A virtual-structure-based type-3 fuzzy system for predictive sensor and actuator fault detection, compensation, and control in nonlinear systems

Abstract

This paper presents an active fault-tolerant control strategy for a class of nonlinear systems subject to both actuator and sensor faults. The system dynamics are assumed to be completely unknown, and only the measured outputs and the applied control inputs are used for system identification. Accordingly, the system is treated as a black-box model. The proposed control framework consists of three integrated subsystems. The first subsystem incorporates a Type-3 fuzzy logic system to model the unknown nonlinear dynamics, in conjunction with a model predictive controller and an adaptive stabilizing unit. The second subsystem is dedicated to sensor fault diagnosis and compensation, employing two Type-3 fuzzy estimators and a supervisory module that generates a corrective signal proportional to the magnitude of the actual sensor fault. The third subsystem constitutes a virtual structure composed of a virtual sensor, a virtual actuator, a Type-3 fuzzy identifier, a virtual model predictive controller, a virtual fault detection unit, an internal adaptive compensator, and a supervisory mechanism.

Introduction

General overview

Real-world systems show nonlinear, unstable behavior, which makes control system stabilization and effective control management extremely difficult, according to¹. The development of control architectures for sensor measurement and actuator-based control strategy implementation helps solve these system challenges. The operational reliability of these systems depends on the continuous correct functioning of sensors and actuators from start to finish of their operational period². The main obstacle in control system development requires the immediate identification of faults that occur in system components.

The research presents an innovative control system that tracks control signals in real-time to identify sensor and actuator faults and perform active compensation for their impact. The literature contains various fault detection and diagnosis methods, yet most approaches need system models for operation and work best for post-analysis purposes. The system requires a model-free real-time solution that operates from input-output data without needing system dynamic information. The system provides important benefits for real-world applications because it does not need system model precision.

The latest developments in Type-3 fuzzy logic systems (T3FLSs) have shown excellent performance when handling extreme uncertainties and external interference, and measurement errors³. The paper implements T3FLSs for system modeling and prediction and closed-loop control operations under both sensor and actuator failure scenarios. The research aims to create a model-predictive control (MPC) system which maintains stability when system faults occur. The following sections provide an extensive evaluation of current fault detection and tolerance methods used in control systems before introducing the proposed solution.

Literature review

The two main fault-tolerant control (FTC) approaches exist as passive (PFTC) and active (AFTC) systems^4,5. The control law in passive frameworks operates with a fixed design that supports pre-defined fault scenarios. The active control approach adjusts its controller design through real-time fault detection to enhance system performance across different fault types^6,7. Nowadays, FTC systems based on fuzzy systems have attracted a lot of attention from researchers⁸. For example, in⁹ and¹⁰, active fault detection approaches of actuator and robust sensor based on fuzzy systems for multivariate systems, and in¹¹, actuator and sensor fault detection based on fuzzy systems under finite time stability conditions are investigated.

MPC stands as a leading practical solution for complex dynamic system management because it represents one of the most effective advanced control approaches¹². The process model inside MPC enables optimization algorithms to generate future control actions. The system solves an optimization problem at each sampling period to find the best control sequence before executing the first sequence element on the system¹³.

The following section provides an extensive evaluation of fault detection systems for actuators and sensors, which includes traditional model-based approaches together with contemporary intelligent solutions.

In¹⁴, an FTC system with MPC and Takagi-Sugeno (T-S) fuzzy control is proposed to manage actuator failures in uncertain systems. Also, a fuzzy controller controls the tracking error. This method requires an explicit model of the system.

In¹⁵, a hierarchical FTC for independent drive systems is presented, which combines MPC and fuzzy logic to compensate for drive failures. This method requires a predefined model of all subsystems. In¹⁶, a hierarchical MPC-based FTC is presented for fault detection of self-driving vehicles. By defining a fault-tolerant range, MPC ensures stability. In hierarchical MPC, the inaccuracy in uncertainty modeling leads to computational complexity and poor fault coverage.

In¹⁷, a stochastic fuzzy MPC for uncertain and delayed nonlinear processes is presented. Through a T-S model for fault detection based on linear matrix inequalities (LMI), stability is guaranteed. However, a predefined model is required for the design. In^18,19, MPC-based FTC and sliding mode control for nonlinear systems are presented. Simulation results show its superiority over the detection method based solely on sliding mode control. However, the vibration and high control effort in the presence of driving faults can pose a major challenge to this method.

In²⁰, a robust MPC with dynamic output feedback is developed to handle actuator failures in uncertain discrete-time industrial control systems. The stability of the system is ensured by solving online LMIs. In this method, computing online LMIs is a challenge. In²¹, a distributed MPC-based FTC is developed for multi-agent systems. A discrete distributed fault observer based on neighboring agent state data is used to detect actuator faults. In this method, a predefined model is required for fault detection.

The research in²² develops a hybrid FTC system for collaborative robots that utilizes MPC to detect faults early and perform compensation. In this method, issues such as covering different types of platforms are not considered, and the design method requires a predefined dynamic.

In²³, an MPC-based FTC ensures the stability of autonomous vehicles by tracking and predicting the path in the presence of uncertainty and actuator faults. However, the problem of various faults in other actuators is not considered, and the model dependency is also explicit.

In²⁴, an MPC-based FTC is proposed to detect sensor faults and external disturbances. The system states are estimated by a Kalman filter. The superiority of the method over conventional MPCs is demonstrated. The dependence of MPC on the predefined model is obvious.

The research by^25,26 develops a T3FLS and MPC system for blood glucose management in type 1 diabetes patients, which operates successfully even when sensors fail. The system combines real-time glucose prediction with fault identification and adaptive correction to achieve stable operation through all system uncertainties and measurement errors.

