Fault-tolerant control of quadrotor unmanned aerial vehicle by using adaptive fuzzy T-S and linear matrix inversion approach

Abstract

In this research, a robust and adaptive fault-tolerant control approach is presented for the rejection of the unknown disturbances and faults existing in the quadrotor unmanned aerial vehicle sensors. The proposed approach is based on the hybrid utilization of the Fuzzy Takagi-Sugeno and linear matrix inversion control. Quadrotors are highly complex and nonlinear dynamic systems, which are vulnerable to unknown faults and disturbances, due to which the system performance may cause instability or degradation. To guarantee the stability of the system, an adaptive fuzzy T-S model is utilized to depict the nonlinear behavior of the system through a set of linear models based on fuzzy rules. The adaptive approach enables the controller to dynamically adjust based on the varying system conditions. A linear matrix inversion (LMI) control strategy is developed based on this model to effectively compute control inputs while maintaining system performance and stability. By integrating the approaches, the real-time faults are compensated without the requirement of fault diagnoses and isolation. The results of the simulation show that the suggested adaptive strategy shows robustness and strong resilience against unknown faults and disturbances, and ensures the stable flight compared to the conventional Fuzzy T-S controller, conventional linear matrix inversion (LMI) controller, and the techniques used in the literature. The results show the efficacy, effectiveness, and efficiency of the proposed approach for real-time UAV applications.To improve the safety and performance of nonlinear systems, it is suggested that the described method be integrated into fault-tolerant control (FTC).

Introduction

Unmanned aerial vehicles (UAVs) have drawn increased interest or attention due to their revolution in numerous industrial applications due to their versatility and efficiency¹, such as agricultural surveying, disaster management, and aerial photography^2,3. UAVs exist in numerous types, and quadrotors have gained more significance because of their ease of utilization, compact design, and the ability to fly with agility. The quadrotor’s maneuverability and stability are enhanced due to its specific configuration of rotors, which makes it more feasible for operations such as agricultural monitoring, rescue and search operations, as well as aerial mapping². Although quadrotors have numerous advantages, they also face several challenges in terms of control, disturbances, and faults that occur during the flight. In the presence of such challenges, there is always a demand for highly advanced control approaches in order to ensure the safety and performance. As a result, fault-tolerant control strategies, which seek to improve the robustness and dependability of quadrotor systems, are becoming more and more popular.

Existing strategies

Designing the advanced or innovative control approaches for the unmanned aerial vehicles (UAVs), numerous strategies have already been proposed in literature considering the challenges such as external disturbances, faults in actuators and sensors, and system nonlinearities. Each technique has its unique restrictions and advantages, depending on its unique methodology, computational necessities, and robustness capabilities in managing the specific dynamics of the UAVs.

In literature, numerous model-based strategies such as model predictive control (MPC), proportional, integral, and derivative (PID) control are proposed for managing the requirements of the UAV control. These strategies are mainly based on the linearized models, which are well known for their practical feasibility and analytical simplicity for ensuring system stability. However, these approaches are restricted in managing the nonlinear and complex dynamics inherent in the motion of the UAVs^4–9.

To manage these restrictions, nonlinear control strategies have gained more attraction and significance due to their capabilities of robustness and flexibility in managing the nonlinear, coupled, and complex dynamics of UAVs. Nonlinear control techniques such as sliding mode control (SMC) improve the robustness by forcing the system into a safe and stable sliding surface, by strengthening it against undesirable and unwanted external disturbances, but may also cause unwanted chattering effects^10–12. Backstepping control technique offers a recursive approach for designing the control, ensuring the stability of the subsystems step-by-step, and is mainly considered suitable for nonlinear and complex systems^8,9,13,14. Furthermore, the linearized feedback techniques transform the nonlinear systems into corresponding linear models, offering accurate control, but demanding a precise model, which is not always possible and feasible^15–17.

In spite of these developments, nonlinear control approaches might face challenges such as chattering effects, calculations of complex control input, and the requirements of a highly accurate model. Due to these challenges, the demand for intelligent and model-free strategies has increased recently, such as neural networks and fuzzy logic control, which ensure that external disturbance and faults are managed more adaptively and flexibly to manage the dynamic variations in the UAV’s environment^6,18–21.

