a sudden fault signal characterized by an amplitude of 0.3[Nm] affecting the longitudinal torque, alongside a repetitive fault signal with an amplitude of 0.1[rad] and a frequency of 0.5[rad/s] impacting the steering angle, As shown by the blue line in Figure 4. As shown in Figure 3c, the estimated unmeasurable premise variables v_y of the AGV rapidly align with their corresponding measurements through the designed fuzzy fault estimation observer, even in the presence of lateral wind disturbances occurring between 10 and 11 s. Furthermore, Figure 4 demonstrates that both the longitudinal torque fault and the steering angle fault can be accurately estimated using the proposed fuzzy fault estimation observer methodology.

4.2 Scenario 2: driving with random lane-changing

In this context, the amplitude of the fault signal is progressively amplified to replicate the actual variation observed in vehicle fault signals. This approach aims to further assess the estimation capabilities of the proposed fault estimation observer without necessitating any information pertaining to the actuator faults in the vehicle's steering system. Accordingly, as depicted in Figure 5a, the vehicle performs a lane change maneuver between 13 and 21 seconds. The initial longitudinal velocity of the vehicle, represented as v_x, is 5[m/s]; the initial lateral velocity, denoted as v_y, is −1[m/s]; and the yaw rate, indicated as r, is 0[rad/s]. Fault signals are represented by blue lines in Figure 6. In alignment with the preceding two test scenarios, Figures 5c, 6 distinctly illustrate that the proposed methodology can not only effectively estimate the dynamic characteristics of AGVs but also accurately quantify the fault signal.

4.3 Scalability discussion

The simulation scenarios in Sections 4.1–4.2 represent worst-case conditions: fault intensity is set to the maximum design limit of 10% of nominal input, faults persist throughout the entire 20s simulation, and both actuators are simultaneously compromised. As demonstrated in Figures 3–6, the proposed observer maintains reliable estimation under these extreme conditions, confirming its scalability to the most demanding fault scenarios in real-world AGV applications.

5 Conclusions

This study focuses on the steering angle control system of AGVs and develops a dynamic model for the AGV system. To address the issue of unmeasured premise variables in the TS fuzzy model, a nonlinear partitioning approach is employed to restructure the TS fuzzy model specific to the AGV dynamics. This restructuring results in an enhanced TS fuzzy model that incorporates localized nonlinear sub-models, termed the N-TS fuzzy model. Furthermore, the differential mean value theorem is utilized to manage the nonlinear components effectively. To mitigate the design's conservatism, fault estimation algorithms are devised using fault signals associated with lateral torque and steering angle, confined to finite frequency domains. The efficacy of these fault estimation algorithms is assessed in relation to the impact of these actuator fault signals on the AGV system. Leveraging Lyapunov stability theory, the LMI conditions for the fault estimation algorithms are formulated and resolved through numerical solvers. Ultimately, co-simulations conducted in MATLAB/Simulink and CarSim demonstrate that the proposed fault estimation algorithms achieve superior performance in fault estimation, particularly when integrated with cloud computing and edge computing frameworks for real-time monitoring and decision-making.

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