The research by²⁷ develops a T3FLS and MPC system for blood glucose management in type 1 diabetes patients through online fuzzy recognition and active fault detection mechanisms. The method achieves successful results through simulations of a modified Bergman model under various disturbance conditions and system uncertainties. In²⁵ and²⁷, despite its dependence on the problem model, the driving fault is not integrated alongside the sensor fault. The following section presents an extensive evaluation of predictive control systems that detect and handle faults that affect sensors and actuators.

The method²⁸ combines an MPC-based AFTC and an extended Kalman filter (EKF) for constrained nonlinear multivariate systems. The method simultaneously identifies actuator and sensor faults. This method requires an explicit model of the system and estimates of all its states. In²⁹, an adaptive MPC-based FTC for multi-agent systems in the presence of sensor and actuator faults is presented. This method maintains a stable cooperative trajectory by estimating the moving horizon in the presence of uncertainty and path disturbance. In this design, the problem of multiple and combined faults is not considered. In³⁰ a sensor-actuator fault-based FTC for DC microgrids is developed. It combines dual EKF state and error estimation with fuzzy MPC for robust voltage regulation. This method is based on state estimation and actually requires a predefined model.³¹ presents an MPC-based FTC for sensor and actuator fault detection in wind turbines in the presence of delay, uncertainty, and disturbance. The system is based on an extended state space model and LMI solutions that maintain stability. The method is conservative and requires a full-state model. In³², an analytical MPC-based FTC and T3FLS for mobile robots in the presence of actuator and sensor faults are presented. Fast calculations in MPC demonstrate the superiority of the method. This method does not consider the modeling of initial uncertainty on the inputs. In³³, a finite-time MPC-based FTC and sliding mode observer for a wind turbine in the presence of sensor and actuator faults is presented. The stability of the system is guaranteed in the presence of uncertainty, disturbance, and full load. The complete dynamic equations of this system are required for design.

Research gap

Most current model-based FTC systems for actuators and sensors use traditional control methods, which need complete system model information and actuator dynamic data. The requirement for system model information makes these methods unsuitable for systems with unpredictable or changing system dynamics. The increasing interest in model-free and model-independent control systems stems from their ability to adapt better in real-world applications. The passive nature of traditional FTC systems makes them susceptible to unanticipated system changes and dynamic system variations. The need for active FTC systems that identify and correct faults in real time has driven researchers to create new fault-tolerance methods. The performance of fuzzy logic-based methods decreases when they need to handle unknown system uncertainties because they depend on fixed models. Research has focused on creating model-independent fuzzy systems to address current system limitations. The basic structure of low-order fuzzy models proves insufficient to represent complex nonlinear system behaviors that include significant uncertainty. The advanced robustness and strong approximation abilities of T3FLS systems make them suitable for controlling complex systems that experience faults and system uncertainties. The current state of FTC research lacks methods that can detect and handle both sensor and actuator faults simultaneously. The development of a single system that detects and handles both sensor and actuator faults stands as a fundamental research challenge. The internal workings of systems remain unavailable during practical operations, so controllers must use available input-output data for both control and identification purposes. The MPC-based strategies without full state information or complete internal models prove essential for achieving effective FTC operations under uncertain conditions.

The research presents an innovative FTC system that uses only input-output data from fuzzy systems to achieve control objectives. The system identification process operates online to handle system uncertainties, which allows researchers to establish stability conditions for both general and asymptotic cases. The proposed method operates without needing direct information about sensor or actuator system dynamics. The system model receives actuator-induced errors through a calculation that measures the difference between actual control inputs and virtual actuator outputs. The main controller receives an additional control signal from a dedicated compensatory structure, which functions to correct sensor faults.

The identification process under uncertain real-time operations becomes possible through T3FLSs because these systems excel at handling complex nonlinear system behaviors and uncertain conditions. The adaptive identification framework with MPC becomes the most appropriate choice because system dynamics remain completely unknown. Fuzzy systems have established themselves as effective tools for nonlinear system modeling, and researchers divide them into four main categories: type-1, interval type-2, general type-2, and T3FLSs. T3FLSs achieve the highest level of modeling accuracy and system stability, which makes them suitable for systems that exhibit strong nonlinear behavior and high levels of uncertainty. Given the identified gap, namely the high dependence of existing FTC methods on precise systems and sensor and actuator dynamics models, and their limited power in dealing with uncertainties, this study presents a T3FLS-based framework that is explicitly independent of the system type and sensor and actuator dynamics modeling. Furthermore, due to the high power of T3FLS in handling severe uncertainties, this framework is considered a suitable option for online and adaptive identification for uncertain dynamics.

Main contributions

Research has proven that T3FLS systems achieve better accuracy and stability than basic fuzzy logic systems. The research develops an integrated control system that uses three interconnected modules to improve system dependability when dealing with system uncertainties and sensor and actuator faults. The system operates through three connected modules, which work together to provide fault-resistant performance.

In fact, the advantage of this method over previous methods is that the system identification is independent of the predefined model and is performed based on measurable input and output information. The fuzzy identifier model is updated adaptively, which handles any changes in the system. By changing the type of system, the structure can be used with only a small adjustment in the fuzzy system. The dynamic model of the sensor and actuator is not required. Therefore, changing the sensor and actuator does not require changing the identifier system. The actuator fault is identified by forming a parallel structure similar to the original structure. In fact, this structure estimates the output of the original actuator well. Also, a similar structure is used for the sensor fault. In the reviewed methods, there is generally a need for a predefined model. The dynamics of the sensor and actuator are important. And the problem of the actuator and sensor fault together is considered in a very limited way. The system contains three essential parts, which include a fuzzy estimation system for sensor data, an MPC, and an adaptive stabilizer for maintaining control accuracy. The system includes two fuzzy subsystems and a supervisory mechanism that produces error correction signals based on measured discrepancies. The system operates through a virtual structure that contains a fuzzy system that produces virtual sensor output and three additional components: a virtual MPC and an adaptive compensator, and a proportional fault-correction block for the virtual sensor. The virtual structure output serves as a fault detection method for actuator problems in the primary system. The system consists of multiple components that work together to create a flexible control framework that preserves stability and precision during system failures and unpredictable system behavior. In general, T3FLS and MPC can lead to an increase in computational burden as the system dimensions increase. However, in the framework proposed in this paper, a compact and adaptive T3FLS structure is used, which is designed only based on effective input-output variables and prevents the exponential growth of fuzzy rules. Also, the MPC is implemented analytically without solving iterative optimization, which significantly reduces the computational complexity. It should be noted that the parameter tuning process is performed online and adaptively, eliminating the need for extensive manual tuning. As a result, the proposed method remains acceptable and scalable for practical implementation in multi-state systems despite its high fault tolerance and uncertainty management capabilities. This research presents the following key findings.