In the analysis of stability, various strategies provide different control accuracy levels over time. The asymptotic stability ensures equilibrium convergence as the time approaches infinity, but it does not guarantee fixed time assurance to reach the equilibrium position²². In contrast, the finite-time stability ensures the equilibrium convergence within the fixed-time frame; however, this concept relies on the initial conditions of the system^23,24. In recent research, the fixed-time stability is considered a useful strategy that guarantees not being too dependent on the initial conditions of the system, as well as ensures the anticipated finite-time frame. This concept of finite-time stability is considered a highly suitable and feasible approach to incorporate in UAVs that ensures certain performance within the specified finite-time frame^25–27.

Fault-tolerant control (FTC) plays a vital role in the control of UAVs and is considered a vital part of UAVs, as the presence of faults in actuators or sensors can significantly degrade the safety and performance of the flight. The capacity of fault-tolerant control (FTC) techniques to preserve system performance and stability in the face of uncertainties, disruptions, and component defects has drawn a lot of attention in recent years. To handle sensor and actuator failures in nonlinear and uncertain systems, a number of adaptive and intelligent control strategies have been put forth^28–31.

Generally, Fault-tolerant control (FTC) strategies are divided into two groups: active and passive control approaches. The passive Fault-tolerant control (FTC) strategies are developed to ensure the safe flight performance in the presence of faults, where non-detection, isolation, and reconfiguration of faults are required. These strategies are considered simpler to employ and implement, which might suggest restricted adaptability and flexibility in managing the numerous unknown faults and disturbances^32–36. Whereas, the active strategies for Fault-tolerant control (FTC) implement and employ the diagnosis, isolation, and detection of faults in order to tune the control approach in consideration of the detected faults. This type of strategy offers a more adaptable, robust, and flexible response to various unknown faults; however, it demands more computational requirements and additional complex algorithms^37–39.

Contributions and suggested strategies

In this research study, the adaptive control approach is presented that manages the several challenges faced by the quadrotor’s nonlinear nature, external disturbances, and manages the actuators and sensors faults. In this study, the intelligent nonlinear approach is combined with the linear control concept in order to ensure robust and safe flight. Because of the consideration of the complex dynamics, the detailed nonlinear dynamical model of the system is developed first. The developed model is then utilized in its condensed form, considering the external disturbances and sensor faults, in order to provide a realistic scenario to the proposed control approach. In order to manage the quadrotor nonlinear dynamics, the linear matrix inversion (LMI), a robust control approach used in literature, is employed in combination with the Fuzzy Takagi-Sugeno (T-S) approach. The integration of the Fuzzy Takagi-Sugeno (T-S) approach with the linear matrix inversion (LMI), having adaptive capabilities, assures the safe and robust flight performance in the existence of external disturbances as well as sensor faults.

The main purpose and contribution of this study is to propose an adaptive fault-tolerant control (FTC) strategy, which assures rapid, accurate tracking and fast convergence in the case of the existence of any unknown fault or disturbances with minimum error. In this study, the proposed control approach, irrespective of the initial conditions, guarantees strong and quick convergence within the specified time compared to strategies presented in^40,41. This concept specifically originated from the need for accurate, robust, and rapid tracking and convergence in UAV applications, specifically in the existence of sensor faults, where delays can cause the degradation of the system performance, which may lead to critical damage. Under such faulty circumstances, the slow response of the system may cause the degradation of the control precision and accuracy, tracking error enhancement, and may cause instability, eventually compromising the efficiency and effectiveness of the FTC (fault-tolerant control) methodology. The concept of the adaptive stability approach assures the system approaches the desired trajectory within the specified upper-bound, which makes it suitable for dynamic environments and safety-critical applications, where abrupt and robust performance is desired. In literature, various approaches use the complex active estimators to manage the sensor faults, due to which their computational requirements are also enhanced⁴², which are difficult to employ³⁰. Whereas the designed proposed adaptive controller is simple, requires reduced computational power, and is easy to employ practically without compromising the efficiency, effectiveness, and robustness. The following are the main contributions of the proposed research study:

• The employment of the linear matrix inversion (LMI) in conjunction with the Fuzzy Takagi-Sugeno (T-S) system, enabling efficient and effective rejection and tolerance of the external unknown disturbances and sensor faults, increases the system’s adaptability under various complex conditions.

• The adaptive and robust stability concept is developed, ensuring that the system approaches the desired state with rapid response.

