Adaptive Integral Sliding Mode Control with Chattering Elimination Considering the Actuator Faults and External Disturbances for Trajectory Tracking of 4Y Octocopter Aircraft

Abstract

This paper presents a control strategy for a 4Y octocopter aircraft that is influenced by multiple actuator faults and external disturbances. The approach relies on a disturbance observer, adaptive type-2 fuzzy sliding mode control scheme, and type-1 fuzzy inference system. The proposed control approach is distinct from other tactics for controlling unmanned aerial vehicles because it can simultaneously compensate for actuator faults and external disturbances. The suggested control technique incorporates adaptive control parameters in both continuous and discontinuous control components. This enables the production of appropriate control signals to manage actuator faults and parametric uncertainties without relying only on the robust discontinuous control approach of sliding mode control. Additionally, a type-1 fuzzy logic system is used to build a fuzzy hitting control law to eliminate the occurrence of chattering phenomena on the integral sliding mode control. In addition, in order to keep the discontinuous control gain in sliding mode control at a small value, a nonlinear disturbance observer is constructed and integrated to mitigate the influence of external disturbances. Moreover, stability analysis of the proposed control method using Lyapunov theory showcases its potential to uphold system tracking performance and minimize tracking errors under specified conditions. The simulation results demonstrate that the proposed control strategy can significantly reduce the chattering effect and provide accurate trajectory tracking in the presence of actuator faults. Furthermore, the efficacy of the recommended control strategy is shown by comparative simulation results of 4Y octocopter under different failing and uncertain settings.

1. Introduction

The control of quadrotors has garnered significant attention in both industrial and academic areas due to its extensive array of applications [1,2,3,4,5,6,7]. These days, quadrotors are frequently utilized in a variety of industries, including film and television aerial photography [8,9], forest fire control [10], rapid delivery [11,12], agricultural precision [13], and other related sectors. Additionally, quadrotors are becoming more widely used due to their advantageous attributes such as simple operation, VTOL (vertical take-off and landing), and flexible movement [14].

Unmanned aerial vehicles (UAVs) have garnered growing attention from industry and academia owing to their appropriateness for monotonous, unclean, hazardous, and intricate tasks, such as infrastructure inspection and surveillance. In addition, the increasing complexity of UAVs has garnered increased attention from academics and industry, leading to heightened concerns regarding their reliability and security [15].

UAVs need more autonomy, more efficiency, and security measures to carry out their assigned jobs, which is a growing worry. The coupled dynamics of this particular vehicle give rise to many intriguing and challenging control problems, and drones are widely used in dangerous and complex work situations [15]. Ensuring the sufficient level of security and reliability of the system to enable indoor flight for the quadrotor while mitigating the potential for significant damage in the event of a collision constitutes a major obstacle to implementing such technologies [16].

The next generation of UAVs must incorporate enhanced levels of autonomy, efficiency, safety, and security measures to effectively fulfill their designated tasks. Several entities are conducting extensive research in quadrotor development, engaging in modeling, designing, and implementing these vehicles [17].

The field of quadrotor development is undergoing extensive study, with several entities actively engaged in modeling, designing, and implementing control laws for these vehicles [16]. The flight control system of the quadrotor serves as the central processing unit of the UAV. It plays a crucial role in ensuring flight operations’ secure, effective, and precise completion. Significant steps have been taken towards autonomous flight; UAVs are expected to be controlled using advanced algorithms that can monitor the health state of UAVs and initiate appropriate measures as necessary [18,19].

In recent years, there have been numerous efforts to improve the control capabilities of controller approaches of quadrotor UAVs; various linear and nonlinear control approaches have been proposed in the literature to regulate and stabilize the movement of quadrotors. Proportional-integral-derivative (PID) controls law [20,21,22] and the linear–quadratic regulator (LQR) algorithm [23] have been commonly employed to introduce traditional linear control techniques. While the standard PID controller is known for its straightforward design, its performance is hindered while dealing with disturbances and uncertainties in the quadrotor attitude system [22]; they cannot provide complete stability of the control system in the presence of external disruptions or uncertainties.

Various control strategies have been suggested to improve the robustness of quadrotor controllers against uncertainty in the model and outside disturbances. Ref. [24] detailed advancements in utilizing sliding mode control (SMC), which helps ensure stability in the position control of a quadrotor subsystem that is fully actuated. In Ref. [25], a quadrotor was equipped with an SMC attitude controller. The system enabled the calculation of a continuous control resistant to uncertainties and external disturbances in the model without the need for significant control gain.