• The system operates through measurable system data without needing system model information for its operation.

• The system combines T3FLS with a main MPC and type-3 fuzzy estimator and adaptive stabilizer and sensor-fault compensation unit, and virtual actuator fault-detection module.

• The sensor-fault compensation unit contains two type-3 fuzzy subsystems, which work with a supervisory mechanism to produce error correction signals based on sensor fault measurements.

• The virtual structure model uses T3FLS and MPC, an adaptive compensator, and a virtual sensor fault-detection block to detect actuator faults.

• The T3FLS parameters receive real-time updates through a Lyapunov-based stability criterion.

• The system achieves global and asymptotic stability through Lyapunov theory, while the adaptive compensator ensures the system converges to zero error.

• The research proves the practicality of the control method through tests that include system parameter changes and model inaccuracies, and external interference.

• The research compares the proposed method to basic fuzzy logic systems to show its superior performance capabilities.

• The developed framework operates effectively with different types of nonlinear systems that contain uncertainties.

The paper continues with Section Problem modeling, which describes the control structure and T3FLS-based problem modeling. Section Main NMPC and virtual NMPC design describes the main and virtual MPCs. Section Main and virtual sensor fault detection unit design explains the design process for main and virtual structures that detect and compensate sensor faults. Section Estimation of the actuator fault coefficient explains the method for detecting actuator faults. Section Stability analysis presents the stability analysis. The paper presents simulation results from different test cases in Section Simulations while demonstrating the superiority of the proposed method against current approaches. The paper ends with Section Conclusion, which summarizes all important research findings.

Problem modeling

This section is organized into two parts. First, it presents the overall problem formulation, and then it proceeds with the development of the T3FLS.

Fuzzy-based modeling approach

In this section, the system is treated as a black box, where both the controller-generated control signal and the system’s sensor output are assumed to be measurable. The study focuses on a single-input, single-output configuration. A control signal is produced and applied to the actuator, which subsequently delivers the actual input to the system. The resulting system response is then captured by the sensor. The overall structure of the system is illustrated in Figure 1. The proposed model will be as follows, where (1) corresponds to the main system and (2) corresponds to the virtual dynamics.

Fig. 1.

A black-box representation of a nonlinear uncertain system.

1

2

Where and are the estimated outputs of the main strcture and the virtual structure, respectively. and denote the actual outputs of the main and virtual structure. and represent the T3FLSs. and are the input vectors to and , respectively. and denote the learnable consequent parameters of and , while and represent the ideal consequent parameters of the fuzzy systems. and indicate the corresponding approximation errors. and denote the main and virtual control signals, respectively. Furthermore, is the estimation of the parameter associated with the actuator fault level. Moreover, is set to and is defined in proportion to the virtual sensor. In practice, the gain is applied to the virtual sensor during the initial time.

The input vectors of the fuzzy systems and are considered as follows:

3

4

Where =1, 2, ..., q. Here, q denotes the total number of samples. In the framework of the proposed control strategy, the control inputs are formulated as follows:

5

6

Where and are the MPCs. and are the adaptive compensators. Also and are the control signal of the main sensor and sensor’s virtual fault compensator, which will be further explained in the actuator and sensor fault detection unit design sections.

Figure 2 shows the proposed control structure in general, and Fig. 3 in more detail. This design is modular, with each subsystem designed independently, and the MPCs implemented analytically without solving iterative optimizations. This approach reduces practical complexity and preserves the benefits of fault tolerance and accurate system performance without unnecessary computational burden. Also, reducing the dimensionality of the inputs and optimizing the structure of the fuzzy rules controls the computational complexity, and simulations show that the system remains suitable and stable for practical implementation.

Fig. 2.

The overall design of the proposed control system.

Fig. 3.

A more detailed description of the proposed control structure.

The estimation technique of the proposed method, compared to classical approaches, does not require explicit knowledge of the system dynamic model and is based on input-output data. Therefore, it can well approximate the nonlinear behavior of the system in the presence of noise, disturbance, and uncertainty. The use of fuzzy techniques enables integration with advanced adaptive MPCs, resulting in less sensitivity to parameter tuning.

The sensor and actuator fault detection mechanism is based on the steady state value of the control signals. Any fault in the actuator and sensor causes a drop in the final control signals obtained from the output of the fuzzy logic systems. By generating compensatory control signals, any faults that causes improper performance at the output is compensated. Also, the MPC controls these faults to some extent using the output prediction model. Finally, the adaptive compensators maintain the stability guarantee. Therefore, different control layers will increase the reliability of the system.

Type-3 fuzzy structure

Over the past five years, T3FLSs have attracted growing attention for their effectiveness in modeling and controlling nonlinear systems²⁵. Owing to their structural characteristics, T3FLSs exhibit superior precision in addressing uncertainty and vagueness in system modeling. This type of fuzzy system has been further advanced through the introduction of distinct membership functions in the fuzzification process³⁴.