• The adaptive strategy ensures the fault-tolerance capabilities, managing the unknown sensor faults and external disturbances.

• In the existence of unknown or undefined faults and disturbances, the faulty system’s stability is ensured.

• The fault-managing efficiency provides robust resilience with the requirements of low computational power, making it suitable for practical incorporation in UAVs.

Remarks

In this research study, the adaptive strategy fault-tolerant control (FTC) is presented by using the Fuzzy T-S (Takagi-Sugeno) model and linear matrix inversion (LMI) having the capability to manage any type of unknown faults or disturbances with efficacy, efficiency and rapid response compared to the techniques used in this research and the techniques used in literature¹.

In the following, the remaining sections of this research study are defined as: in Sect. 2 system dynamical model is presented, the problem formulation is presented in Sect. 3, Sect. 4 presents modeling of Fuzzy T-S (Takagi-Sugeno) and controller design based on linear matrix inversion (LMI), the simulation results are presented in Sect. 5 and the Sect. 6 presents the conclusion of this research study.

System dynamical model

The system dynamical model is introduced in this section, along with two groups of dynamics. The system’s body dynamics are introduced in the first portion, while the motor dynamics are introduced in the second.

Quadrotor body dynamics

In this part, the quadrotor nonlinear body dynamics are presented, which are the essential part for designing an efficient and effective fault-tolerant control (FTC) strategy.

1

2

3

The letters stand for the quadrotor’s roll, pitch, and yaw rates [rad/sec]s along the body axes. The are moment of inertia, while the roll, pitch, and yaw angles are represented by Euler angles [rad or deg] respectively. The represents the roll, pitch, and yaw moments, and the velocities [m/sec] along the axes, and F represents the motor’s thrust force.

Quadrotor motor dynamics

The quadrotor is made up of four rotors that are coupled so that each pair spins in the opposite direction to counteract the torque effect and are aligned across each other as shown in Fig. 1. Roll, pitch, yaw, lift force, and other required moments are produced by variations in the rotor speed. Inflow ratio, rotor speed,, angle of attack, propeller airfoil, and other factors all affect the aerodynamic forces and moments generated by rotors. This article describes how the thrust force. and the torque relate to the angular velocity ⁴²:

4

Fig. 1.

Quadrotor Free Body Diagram.

5

The represents the thrust force anti-torque coefficients respectively, and denote four rotors. The Eq. (5) makes it evident that the lift force and moments depend on the rotor speed; as a result, we get.

6

7

8

9

The symbol indicates the length of the arm of the lever. The following is a matrix representation of Eqs. (6–9):

10

Problem Formulation

The nonlinear system in the presence of disturbances and faults is described as follows:

11

The input, output, and state vectors are used to indicate . The symbol shows the nonlinear functions that are known. are the system uncertaintiesand un-modeled dynamics,illustrates uncertainties and sensor noises, represents the actuator’s unknown fault and represents the sensor’s unknown fault. If the term Ax is subtracted and added on both sides of Eq. (11), where the matrix A is deliberated to be Hurwitz, then Eq. (11) is defined as;

12

Where . To create a fault detection (FD) system that is efficient, accurate, and quick, several presumptions are made:

Assumption 1

The key gyro rate variables () are guaranteed to be available for fault discovery and approximation.

Assumption 2

It is intended that the system state variables x(t) be quantifiable.

Assumption 3

The system uncertainties and sensors are assumed to be uniformly bounded;

13

Assumption 4

The maximum magnitude of the unknown sensor faults limits them.

Assumption 5

Since the nominal closed-loop system is stable, the asymptotic stability of the system should be guaranteed by some control rule.

Modeling of Fuzzy T-S (Takagi-Sugeno) and Controller Design based on Linear Matrix Inversion (LMI)

Linear matrix inversion (LMI) is considered one of the most powerful control techniques in the control systems field. Various synthesis and control theory problems can be simplified into their equivalent LMI problem. Fuzzy T-S is the most effective strategy in the field of control theory to simulate and analyze system nonlinearities. It is established that the fuzzy T-S model with a linear back portion may simulate any real plant with arbitrary precision while fully utilizing the fuzzy rules derived from the system’s local data and expert control experience. The Fuzzy T-S system stability condition is depicted in the linear matrix inversion (LMI) form. The goal of this study is to develop a fault-tolerant control (FTC) approach based on the T-S fuzzy model that is resilient, stable, and adaptive in order to guarantee the quadrotor UAV’s performance and safety.