SMC is extensively employed in many domains because of its fast reaction, high robustness, and straightforward implementation [22]. However, the conventional sliding mode approach has drawbacks, including the issue of chattering, the need for more robustness during the attainment stage, and the necessity for awareness of the maximum level of disturbances and uncertainties [26]. Various techniques have been employed to address the issue of chattering in the quadrotor, including the utilization of the saturation function technique [27], the implementation of continuous method SMC [28], and the application of the sliding mode at high order [29].

Second-order sliding mode control was presented in Refs. [30,31,32,33], backstepping and integral sliding mode control were presented in Refs. [34,35], and adaptive sliding mode control was explored in Refs. [36,37]. In Ref. [38], a second-order sliding mode control is developed for a quadrotor UAV with an imbalanced load. The objective is to enhance the resilience of the flight control systems, considering complex flying situations. The nonsingular terminal sliding mode control (NSTSMC) is a robust sliding mode control technique; as described in Ref. [39], this technique is particularly intriguing since it guarantees both the finite-time stability of the sliding surface and the convergence of system states to desired trajectories. In Ref. [40], a sliding mode control approach using backstepping, along with a cascade active perturbation rejection control, is developed. The purpose is to guarantee the desired tracking performance of a quadrotor drone, even in the presence of errors in the UAV’s model and external disturbances.

To ensure the quadrotor UAV’s safety and reliability, its control system must be designed to withstand actuator defects, external disturbances, and uncertainties in the model system. This should be done without compromising the control system’s functionality or posing any risks to its surroundings. The authors of [41] developed and implemented adaptive rules and control laws utilizing adaptive techniques. The researchers in Ref. [42] suggest a nonlinear adaptive fault-tolerant controller for a quadrotor UAV. This controller is based on an immersion and invariance observer and aims to address the reduced efficiency caused by actuator malfunction. The authors in Ref. [43] use a retrofit fault-tolerant monitoring control scheme with an adaptive fault absorber in a quadrotor UAV application to address actuator failures. An advanced adaptive fault-tolerant control system is developed for the attitude subsystem of a quadcopter. This system is designed to achieve the desired formation trajectory even in the presence of actuator failures. The issue of fault-tolerant control for quadrotors was tackled in Ref. [44] by implementing a backstepping control technique. The adaptive control rule used in Ref. [45] has been included in every level of the backstepping control architecture, enhancing the controller’s tolerance.

In the last ten years, the backstepping control method has been shown to be an effective way to deal with the underactuated issue in UAV control. The traditional backstepping design begins with the nonlinear coupling term being used as the virtual control input to stabilize the translational (outer-loop) tracking error dynamics. Subsequently, the inner-loop control is initiated to regulate the discrepancy between the coupling nonlinearity and the virtual controller to achieve convergence to zero [22]. The backstepping method is widely recognized for its efficiency in the group of robust control approaches for complicated engineering models, particularly for unmanned aerial vehicles, thanks to its advantageous recursive structure. While previous studies have addressed the tracking control problem in attitude stabilization and position control [46,47,48], the control performance is compromised in different working conditions due to the difficulty in accurately estimating uncertainties, external disturbances, and actuator faults. To address this issue, alternative extended backstepping controllers incorporating a disturbance observer and fault estimate have been utilized [49,50,51]. In Ref. [52], an adaptive backstepping control is developed for position control of a quadrotor, while an adaptive backstepping fast terminal sliding mode control is designed for attitude control. In Refs. [53,54,55], a passive fault-tolerant control (FTC) method for multi-rotor helicopters was introduced based on a backstepping control approach. Formulating an adaptive fuzzy backstepping control rule involves integrating a mass observer and a fuzzy control method with proposed backstepping control in a quadcopter drone’s trajectory control strategy [56].

The 4Y octocopter is an underactuated aerial vehicle, meaning it has fewer control inputs than degrees of freedom. It is a modified version of the quadrotor aircraft. Figure 1 displays the vehicle [16]. The system comprises a central framework designed to support a range of sensors and an autonomous piloting system. The object features four inwardly positioned arms, each joined to two outwardly positioned arms, creating a Y-shaped structure. A motor is placed at the end of each outboard arm. Every motor is equipped with a rotor that has a constant pitch, resulting in the generation of thrust. This design offers built-in hardware redundancy. In the event of a rotor failure, the octorotor can still maintain stability. Indeed, depending on the specific arrangement of malfunctioning rotors, it has the capacity to withstand as many as four rotor failures.