Several recent studies have demonstrated the versatility of T3FLSs across diverse application domains. In³⁵, the modeling and experimental realization of a DC motor control system under uncertainty are investigated. Decision-making challenges in complex systems are analyzed in³⁶. A diagnostic modeling framework for identifying mental disorders in patients is presented in³⁷, while a related study in³⁸ introduces a method for diagnosing vascular diseases by modeling the heart’s blood vessels. In³⁹, a modeling and optimization approach based on the bee colony algorithm is proposed for solving mathematical functions. The work in⁴⁰ focuses on modeling and identification techniques for brain tumor analysis.

Further, in⁴¹, the prediction capability of T3FLSs is evaluated on a management strategy. In⁴², a method for modeling and achieving synchronization in chaotic financial systems is proposed. A modeling and control strategy incorporating soft-switching techniques for non-holonomic robots driven by electric actuators is introduced in⁴³. Moreover,⁴⁴ provides an extensive review of additional T3FLS applications in system modeling and control, underscoring their broad utility and effectiveness in addressing complex, uncertain systems. The following section details the development procedure of the T3FLS employed in this study (see the general structure on Figure 4). To compute the membership functions, the input variables are first transformed according to relations (11)–(14). The corresponding expressions for the membership functions are then derived as follows (see Figure 5).

7

8

9

10

where,

11

12

13

14

In which and are memberships corresponding to the left and right sides of the upper bound of . and are memberships corresponding to the left and right sides of the upper bound of . and are the widths for the right-left side of the upper bounds of . and are the widths for the right-left side of the lower bounds of . describe the degree of uncertainty. is the center of . For fuzzy systems, the rule are defined as follows:

Fig. 4.

Internal configuration of the T3FLS.

Fig. 5.

The general form of T3FLS memberships.

then

15

where, and . Also, , , , and are the rule parameters. The upper and lower rule firing levels are expressed as:

16

17

18

19

Next, the output value of the fuzzy system is calculated as follows:

20

where

21

and

22

Also and are calculated as:

23

24

In the proposed framework, the T3FLS matching rules are designed to ensure signal boundedness and transient stability even in the presence of non-ideal initial settings. In addition, the MPC, in the presence of the adaptive compensator, plays a compensatory role in the transient phase and prevents the occurrence of instability. The presented simulation results also show that the proposed system has low sensitivity to the choice of initial parameters and achieves acceptable convergence. Regarding uncertainties and disturbances, including noise, the T3FLS is well able to estimate it. This estimate appears as a high-frequency disturbance to the MPC and the adaptive compensator. The adaptive compensator, as a parallel conservative controller, ensures stability in noisy conditions. In fuzzy systems of the fault detection part that are not updated adaptively, an EKF is used, which is robust against white and colored noise.

The number of fuzzy rules is determined according to the number of input and output samples of the system. This selected number of inputs is a logical compromise between the model approximation and the computational volume. In fact, the database is selected as compact as possible to cover nonlinearities, disturbances, and uncertainties, while maintaining the possibility of real-time implementation in the NMPC framework. The shape of the membership functions is selected as smooth and symmetric functions with more uncertainty on the inputs. The parameters of the membership functions are selected in the working range of the input variables, which is a common approach in fuzzy systems. The uncertainty range of the parameters is determined based on the knowledge and experience of the systems. Finally, due to the adaptive update of the output parameters of the fuzzy system, there will be less sensitivity than with the initial selection. However, all the parameters of the fuzzy system can be optimized.

The black-box system identification method in this paper is based on the input-output data of the system, without the need for explicit knowledge of the internal dynamic structure of the system. T3FLS, as general estimators, are able to learn the dynamic behavior of the system well. It is also assumed that uncertainties such as noise, disturbances, and modeling errors are bounded. Ultimately, the reliability of the black-box identification method depends on the actual choice of parameters of the T3FLS.

Main NMPC and virtual NMPC design

In this section, the design of the main and virtual MPC is presented. At this stage, the focus is solely on developing the MPC. At the same time, the main sensor fault compensator, virtual sensor fault compensator, the adaptive stabilizer, and the virtual adaptive compensator will be designed separately and later integrated into the main and virtual MPC. The analytical design of the MPC and the reduction of the dimension of the fuzzy rules limit the memory and computation consumption, and simulations show that the required resources are within acceptable practical limits. The slight increase in computations brings the advantage of detecting actuator and sensor faults. Considering the system dynamics in (1) and (2), the discrete-time formulation of the outputs from the main and virtual estimator can be expressed as follows:

25

26

According to (25) and (26), assuming that , , and remain constant, the prediction over the horizon for the main estimator can be formulated as:

27

Also, the prediction over the horizon for the virtual estimator can be formulated as:

28

Equation (27) and (28) can be rewritten in the following form:

29

30

in which:

The MPC signals are obtained by formulating the following cost functions:

31

32

Here, represents the reference input, while denotes the vector of reference inputs over the prediction horizon and , which is defined as follows:

33

In addition, and denotes the steady-state component of the main and virtual control input, where and are a positive-definite matrix, and and are a positive semi-definite matrix. The steady-state value of the main and virtual control signals, derived from the dynamics of the Estimator (1) and (2), can be expressed as:

34

35

Finally, using the dynamics described in (1), (2), (36) and (37), the steady-state expression for the control signal can be formulated as:

36

37

Consequently, the vector and are computed as follows:

38

39

Ultimately, the main and virtual MPC signal vectors are obtained by minimizing the cost functions defined in (31) and (32), as expressed below:

40

41

The initial element of this vectors are applied to the system, after which the procedure is repeated.