The fuzzy T-S type model’s rules for continuous-time nonlinear systems with n outputs and m inputs can be represented as follows:

14

Then.

15

Where is the state, is the membership function, is the state vector, , is the control input vector, , ,

Referring to the defuzzification rule of the fuzzy system, the fuzzy model’s total output is defined as;

16

Where represents the membership function for the fuzzy rule i, , for example, there are four rules, suppose that represents rule 1, then we have .

For each rule of the fuzzy T-S model, the state feedback approach is utilized to design r types of fuzzy controller rules defined as;

17

Then.

18

Sugeno, etc⁴³. proposed the PDC (Parallel distributed compensation) method, which, according to the model, is regarded as a form of fuzzy controller; the following provides the theorem and proof of the PDC. PDC is used as the controller to handle the nonlinear systems control problem using fuzzy T-S modeling⁴⁴.

Theorem 1

Suppose the discrete system is defined as.

19

Where is a function vector with the property that.

Assume that a scalar function is continuous in such a way.

a)

b)

c)

d)

Consequently, the Lyapunov function is the equilibrium state for all t, and it is asymptotically stable in the large.

Lemma 1

If P is a matrix that is positive and definite, then

20

Where therefore.

21

Proof

22

The Lemma’s conclusion follows as a result.

Theorem 2

If there is a common positive definite matrix P for each of the subsystems, then the equilibrium state of the fuzzy system as described in (16) is globally asymptotically stable.

23

Proof

Consider the scalar function such that.

24

In this case, P represents a positive definite matrix. The following characteristics of this function are fulfilled:

a)

b)

c)

Next,

25

Where.

The is a Lyapunov function according to Theorem 1, and the fuzzy system in Eq. (16) is globally asymptotically stable.

Considering the fuzzy rules in Eq. (17) and the system in Eq. (14), the controller T-S fuzzy is designed by utilizing the PDC approach⁴⁵;

26

Linear matrix inversion (LMI) design and analysis

Based on Theorem 3 given below, Q is a positive definite matrix. The fuzzy T-S system given in Eq. (14) is considered to be asymptotically stable if the following conditions are fulfilled:

27

28

29

Based on the Eq. (28), the used in controller used (26) is obtained by using the linear matrix inversion (LMI). Theorem 5.1, given in⁴⁶, the complete proof of the theorem is presented below:

Proof

The following defines the candidate Lyapunov function:

30

Where P is a symmetric, positive matrix. Consequently.

Submitting the controller given in Eq. (26) in the above Eqn.

31

As a result, the following inequality is satisfied:

32

As, where . Considering that , therefore.

33

Where .

If the subsequent inequality is fulfilled.

34

Then, we have .

From Eq. (34), if then , according to LaSalle’s invariance principle, when , as given below:

Theorem 3

Suppose the continuous autonomous system is such that.

35

Suppose that is a positively invariant and compact set. Suppose is a continuous function that is differentiable such that.

36

Define the set.

37

Suppose that is the largest invariant set in E. For any solution that starts with fulfilling:

38

Proof

As for all, the function is bounded and non-decreasing. Therefore, for some as .

Illustrate the limit of the omega set for the trajectory, defined as:

39

is invariant, compact, and nonempty.

, as the is positively invariant and compact.

For any , by the continuity of v, and therefore

Hence,

Therefore, it is invariant and contained in E, and it must be contained in M. Therefore,

40

Linear Matrix Inversion Transformation

In this section, the transformation of the linear matrix inversion is presented.

Suppose that , , such that , where Q is a positive and symmetric matrix, suppose that , therefore

41

By multiplying the Eq. (40) with on both sides, we get.

42

Therefore.

43

And.

44

Consider that , . Suppose that and , therefore.

45

Multiplying the above inequality in Eq. (45) with on both sides, and consider that we get.

46

That is.

47

Then.

48

That is.

49

Strategy of fuzzy rules

In this section, the rules adopted for the adaptive control strategy are presented:

Rule 1

IF states are about O, THEN

Rule 2

IF states are about, THEN.

Rule 3

IF states are about, THEN.

Rule 4

IF states are about, THEN.

Rule 5

IF states are about, THEN.