The complete failure of a rotor or propeller in systems with an inadequate number of rotors might result in the loss of controllability of the system. A potential way to address this issue is the implementation of redundant rotors inside the system. This approach’s primary advantage, albeit resulting in a more complex system design, is its capacity to endure several rotor failures in flight while maintaining complete controllability. Consequently, a more secure and dependable UAV may be achieved regarding steady flying and mission execution. The authors in Ref. [57] presented a reconfigurable hexarotor UAV to optimize performance in the event of complete rotor failure. They determined that maneuverability in the event of rotor failure is much superior to conventional hexarotor UAVs. Simple controllers such as a proportional–integral–derivative (PID) controller were proposed in Ref. [58] in order to give stable flight and task completion with even multiple rotor failures by using a number of redundant rotors. The authors in Ref. [59] examined the design of a unique hybrid-parallel, fixed-wing UAV equipped with VTOL capabilities. The objective of this concept is to create a system that integrates the hovering and maneuverability of a rotary-wing system with the enhanced speed, efficiency, and autonomy provided by fixed-wing alternatives. Additionally, several redundancies in the power and propulsion systems provide optimum problem mitigation and the high degree of dependability necessary for operation in perilous and extreme situations.

Using the discoveries given above, a technique for developing adaptive type-2 fuzzy fault-tolerant control is created. This approach employs sliding mode control and a type-2 fuzzy inference system. The objectives are to ensure the safety and reliability of the 4Y octocopter system, even in the event of actuator failures, and to maintain the system’s ability to accurately track its intended trajectory while minimizing the effects of external disturbances. Nevertheless, the 4Y octocopter input control may be plagued by the unwanted occurrence of chattering phenomena. To address this problem, type-1 fuzzy hitting control laws are integrated into the proposed control method, where stability in a closed loop has been demonstrated by utilizing Lyapunov’s theorem. The primary contributions of this study may be succinctly summarized as follows:

• (1). The presented approach of adaptive type-2 fuzzy fault-tolerant control has the capacity to adaptively generate control signals that can successfully adjust to both actuator failures and external disturbances. This is accomplished without requiring prior knowledge of defect information or perturbation bounds. In theory, the closed-loop system will always exhibit stability.

• (2). Contrary to the current fault-tolerant control methods for quadrotor UAVs in Refs. [60,61,62], which consider the combined impact of actuator problem and external disturbances as a single disturbance, leading to a cautious design approach, this paper addresses the issue of conservativeness by separately considering and accommodating actuator faults and external disturbances. The proposed adaptive type-2 fuzzy inference system schemes compensate for actuator faults, while a nonlinear disturbance observer is designed to compensate for external disturbances.

• (3). The integration of type-1 fuzzy hitting control laws and adaptive parameters in both the continuous and discontinuous control sections of sliding mode control reduces the discontinuous control gain to compensate for actuator defects, therefore alleviating unexpected control chattering. Furthermore, the use of estimated external disturbances by the disturbance observer allows for a further reduction in the size of the discontinuous control gain without depending only on the inherent robustness of sliding mode control.

This paper is arranged as follows: In Section 2, 4Y octocopter aircraft modeling and problem formulation are presented. The design of conventional integral sliding mode control is presented in Section 3. In Section 4, the background of the type-2 fuzzy logic inference system is synthesized. Type-2 fuzzy adaptive integral sliding mode control synthesis and stability analysis are then introduced in Section 5. Section 6 exposes the disturbance observer based on the type-2 fuzzy adaptive integral sliding mode control strategy. The simulation results are presented and analyzed in Section 7. Finally, the conclusions of the present paper are presented in Section 8.

2. 4Y Octocopter Aircraft Modeling and Problem Formulation

2.1. Modeling of 4Y Octocopter Aircraft

Figure 1a depicts a helicopter with 4Y octocopter that is an underactuated system, multivariable, tightly coupled, and highly nonlinear. The propellers are responsible for producing the primary forces and moments that are exerted on the helicopter.

The following are some of the assumptions that we will use in order to construct the dynamic model of the 4Y octocopter aircraft [16].

• -. The aircraft’s construction is sturdy and symmetrical.

• -. The propellers are rigid.

• -. Drag and thrust are proportional to propeller speed squared.