Main and virtual sensor fault detection unit design

In this section, two units are designed to compensate for the faults of the main and virtual sensors. As shown in²⁷, when a sensor fault occurs, the effect of the fault can be compensated for by creating a compensation structure through a control expression. It should be noted that the virtual sensor fault detection mechanism causes the virtual actuator to generate a control signal proportional to the reference input and disturbances entering the system. This virtual actuator signal is used to detect the fault of the main actuator. Based on this idea, the following reference signals are defined:

42

43

Let and are represent the reference control signals, and define and in a manner similar to (38) and (39). The quantities and correspond to a time-varying gain whose actual value cannot be measured directly. Therefore, it must be inferred through estimation. To do this, an estimate of and are first computed, which are subsequently used to approximate and . Two T3FLS and are utilized to estimate and , taking the following input vectors:

44

45

To estimate and , two T3FLS and are employed, which takes the following variables as its inputs:

46

47

The structures of , , , and along with their membership functions, as well as the fuzzification and defuzzification procedures, are designed following the same principles as those used for the fuzzy systems and in Section Type-3 fuzzy structure. Moreover, , , , and represent previous samples of the corresponding signals. When training the fuzzy system and , the target values are set to and . In contrast, the desired values used to train the fuzzy systems and are defined as follows:

48

49

As stated in (50) and (51), the estimated steady-state control signal remains non-zero, owing to the alignment of the system output with the reference input. In this section, the fuzzy systems , , , and are trained using the EKF algorithm⁴⁵. The one-step-ahead prediction of and , represented by and , can be expressed as:

50

51

Ultimately, a supervisory mechanism is utilized to produce the control signals, which compensates for the main and Virtual sensor fault, as expressed below:

52

53

Here, and represents the control gains. In (52) and (53), and takes a specific value when a main and virtual sensor fault occurs, and this estimated value are employed to detect the presence of such a fault. The target output of T3FLS is defined to be only one step ahead of the input vectors. Therefore, the estimation process is only accompanied by one unit of time delay. In addition, the sensor fault value is estimated as a step-forward prediction, which allows for fast error detection and compensation even in the presence of high-frequency disturbances. Simulation results also show that this limited delay does not have a noticeable impact on the sensor fault detection and compensation performance.

Estimation of the actuator fault coefficient

In this part, the estimated value of the parameter , introduced in (1), is calculated. In practice, the actual value of the parameter corresponds to the difference between the control input and in (5) and (6). However, since the actuator’s output cannot be directly measured, this value is approximated using the virtual structure introduced earlier. By comparing the estimated virtual control signal with the value in (6), the parameter is ultimately obtained as follows:

54

Where and are introduced in (5) and (6). Subsequently, the stability analysis of the closed-loop system is carried out, and the control signals for both the primary and virtual compensators are derived.

In classical approaches, it is essential to have an explicit model of the system and sometimes the actuator dynamics. Any inaccuracy of the model in the presence of uncertainty and disturbance will cause a decrease in the efficiency of actuator fault detection. Also, unlike classical methods, there is no need to reconfigure the controller and the fault is compensated by modifying the fuzzy model online. In fact, the actuator fault detection system in this paper is itself a control loop that acts like the original structure that imitates the behavior of the original system in the presence of a virtual actuator and a virtual sensor.

Stability analysis

This section establishes the asymptotic stability based on Theorem 1. The stability of both the virtual structure loop and the main structure is guaranteed by employing the virtual and main adaptive compensators. The selected Lyapunov function is defined as a combination of estimation error, tracking error, adaptive fuzzy system parameters, and approximation error. The overall dynamics of the system are uncertain, and the real and virtual structures are modeled by first-order nonlinear estimators. Therefore, the dynamics that must be stable and convergent are tracking errors, estimation errors, adaptive fuzzy system parameters, and approximation errors. Including the fuzzy system parameters in the Lyapunov function prevents parameter drift and ensures that the adaptive coefficients are bounded. Also, considering the approximation error and its adaptive update allows compensating for the inherent limitations of the fuzzy system in approximating unknown dynamics and increases the robustness of the control structure against uncertainties. Therefore, from a practical point of view, the selection of this Lyapunov function is practical and can be generalized to various systems.

Asymptotic stability

Theorem 1

As shown in Section Problem modeling, the control system–consisting of the uncertain dynamic model (Figure 3), the estimator models (1) and (2), along with the controllers (5) and (6)–is guaranteed to be asymptotically stable, provided that the compensators and , the adaptation laws and , as well as and , are specified as follows:

55

56

57

58

59

60

where and denote the tracking errors. The parameters , , , and are predefined positive constants. The terms and represent the estimated upper bounds of and , respectively. Furthermore, the coefficients satisfy , , , and .

Proof

Based on the definitions of T3FLSs in (1) and (2), the estimation errors are defined as and , respectively, which leads to:

61

62

Where and . Also, for the tracking errors we have:

63

64

By defining and , V is candidate as below:

65

Then, we have:

66

According to (61), (62), (63), and (64) we have:

67

Now, according to (57) and (58), along with placement in the first and second sentences (67), we have:

68

Also, according to (59) and (60), by substituting into the third and fourth terms in (68), and considering the positive values of the terms of the first and second sentences, we have:

69

Considering the maximum bandwidth of the signals , , and as , , and , equation (69) can be rewritten as follows:

70

With a few changes in (70) and rewriting it, we will have:

71

Assuming and , Equation (71) will be as follows:

72

By substituting (55) and (56) into (72), we have:

73

With a slight change in (73), we will have:

74

Also, by sorting (74), we have:

75

So, with assumptions and , we have:

76

77

Finally, it is proven that:

78

To show asymptotic stability, according to (75) with the assumption:

79

80

81

82

The inequality (75) will be as follows:

83

With assumptions and , we have:

84

Considering (63), (64), and (82), we have:

85

Assuming that all signals and parameters in (85) are bounded, it follows that is also bounded. Similarly, under the conditions and , the boundedness of still holds. Hence, by applying Barbalat’s Lemma, one can conclude that is negative semi-definite, ensuring the asymptotic stability of the system. This completes the proof of Theorem (1). The adaptive compensator plays a key role in ensuring the boundedness of all signals and the convergence of the system. The overall stability of the system in the closed loop was proved using the combined Lyapunov function that includes the tracking error, the estimation error, the fuzzy system parameters, and the adaptive approximation error. Using the Barbalat’s Lemma and a supplementary proof, the asymptotic stability was also shown. Thus, the stable operation of the closed loop is guaranteed under the assumed conditions. The simulation results also confirm these proofs.