Similarly, the control rules are designed as:

Rule 1

IF states are about o, THEN

Rule 2

IF states are about, THEN.

Rule 3

IF states are about, THEN.

Rule 4

IF states are about, THEN.

Rule 5

IF states are about, THEN.

Based on the fuzzy rules, the membership function is also designed. The Triangular membership is utilized to fuzzify the as depicted in Fig. 2, given below:

Fig. 2.

Membership Function used for Fuzzification.

Figure 3 shows the anticipated adaptive control strategy established on the Fuzzy T-S model and linear matrix inversion (LMI) for the aim of fault tolerance control (FTC):

Fig. 3.

Framework of the suggested adaptive controller approach.

Step 1

This stage establishes the starting trajectory that the system will adhere to.

Step 2

This step involves implementing the quadrotor unmanned aerial vehicle’s 6-DOF nonlinear dynamics.

Step 3

In this step, various fuzzy rules are defined and implemented, based on which the proposed controller is designed.

Step 4

In this step, based on the various rules defined in step 3, various state space models are computed.

Step 5

In this step, the defuzzification is defined.

Step 6

In this step, the linear matrix inversion (LMI) rules are defined.

Step 7

The controller gains are computed based on the covariance matrix obtained from linear matrix inversion (LMI).

Step 8

Computation of Fuzzy Controller based on PDC (Parallel Distributed Compensation).

Simulation results and discussion

In this part, the simulated results are presented. In order to prove the efficiency and efficacy of the anticipated strategy based on the Fuzzy T-S model in integration with linear matrix inversion (LMI), the results of the simulation are presented. For verifying the robustness and performance of the suggested approach, the dynamics of the quadrotor are used in the occurrence of unknown external disturbances and faults. The advantages and efficacy of the anticipated strategy are discussed and compared with the conventional Fuzzy T-S model, conventional linear matrix inversion (LMI), as well as with the approach used in¹. The proposed approach, based on the adaptive Fuzzy T-S and linear matrix inversion (LMI) is depicted in Fig. 4.

Fig. 4.

General structure of fault-tolerant control (FTC).

For the performance validation, effectiveness, efficacy, and analysis of the proposed strategy, unknown disturbances or faults are deliberated in the system’s nonlinear dynamics. The considered disturbance or faults used in this research are given below in Table 1:

Table 1.

Fault Classifications.

Time Period	Symbols	Fault Magnitude	Fault Category
			Cosine
and		0.06 and −0.06	Rectangular

In order to demonstrate the effectiveness of the anticipated approach, these faults or disturbances are considered to be unknown to the system.

The specifications of the system are presented in Table 2, given below:

Table 2.

Quadrotor Specifications.

Parameter	Description	Value
m	mass	0.19 kg
L	The arm lever’s Length	0.26 m
	Force Coefficient	3.14 × 10− 5
	Torque Coefficient	7.6 × 10− 7
	body x-axis Inertia	2.6 × 10− 4 kgm2
	body y-axis Inertia	2.41 × 10− 4 kgm2
	body z-axis Inertia	3.837 × 10− 4 kgm2

In order to demonstrate and demonstrate the efficacy and efficacy of the suggested adaptive controller compared to conventional techniques used in literature¹, the results are presented in two steps. The first step is comprised of the time duration from take-off to the 5 s into the flight, i.e. , during this time duration, the effect of the external disturbances and unknown faults is considered to be zero . The second step is comprised of various faults at random time durations, in which the effect of the disturbances and unknown faults is taken into account in the system’s nonlinear dynamics based on Table 1. To demonstrate the efficacy of the suggested approach, it is ensured that these faults might occur at random rather than at predetermined intervals. It is guaranteed that the system is unaware of any faults.

To prove the efficacy, usefulness, and performance of the anticipated adaptive controller, the performance of the conventional controllers is also simulated under the same conditions. These conventional controllers are conventional Fuzzy T-S and conventional linear matrix inversion (LMI). By comparing the results given in Figs. 5 and 6, it can be observed that during the flight time (), which is considered to be the disturbance free zone, the performance of the attitude (roll, pitch and yaw) in case of conventional Fuzzy T-S and the proposed adaptive controller are identical and has no oscillations, both in tracking performance and in time-course of the response, whereas the conventional linear matrix inversion (LMI) is showing oscillations. The adaptive controller and the conventional Fuzzy T-S rapidly track the desired state without any steady state error, whereas the conventional linear matrix inversion (LMI) does not track the desired state well and shows steady state error.