In Figure 1b, E(X_E, Y_E, Z_E) denotes an inertial frame, and B(X_B, Y_B, Z_B) denotes a firmly connected frame for the 4Y octocopter aircraft. The rotors are arranged in pairs as follows [16]:

$\mathrm{Pair} \mathrm{A} \mathrm{rotor} 1 \mathrm{with} \mathrm{rotor} 2 \left({\mathsf{\Omega }}_1={\mathsf{\Omega }}_2={\mathsf{\Omega }}_A\right)$

$\mathrm{Pair} \mathrm{B} \mathrm{rotor} 3 \mathrm{with} \mathrm{rotor} 4 \left({\mathsf{\Omega }}_3={\mathsf{\Omega }}_4={\mathsf{\Omega }}_B\right)$

$\mathrm{Pair} \mathrm{C} \mathrm{rotor} 5 \mathrm{with} \mathrm{rotor} 6 \left({\mathsf{\Omega }}_5={\mathsf{\Omega }}_6={\mathsf{\Omega }}_C\right)$

$\mathrm{Pair} \mathrm{D} \mathrm{rotor} 7 \mathrm{with} \mathrm{rotor} 8 \left({\mathsf{\Omega }}_7={\mathsf{\Omega }}_8={\mathsf{\Omega }}_D\right)$

By adjusting the speed of the pair B and pair D propellers, the opposite effect of roll rotation and lateral motion may be achieved. The pitch rotation and lateral motion are caused by the reversal of the speed of the propellers in pairs A and C. The difference in counter-torque between the two pairings B, D and A, C causes yaw rotation, which is more modest.

The Newton–Euler formalism yields the following 4Y octocopter aircraft dynamic model [16]:

(1)$\left\{\begin{array}{l}\ddot{z}={\displaystyle \frac{1}{m}}\{\left(cos\left(\phi \right) cos\left(\theta \right)\right)U_1-k_{ftz}\dot{z}\}-g\\ \ddot{x}={\displaystyle \frac{1}{m}}\{\left(cos \left(\phi \right) cos\left( \psi \right) sin\left(\theta \right)+sin \left(\psi \right) sin \left(\phi \right)\right) U_1-k_{ftx}\dot{x}\}\\ \ddot{y}={\displaystyle \frac{1}{m}}\{\left(cos \left(\phi \right) sin \left(\psi \right) sin \left(\theta \right)-cos \left(\psi \right) sin \left(\phi \right)\right) U_1-k_{fty}\dot{y}\}\\ \ddot{\phi }={\displaystyle \frac{1}{I_x}}[\dot{\theta } \dot{\psi }(I_y-I_z)+ U_4-k_{fax}{\dot{\phi }}^2]\\ \ddot{\theta }={\displaystyle \frac{1}{I_y}}[\dot{\phi } \dot{\psi }(I_z-I_x)+ U_5-k_{fay}{\dot{\theta }}^2]\\ \ddot{\psi }={\displaystyle \frac{1}{I_z}}[\dot{\phi } \dot{\theta }(I_x-I_y)+U_6-k_{faz}{\dot{\psi }}^2]\end{array}\right.,$

where $\left(x,y,z\right)$ are three positions, and $\left(\phi ,\psi ,\theta \right)$ are three Euler angles, denoting pitch, roll, and yaw, respectively. $K_{ftx}, K_{fty},K_{ftz}$ are the translation drag coefficients, $K_{fax}, K_{fay},K_{faz}$ denotes the friction aerodynamic coefficients, $\left(I_x,I_y,I_z\right)$ represents the aircraft inertia moments, the aircraft system’s overall mass is denoted by $m$, $l$ is the distance from the rotor, and the center of mass of the aircraft is computed as $l=L+l_1 \cos \left({\displaystyle \frac{\alpha }{2}}\right)$ (Figure 1a). l₁ is the length of the outboard arm, L is the length of the inboard arm, α is the angle between outboard, and $U={\left(U_1 U_4 U_5 U_6\right)}^T$ is the virtual control input vector defined as follows:

(2)$U=A K u,$

where: $A=\left[\begin{array}{cccccccc}1& 1& 1& 1& 1& 1& 1& 1\\ 0& 0& - l& - l& 0& 0& l& l\\ - l& - l& 0& 0& l& l& 0& 0\\ -{\displaystyle \frac{K_d}{K_p}}& -{\displaystyle \frac{K_d}{K_p}}& {\displaystyle \frac{K_d}{K_p}}& {\displaystyle \frac{K_d}{K_p}}& -{\displaystyle \frac{K_d}{K_p}}& -{\displaystyle \frac{K_d}{K_p}}& {\displaystyle \frac{K_d}{K_p}}& {\displaystyle \frac{K_d}{K_p}}\end{array}\right]$,