In the proposed method, the stability and robustness of the system are ensured by T3FLS, the MPC controller, and adaptive compensators. T3FLS, with high estimation accuracy under conditions of disturbance, uncertainty, and time-varying dynamic changes, models the collective effect of these factors as an online adaptive estimator. MPC generates the optimal control signal with respect to this estimation accuracy. Finally, by approximating the upper bound of the modeling error, real and virtual compensators ensure robust stability in the presence of faults and in fault-free conditions.

Also, this system is model independent and depends only on the input and output information of the system. Therefore, increasing the internal dimension of the dynamic system has no effect on the calculations and stability. In fact, the system is a black box, only its input and output are important.

Simulations

The following section demonstrates the effectiveness of the proposed solution through two separate case studies. The first case study investigates robotic arm control systems, while the second case study examines PH control systems in chemical processing. The system requires a control strategy that operates based on measurable input-output data without needing to know the system’s internal structure or dynamic behavior.

The system dynamics remain completely unknown throughout this research. The control risk measure helps evaluate the proposed method across different operational scenarios. The performance evaluation of control systems becomes more accurate through this measure because it considers both tracking precision and control signal magnitude. The measure shows how well tracking performance relates to control signal requirements.

The process failure probability defines the risk factor in this situation according to⁴⁶. The risk assessment includes two essential elements that identify potential system breakdowns and their resulting expenses. The risk function accepts tracking error and control signal as its input elements to produce a risk output.

86

Where e(t) and u(t) represent the tracking error and control signal, respectively, it is worth noting that both the tracking error and the control signal are normalized within the range of 0 to 1. Accordingly, based on the defined risk criterion, the following index is employed for comparative analysis:

87

Where represents the total number of signal samples over the entire time horizon, the parameters used in all scenarios–kept consistent throughout the analysis–are summarized in Table 2. The risk index is a combination of tracking error and control effort, mainly reflecting the overall trend of system performance and may not directly reflect severe momentary faults. However, this index is a good measure for comparing the overall performance and stability of the system over time, and its limitation is due to the nature of averaging and normalization. However, the error signals and momentary control effort can be considered as complementary tools for identifying severe short-term events. In addition, the simulations evaluate several performance indices, whose definitions and interrelations are described below.

88

In this context, represents the number of data points, denotes the desired signal, and refers to the reference signal. The adaptive design of T3FLS and MPC, together with the adaptive compensator, effectively limits unexpected transient and nonlinear behaviors. Simulations and experimental results show that the proposed system maintains stability and acceptable performance even in the presence of severe uncertainties and nonlinearities.

Table 2.

Parameters of the control structure for case study 1.

Parameter	Value	Number of Equation
L	16	(15)
	(, ), (, )	(7), (8), (9), (10)
, ,	1, 100, 0.1	(87), (8) , (9), (10)
	1000, 1800, 1000, 1800	(55), (56)
, , ,	0.01, 0.01, 0.001, 0.001	(57), (58), (59), (60)
,	100, 100	(52), (53)

In the proposed method, there are key parameters whose correct selection is of great importance. The basis of the work is the online identification of the uncertain system. Therefore, the correct selection of the parameters of the fuzzy system is important for the accuracy of the estimation and convergence. For example, the center of the membership functions should be within the range of changes in the input signals. Deviation from these values reduces the sensitivity of the fuzzy system in output estimation and increases the modeling error. The update rate of the fuzzy system should not be too large or too small so that the appropriate estimate with the least error is always given. The update rate of the modeling error estimate should be less than the update rate of the parameters of the fuzzy system. This choice prevents output oscillation and improves stability. The robustness of the proposed method is ensured by adaptive compensators. In fact, the compensators operate in parallel with the MPCs and ensure the stability of the closed-loop in conditions of disturbance, noise and uncertainty. This combination has resulted in the structure being robust under a wide range of different conditions such as parameter uncertainty, disturbance, changes in initial conditions, etc.

In the method of this paper, the parameters of the case studies are assumed to be uncertain according to the initial assumptions of the problem. The adjustment of the fuzzy systems in both case studies has a great impact on the estimation accuracy. These adjustments are given in the simulation sections in Table 2 and Table 5. Therefore, despite the parametric uncertainties, only by changing some parameters of the fuzzy system in accordance with the constraints of the case studies, the stability of the closed-loop system will be maintained in the presence of adaptive compensators.

Table 5.

Parameters of the control structure for case study 2.

Parameter	Value	Number of Equation
L	16	(15)
	(0, 5), (0, 14)	(87), (8), (9), (10)
, ,	1, 100, 0.1	(7), (8) , (9), (10)
	1000, 1800, 1000, 1800	(55), (56)
, , ,	0.01, 0.01, 0.001, 0.001	(57), (58), (59), (60)
,	100, 100	(52), (53)

Case study 1

In this study, the control problem of a single-arm robot system is examined. Several control strategies have been developed for this class of systems, most of which rely on predefined mathematical models. To evaluate the effectiveness of the proposed approach, the system is analyzed under conditions of parametric uncertainty and external disturbances. The dynamic equations governing the single-arm robot system⁴⁷ are expressed as follows:

89

where , , , and represent the link angle, link angular velocity, motor angle, and motor angular velocity, respectively. The parameters of the model (89) are summarized in Table 1. The control performance strongly relies on the accuracy of the fuzzy identifier, as shown in the simulation results. For the simulation, the reference was set at . Table 2 presents the T3FLS and controller parameters, which were first initialized using input bounds.