Fig. 5.

Comparison of adaptive control strategy performance with other strategies (Scenario 1).

Fig. 6.

Comparison of adaptive control strategy performance with other strategies (Scenario 2).

Scenario 1

In this step, the cosine fault is injected into the system for the time duration.

After the time (), after applying the disturbance or faults in the roll, it is observable from Fig. 5, the anticipated control strategy shows significant performance compared to conventional Fuzzy T-S and conventional LMI control strategies, which show large oscillations. It is evident that the effectiveness of the conventional strategies is degraded with the passage of time, and the oscillations become larger and larger. In the case of conventional Fuzzy T-S, the system shows larger oscillations initially; however, the system converges to a stable phase with the passage of time. While in the case of conventional LMI, the system becomes unstable in the later stages, which shows that the conventional LMI has no capability of stabilizing the system in case of the occurrence of a disturbance of unknown faults. It is observable that the anticipated adaptive controller is showing slight oscillations after the occurrence of disturbance or unknown faults, and it is observable that it can stabilize the system within 1s, which shows that the anticipated adaptive control approach has robust fault-tolerance capabilities compared to other conventional control strategies and the control strategy adopted in literature¹.

Similarly, in the case of applying disturbance or faults in pitch angle shown in Fig. 4, it can be seen that the conventional LMI and Fuzzy T-S exhibit much higher oscillations compared to the proposed adaptive controller. The convergence time of conventional LMI and Fuzzy T-S is also much higher, or slow convergence response, while the adaptive controller is showing a much faster response in convergence after the existence of the faults or disturbance. By comparing the responses of these three controllers and the controller used in literature, it has been proven that the adaptive controller is the most efficient and robust controller for any kind of disturbances and unknown faults.

In case of applying disturbance or faults in the yaw angle given in Fig. 5, it can be observed that the performance of the conventional Fuzzy T-S is much degraded compared to the conventional LMI and proposed adaptive controller. The conventional Fuzzy T-S has larger oscillations as well as much higher convergence time or slow response in achieving the equilibrium position. The conventional LMI is also showing oscillations and a slow convergence response. The proposed adaptive controller has no significant oscillations and also shows a much faster response in stabilizing the system compared to conventional techniques and the approach used in literature¹.

Scenario 2

In this step, the rectangular fault is injected into the system in the time duration of and.

In Fig. 6, the results are presented based on the injection of the rectangular fault. By observing the result of roll rate sensor during the time duration of , it is observable that anticipated strategy has the negligible deviation from the actual path and correspondingly during the time duration of the performance of the conventional Fuzzy T-S and the conventional LMI are not reasonable and having substantially higher deviations in the presence of the occurrence of the fault as well as having much higher error in tracing the desired path, while the performance of the proposed adaptive controller is much effective, precise and efficient in tracking the path even in the presence of the unknown fault.

Similarly, in the case of the pitch rate sensor and yaw rate sensor during the time duration of of and, by comparing the results it can be seen that the performance of the anticipated adaptive controller is much effective in fault-tolerance control (FTC) compared to conventional techniques and the method used in¹, consequently, it is determined that the suggested adaptive controller is the most precise, effective and robust technique for managing the unknown faults.

In the following Table 3, the numerical results comparison is presented based on both scenarios, it can be observed for the scenario 1 that in case of roll angle, the conventional LMI reaches up to , the conventional Fuzzy T-S reaches up to , while the proposed adaptive controller has maximum deviation of , which is lowest deviation compared to conventional techniques and the approach used in literature¹. In case of pitch angle, the conventional LMI has a deviation of , the Fuzzy T-S has a deviation, and the adaptive controller has a maximum deviation of . Similarly, in the case of yaw angle, the maximum deviation is based on the Fuzzy T-S controller, and the minimum deviation is . Similarly, by comparing the results for scenario 2, it is observable that the anticipated adaptive control approach has the lowest deviation of , and for roll, pitch, and yaw angle, respectively, compared to conventional Fuzzy T-S, conventional LMI, and the techniques used in the literature¹. Therefore, by comparing all these mathematical results, it has been proved that the proposed adaptive controller shows the minimum deviations in all cases in the existence of disturbances and faults, which proves the efficacy and efficiency of the proposed adaptive controller.