$u={\left(F_1 F_2 F_3 F_4 F_5 F_6 F_7 F_8\right)}^T$

$K=diag\left(k_1,k_2,k_3,k_4,k_5,k_6,k_7,k_8\right)$, $F_i (i=1\dots 8)$ are the forces generated by the 4Y octocopter obtained by $F_i=K_p {\mathsf{\Omega }}_i^2$, $K_p$ is the thrust factor, $K_d$ represents the drag factor, and $k_i (i=1\dots 8)$ stands for the ith motor’s control efficacy level.

2.2. Problem Description and System Design

Let us contemplate a nonlinear affine system that is perturbed by external disturbances, actuator defects, and model uncertainties [63].

(3)$\left\{\begin{array}{l}\dot{X}=f\left(X\right)+g\left(X\right) U+d\\ U=A K u\end{array}\right.,$

With $X\in {\Re }^n$ as the state vector, $u\in {\Re }^m$ represents the control input vector, $U\in {\Re }^n$ denotes the intermediate virtual control input vector, and $f\left(X\right)\in {\Re }^n$ and $g\left(X\right)\in {\Re }^n$ stand for nonlinear functions that include model uncertainty. The limits of these uncertainties are not known in advance. $d$ symbolizes perturbations and external disturbances that are unspecified, i.e., $d={\left({\mathsf{\Delta }}_1 {\mathsf{\Delta }}_2 {\mathsf{\Delta }}_3 {\mathsf{\Delta }}_4 {\mathsf{\Delta }}_5 {\mathsf{\Delta }}_6\right)}^T$, $A\in {\Re }^{n\times m}$ is the transformation matrix, $K=diag\left(k_1,k_2,\dots ,k_m\right)$ stands for the actuators’ control efficacy level, where the scalar $k_j$ $\left(j=1,2,\dots ,m\right)$ satisfies $0\le k_j\le 1$. If $k_j=1$, then the jth actuator is functioning properly. If it is not, then the jth actuator is experiencing some amount of malfunction, and, in the worst-case scenario, $k_j=1$, it means the jth actuator has failed entirely [63].

Assumption 1.

The disturbances considered in this work are bounded as $\left|{\mathsf{\Delta }}_j\right|\le D_j$, $\forall j=1,2,\dots 6$ in which, $D_j$ are unknown constants.

With respect to the 4Y octocopter aircraft’s fault-tolerant control, the state vector is denoted as follows:

(4)$\begin{array}{cc}X& ={\left( z \dot{z} x \dot{x} y \dot{y} \phi \dot{\phi } \theta \dot{\theta } \psi \dot{\psi }\right)}^T\hfill \\ & ={\left( x_{11} x_{21} x_{12} x_{22} x_{13} x_{23} x_{14} x_{24} x_{15} x_{25} x_{16} x_{26}\right)}^T,\hfill \end{array}$

Within the 4Y octocopter aircraft’s dynamic model, there are six outputs $\left( z x y \phi \theta \psi \right)$ and four inputs $\left(U_1 U_4 U_5 U_6\right)$. Because of this, the 4Y octocopter aircraft may be considered an underactuated system. We cannot possibly govern every state simultaneously [64]. The system may attain stable zero dynamics by stabilizing the roll and pitch angles $\left(\phi \theta \right)$ and combining the regulated outputs $\left( z x y \phi \theta \psi \right)$ to track the required locations. The current four control inputs of the 4Y octocopter aircraft need to be expanded to incorporate two more virtual control inputs $\left(U_2 U_3\right)$ to solve this issue. This will make it possible to handle each output separately. The link between pitch-x movement and roll-y movement is represented by the virtual control inputs $\left(U_2 U_3\right)$ [65].