Table 1.

Model parameters introduced in (89).

Parameter	Value
m	0.12kg
g	9.88
l	0.53m
I	0.083
k	3.33
B
J
R

The matching coefficients and are chosen as small and positive values to achieve a good balance between the speed of parameter convergence and the prevention of oscillations and noise sensitivity in adaptive type 3 fuzzy systems. Therefore, the value of 0.01 is considered as a common, stable and practical choice in Lyapunov-based fuzzy systems. The system uses to represent its uncertain system parameters. The system receives disturbances at two specific time points, which are and . The system experiences an actuator fault at time , which causes the actuator efficiency to decrease smoothly until it reaches 50% loss. The system experiences a sensor fault with a gentle slope when time reaches 10. The simulation results appear in Figures 6, 7 and 8. The control system successfully controls disturbances while preserving system stability according to Figure 6. The system tracks its reference input while staying stable even though it experiences both actuator and sensor faults. The controller produces control signals which appear in Figure 7 together with the actual control signal from the actuator and its predicted value. The estimated signal tracks the actual signal with high precision, which demonstrates good estimation performance. The figure demonstrates that the system accurately determines the strength of the actuator fault. The system demonstrates stability through both RMSE and risk value measurements, which stay at low levels even when sensors fail and actuators malfunction, external disturbances occur, and system parameters become uncertain. The system maintains stability through all operational conditions according to the results.

Fig. 8.

True parameter of the actuator fault and its estimation for a 90 reference input under actuator and sensor faults in Case Study 1.

Fig. 6.

Output response and its estimated value for a 90 reference input under actuator and sensor faults in Case Study 1.

Fig. 7.

The original control signal, the actual actuator signal, and its estimated value for a 90 reference input under actuator and sensor faults in Case Study 1.

Table 3 summarizes the system performance under various reference inputs and presents the outcomes obtained using the proposed control and fault diagnosis approach⁴⁸. To assess the effectiveness, several performance metrics were employed to examine its stability/robustness. The findings indicate that, despite variations in reference inputs, external disturbances, and parameter uncertainties, the RMSE and risk values remain consistently low. These results confirm the capability of the proposed closed-loop control scheme to sustain stable and reliable performance under diverse and uncertain operating conditions.

Table 3.

Parameters of the control structure for case study 1.

Reference Input (Angle)		RMSE,	RMSE,	RMSE,
10	0.0023	0.57	0.43	2.03
20	0.0026	0.60	0.42	2.04
30	0.0027	0.64	0.42	2.05
40	0.0031	0.67	0.42	2.05
50	0.0033	0.70	0.43	2.07
60	0.0034	0.72	0.45	2.09
70	0.0036	0.75	0.45	2.09
80	0.0041	0.77	0.46	2.10
90	0.0049	0.81	0.46	2.11

Case study 2

In this study, the PH control system in chemical processes is examined. Although the dynamics of such processes are generally multi-input, multi-output, under conditions where the control input is base, the acid input remains constant, and the buffer is considered as disturbances, the system can be approximated as a single-input, single-output process. The dynamic equations governing this system are as follows⁴⁹:

90

Here, h denotes the tank height, is a coefficient, A represents the cross-sectional area, and and are the flow rates of the acid, base, and buffer streams, respectively, , , , and indicate the concentration of substances. and show dynamic changes. Finally, the output value is determined by solving the following equation and considering its real response.

91

Then

92

As reported in (90), , , and represent the equilibrium constants for the equilibrium reactions occurring in this process. The parameters associated with Model (90) are summarized in Table 4. In this section, the uncertain parameters are considered to be identical to those defined in case study 1.

Table 4.

Parameters of the model (90).

Parameter	Value
	0
A

Table 5 also summarizes the T3FLS and controller parameters corresponding to the case study 2.

The simulation ran for 120 minutes in total duration. The simulation introduced an actuator fault at minutes before adding a sensor fault at minutes. The process maintained its reference PH value at throughout the entire operation. The system starts with an initial value of 4. The results from this case study appear in Figures 9, 10, 11. The virtual framework tested the fault tolerance of its control system through an artificial sensor failure which occurred during the initial operational phase. The system operated under this fault condition for less than one minute based on its response time. The proposed controller maintains process PH tracking and regulation while maintaining precise prediction accuracy through both sensor and actuator faults as shown in Figure 9. The control input and actual control signal and their estimated values appear in Figure 10 while Figure 11 shows the estimated parameters that match the actual actuator fault behavior. The system performance under different initial conditions appears in Table 6 which demonstrates the results of the proposed control system and fuzzy diagnosis framework. The closed-loop system required multiple performance indicators to determine its operational stability and resistance to disturbances. The system performance indicators show that the RMSE and Risk values stay at low levels regardless of the initial conditions and external disturbances and parameter uncertainties. The proposed closed-loop system demonstrates its ability to operate stably through various uncertain situations according to the obtained results.

Fig. 9.

Output response, estimated output, and sensor output for initial conditions of pH = 4 in Case Study 2.

Fig. 10.

The original control signal, the actual actuator signal, and its estimate for initial conditions of PH = 4 in Case Study 2.

Fig. 11.

Real actuator fault and its estimation for initial conditions of PH = 4 in Case Study 2.

Table 6.

Parameters of the control structure for case study 2.

Initial condition (PH)		RMSE,	RMSE,	RMSE,
2	0.0087	1.09	0.92	5.10
3	0.0085	1.07	0.89	5.08
4	0.0084	1.05	0.87	5.07
5	0.0083	1.04	0.85	5.04
6	0.0083	1.03	0.85	5.03
7	0.0082	1.03	0.87	5.03
8	0.0081	1.00	0.87	5.00
9	0.0082	1.01	0.89	5.00
10	0.0083	1.01	0.89	4.99
11	0.0084	1.02	0.90	5.05
12	0.0084	1.04	0.91	5.06

Comparison

In this section, the closed-loop control system is evaluated using different types of fuzzy systems and also compared with other methods. It should be noted that both case studies 1 and 2 are compared under similar conditions.