Table 3.

Control strategies performance comparison.

Controller	Scenario 1			Scenario 2
	Roll ()	Pitch ()	Yaw ()	Roll ()	Pitch ()	Yaw ()
Proposed Adaptive Controller	0.02	0.029	0.019	0.023	0.032	0.014
Conventional Fuzzy T-S	0.585	0.143	0.356	0.428	0.206	0.369
Conventional LMI	0.666	0.262	0.113	0.964	0.799	0.560

In the subsequent section, the control strategies are compared based on the controllers for scenario 1 and scenario 2, given in Figs. 7 and 8. Based on the comparison of controller performance, it is observable that the anticipated adaptive controller is showing negligible oscillations in the occurrence of the faults and disturbances compared to both conventional LMI and Fuzz T-S strategies, and the strategy adopted in¹. The conventional strategies are showing much higher oscillations in the occurrence of the faults or disturbances. Results prove that the proposed adaptive controller recovers the steady state abruptly after the occurrence of the faults or disturbances. It can be observed that the adaptive controller compensates for the effects immediately, which proves the efficiency and efficacy of the adaptive strategy.

Fig. 7.

Comparison of adaptive control strategy performance with other strategies based on U₁, U_2, and U₃ Controller (Scenario 1).

Fig. 8.

Comparison of adaptive control strategy performance with other strategies based on U₁, U_2, and U₃ Controller (Scenario 2).

Remarks

The simulation results exhibit that all three tested techniques showed equally significant performance in the absence of any external disturbances or faults, and they showed good tracking and rapid response. The performance differences appeared after applying the disturbance or faults to the system. The conventional strategies, such as linear matrix inversion (LMI) and Fuzzy T-S, showed degraded performance and did not tackle these perturbations, resulting in much larger oscillations in pitch, roll, and yaw. The proposed adaptive controller exhibited the best performance in tackling the external perturbations and disturbances. The overall best performance was shown by the proposed adaptive controller, which guaranteed the abrupt adaptation, showing rapid and robust response, and ensured the stability of the system.

Conclusion

In this research study, a hybrid adaptive control strategy based on the integration of the Fuzzy T-S model and linear matrix inversion (LMI) is introduced to enhance the tracking performance of the attitude of the quadrotor as well as to manage the unknown disturbances and faults. It has been shown that the adaptive controller strategy expressively improves the current state-of-the-art. It has been proven that the proposed approach is capable of handling any type of unknown disturbances and faults of any unknown duration and magnitude. It has been made possible to compensate for the variations in the duration and magnitude of the faults or disturbances by using the adaptive controller, which adjusts itself according to the environment. Numerical simulation results are presented, and the effectiveness, efficiency, and efficacy of the proposed strategy are proven compared to conventional techniques and the techniques existing in the literature¹. The proposed approach has been validated based on various unknown faults or disturbances, and it is proposed that it is the best approach for fault-tolerance. Consequently, it is suggested that the proposed strategy be integrated into nonlinear systems for improving the system performance as well as the safety level.

Future work

The suggested approach will be practically employed on a quadrotor unmanned aerial vehicle in order to validate the proposed fault-tolerant control (FTC) strategy experimentally. The size, weight, and cost play an important role in the design of unmanned aerial vehicles; the hardware redundancy is not encouraged for practical implementation. Therefore, the analytical redundancy is proposed as the alternative solution for the implementation of the proposed strategy. The analytical redundant approaches are based on the mathematical model of the system; therefore, such methods are called model-based fault-tolerant control methods. From the research, it is evident that the analytical techniques can detect any type of unknown faults and disturbances.

Author contributions

Conceptualization, M.T. and X.L.; methodology, S.B.; software, C.S.; validation, U.H.S., I.H., and H.W.; formal analysis, M.T. and X.L.; investigation, Z.Z.; resources, X.L. and I.H.; data curation, M.T.J., Z.Z., and H.W.; writing-original draft preparation, M.T.; writing-review and editing and funding acquisition, I.H. and U.H.S. All authors have read and agreed to the current version of the manuscript.

Funding

This work was supported by Khalifa University in collaboration with Dubai Future Labs through the Project "Underwater Maritime Monitoring" under Award 8475000016 and Award RC1-2018-KUCARS-8474000136.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author(s) on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.