(5)$U_2=\sin \left(x_{16}\right)\sin \left(x_{14}\right)+\cos \left(x_{16}\right)\sin \left(x_{15}\right)\cos \left(x_{14}\right),$

(6)$U_3=\sin \left(x_{16}\right)\sin \left(x_{15}\right)\cos \left(x_{14}\right)-\cos \left(x_{16}\right)\sin \left(x_{14}\right),$

By utilizing Equations (5) and (6), we obtain the desired roll and pitch angles, as follows:

(7)$\left\{\begin{array}{l}{\phi }_d=\arcsin \left(U_5 \sin \left(x_{16}\right)-U_6\cos \left(x_{16}\right) \right)\\ {\theta }_d=\arcsin \left({\displaystyle \frac{U_5 \cos \left(x_{16}\right)+U_6\sin \left(x_{16}\right)}{\cos \left({\phi }_d\right)}} \right)\end{array}\right.$

Virtual controls are interpreted as roll and pitch angles that dictate horizontal movement in the $x$ and $y$ axes. For precise horizontal movement control, set the rolling and tilting actions to the necessary angles $\left({\phi }_d {\theta }_d\right)$. Figure 2 shows the actuated 4Y octocopter aircraft with six inputs $\left(U_1 U_2 U_3 U_4 U_5 U_6\right)$, together with six outputs $\left( z x y \phi \theta \psi \right)$.

The 4Y octocopter aircraft’s nonlinear dynamic equations in Equation (1) may be resolved into state-space subsystems as follows:

(8)$\mathrm{z} \mathrm{position} \mathrm{subsystem} \left\{\begin{array}{l}{\dot{x}}_{11}=x_{21}\\ {\dot{x}}_{21}=f_1\left(X\right)+g_1\left(X\right) {\vartheta }_1+d_1\end{array}\right.,$

where: $f_1\left(X\right)={\displaystyle \frac{-k_{ftz}}{m}} x_{21}-g$, $g_1\left(X\right)={\displaystyle \frac{cos\left(x_{14}\right) cos\left(x_{15}\right)}{m}}$, ${\vartheta }_1 = U_1$ and $d_1={\mathsf{\Delta }}_1$

(9)$\mathrm{x} \mathrm{position} \mathrm{subsystem} \left\{\begin{array}{l}{\dot{x}}_{12}=x_{22}\\ {\dot{x}}_{22}=f_2\left(X\right)+g_2\left(X\right) {\vartheta }_2+d_2\end{array}\right.,$

where: $f_2\left(X\right)={\displaystyle \frac{-k_{ftx}}{m}} x_{22}$, $g_2\left(X\right)={\displaystyle \frac{U_1}{m}}$, ${\vartheta }_2 = U_2$ and $d_2={\mathsf{\Delta }}_2$

(10)$\mathrm{y} \mathrm{position} \mathrm{subsystem} \left\{\begin{array}{l}{\dot{x}}_{13}=x_{23}\\ {\dot{x}}_{23}=f_3\left(X\right)+g_3\left(X\right) {\vartheta }_3+d_3\end{array}\right.,$

where: $f_3\left(X\right)={\displaystyle \frac{-k_{fty}}{m}} x_{23}$, $g_3\left(X\right)={\displaystyle \frac{U_1}{m}}$, ${\vartheta }_3 = U_3$ and $d_3={\mathsf{\Delta }}_3$

(11)$\mathrm{Roll} \mathrm{subsystem} \left\{\begin{array}{l}{\dot{x}}_{14}=x_{24}\\ {\dot{x}}_{24}=f_4\left(X\right)+g_4\left(X\right) {\vartheta }_4+d_4\end{array}\right.,$

where: $f_4\left(X\right)=\left({\displaystyle \frac{I_y-I_z}{I_x}}\right) x_{25} x_{26}-\left({\displaystyle \frac{k_{fax}}{I_x}}\right) x_{24}^2$, $g_4\left(X\right)={\displaystyle \frac{1}{I_x}}$, ${\vartheta }_4 = U_4$ and $d_4={\mathsf{\Delta }}_4$

(12)$\mathrm{Pitch} \mathrm{subsystem} \left\{\begin{array}{l}{\dot{x}}_{15}=x_{25}\\ {\dot{x}}_{25}=f_5\left(X\right)+g_5\left(X\right) {\vartheta }_5+d_5\end{array}\right.,$

where: $f_5\left(X\right)=\left({\displaystyle \frac{I_z-I_x}{I_y}}\right) x_{24} x_{26}-\left({\displaystyle \frac{k_{fay}}{I_y}}\right) x_{25}^2$, $g_5\left(X\right)={\displaystyle \frac{1}{I_y}}$, ${\vartheta }_5 = U_5$ and $d_5={\mathsf{\Delta }}_5$