Scenario 1

In this scenario, the closed-loop structure is compared by considering different fuzzy systems of type 1, type 2, and type 3. The comparison results are summarized in Table 7. As observed, the control structure designed based on T3FLSs demonstrates superior performance. Specifically, according to the selected evaluation metrics, the T3FLSs design yields smaller tracking and estimation errors, as well as lower control risk, thereby enhancing the overall reliability of the control framework.

Table 7.

A comparison of closed-loop control systems based on various fuzzy system.

	FLS-type
	1	2	3
in case study 1	0.174	0.034	0.0049
in case study 2	0.112	0.0132	0.0084
RMSE of in case study 1	3.43	1.87	0.81
RMSE of in case study 2	4.12	2.95	1.05
RMSE of in case study 1	2.89	1.84	0.46
RMSE of in case study 2	2.13	1.23	0.87

According to Table 7, the risk index and RMSE are presented for both the case studies, and the results show that the T3FLS-based design performs better than the low-order fuzzy systems in managing errors and reducing the effect of disturbances. This advantage is due to the ability of T3FLS to better model the uncertainties and disturbances of the system and reduce the effect of approximation error, which leads to a more stable control response and more reliable convergence. Therefore, the table not only shows the numbers but also explains the superior performance of T3FLS.

Scenario 2

In this scenario, the proposed method of this paper is compared with the methods of²⁸ and³². The method of²⁸ presents the problem of detecting and compensating for actuator and sensor errors by considering the complete system model with classical MPC control. The method of³² presents a fast MPC analytical approach based on type 3 fuzzy systems. For a better comparison, case studies 1 and 2 are investigated under similar initial conditions but with more complex noise conditions, more uncertainty, and larger disturbances. In fact, more severe noise is assumed in addition to the process dynamics for the sensor and actuator. Unlike in case study 1 and case study 2 simulation , parametric uncertainties of more than 50% uncertainty are considered. The amplitude of disturbances is also considered differently. The results of this comparison in terms of RMSE and risk index are reviewed in Table 8. As can be seen, the proposed structure is more robust to noise, disturbance, and parameter uncertainty, and the RMSE error index is always lower. The proposed structure also has lower control risk than similar approaches.

Table 8.

Comparison of closed-loop control systems with methods²⁸ and³².

	Reference number
	28	32	Proposed Method
in case study 1	0.47	0.17	0.11
in case study 2	0.38	0.13	0.09
RMSE of in case study 1	5.87	2.37	1.74
RMSE of in case study 2	5.21	2.26	1.56

For future studies, the advanced leaning methods such as deep learning methods⁵⁰ can be used. Also, the effect of time-delay can be considered⁵¹.

Conclusion

This paper presented a control system for a class of nonlinear systems with uncertain dynamics. The system was treated as a black box, and only input–output data was used. The proposed closed-loop configuration consisted of two subsystems: the main and virtual structures. The main structure included a T3FLS to estimate the system’s dynamics, an MPC, and an adaptive compensator to guarantee the stability. The virtual structure was introduced to construct an actuator fault detection mechanism, using T3FLSs. By defining a virtual sensor and injecting a virtual fault, the actuator control signal was accurately estimated through this virtual structure, allowing for dynamic fault detection and compensation via adaptive tuning of the T3FLS model. A complementary mechanism was also designed for sensor fault detection, with errors mitigated through an additional compensatory control term.

To examine the proposed scheme, two case studies were used: (1) control of a single-arm robotic manipulator and (2) PH regulation in chemical processes. In both cases, the system dynamics are considered to be completely unknown, and the control strategy relies just on measurable input–output data. The controller was tested under different conditions, including uncertainties, external disturbances, actuator/sensor faults, and time-varying reference signals or initial conditions. The results demonstrated that the introduced T3FLS-based control scheme achieves high accuracy and robust fault-detection capability. In all scenarios, the closed-loop system was stable, and RMSE/risk values remained consistently low, confirming the resilience and reliability of the designed controller. Comparative analyses with other related controllers further highlighted the superior robustness/response of the suggested approach.

As mentioned, the results of the paper are based on simulations, and practical hardware limitations, including real-time computation, sensor noise, and actuator saturation, have not been directly investigated in this study. However, the design of the adaptive compensator and the fuzzy NMPC with short-horizon prediction has reduced the computational load and increased the system’s resistance to disturbances and sensor errors. In the systems studied, both fast and relatively slow dynamics have been considered, and the closed-loop response is acceptable in both cases.

The method in this paper is model-independent and allows generalization to multivariable systems by considering virtual structures that behave similarly to real dynamics. The method is also model-independent and can be generalized to more complex systems because it relies only on input and output information.

Future research will focus on extending this approach to more complex nonlinear systems. In addition, alternative MPC architectures and fuzzy systems based on hybrid learning will be investigated. Also, a more advanced architecture will be designed in which the magnitude of errors due to actuator and sensor faults is simultaneously detected and compensated.

Author contributions

Conceptualization: X. H., N. Z., D. Y., A. K., A. M, and C. Z.; Writing initial draft: X. H. and A. K.; Review-Editing: N. Z., D. Y., A. M, and C. Z.; Simulations: A. K.; Fund acquisition: N. Z. and D. Y.

Funding

This work was supported by the Ministry of Science and Higher Education of the Republic of Kazakhstan “Intelligent Models and Methods of Smart City Digital Ecosystem for Sustainable Development and the Citizens’ Quality of Life Improvement” under Grant BR24992852.

Data availability

All data generated or analysed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.

Ethical approval

No ethics approval was required.