(13)$\mathrm{Yaw} \mathrm{subsystem} \left\{\begin{array}{l}{\dot{x}}_{16}=x_{26}\\ {\dot{x}}_{26}=f_6\left(X\right)+g_6\left(X\right) {\vartheta }_6+d_6\end{array}\right.,$

where $f_6\left(X\right)=\left({\displaystyle \frac{I_x-I_y}{I_z}}\right) x_{24} x_{25}-\left({\displaystyle \frac{k_{faz}}{I_z}}\right) x_{26}^2$, $g_6\left(X\right)={\displaystyle \frac{1}{I_z}}$, ${\vartheta }_6 = U_6$ and $d_6={\mathsf{\Delta }}_6$.

The unknown perturbations induced by the external environment are represented by $\left({\mathsf{\Delta }}_1,\dots ,{\mathsf{\Delta }}_6\right)$. Based on the subsystems above, the 4Y octocopter aircraft’s nonlinear system in state space can be rewritten as follows:

(14)$\left\{\begin{array}{l}{\dot{x}}_{1i}=x_{2i}\\ {\dot{x}}_{2i}=f_i\left(X\right)+g_i\left(X\right) {\vartheta }_i+d_i\end{array}\right.,$

where $i=1,2,\dots 6$ denotes each subsystem, $x_{1i}={\left( z x y \phi \theta \psi \right)}^T$, $x_{2i}={\left( \dot{z} \dot{x} \dot{y} \dot{\phi } \dot{\theta } \dot{\psi }\right)}^T$, and ${\vartheta }_i={\left({\vartheta }_1 {\vartheta }_2 {\vartheta }_3 {\vartheta }_4 {\vartheta }_5 {\vartheta }_6\right)}^T={\left(U_1 U_2 U_3 U_4 U_5 U_6\right)}^T$.

To ensure accurate trajectory tracking, the following assumptions have been made:

Assumption 2.

The reference signals $z^{*},x^{*}, y^{*},{\phi }^{*}, {\theta }^{*}$ and ${\psi }^{*}$, together with their first derivatives, are limited and exhibit continuity.

Assumption 3.

All three coordinates $\left(z,x,y\right)$ and all three angles $\left(\phi ,\theta ,\psi \right)$ may be measured.

Assumption 4.

In the state-space model (17), the nonlinear functions $f_i\left(X\right)$ and $g_i\left(X\right)$ remain unknown.

This work develops an adaptive fault-tolerant control technique for the studied 4Y octocopter aircraft to tolerate actuator defects and disturbances. Figure 3 illustrates the suggested control system. Two issues must be addressed for optimal control performance: Building a conventional integral sliding mode control to ensure system tracking performance under nominal conditions is the first task. The second is to investigate type-2 fuzzy adaptive control methods and build a disturbance observer using integral sliding mode control to adjust actuator defects and disturbances.

3. Design of Conventional Integral Sliding Mode Control

Sliding mode controllers are designed in two phases. A sliding surface must be built first to maintain system performance. Second, a control law is created to push the sliding variable to approach the desired surface and maintain its proximity.

For the reference signals $x_{1i}^{*}$, we may establish the following definitions for the tracking error variables and their temporal derivatives:

(15)${\tilde{x}}_{1i}=x_{1i}^{*}-x_{1i} \mathrm{for} i=1,2,\dots 6,$

(16)${\dot{\tilde{x}}}_{1i}={\dot{x}}_{1i}^{*}-{\dot{x}}_{1i} \mathrm{for} i=1,2,\dots 6$

The sliding surfaces can be defined as follows:

(17)$s_i={\dot{\tilde{x}}}_{1i}+{\lambda }_i {\tilde{x}}_{1i}+{\beta }_i {\displaystyle \underset{0}{\overset{t}{\int }}{\tilde{x}}_{1i}\left(\tau \right) d\tau },$

with

(18)${\left(x_{11}^{*} x_{12}^{*} x_{13}^{*} x_{14}^{*} x_{15}^{*} x_{16}^{*}\right)}^T={\left( z^{*} x^{*} y^{*} {\phi }^{*} {\theta }^{*} {\psi }^{*}\right)}^T,$

and we admit that

(19)$x_{14}^{*} ={\phi }^{*}= 0 ,$

(20)$x_{15}^{*} ={\theta }^{*}= 0 ,$

${\lambda }_i$ and ${\beta }_i$ are positive design parameters.

Once the sliding surface has been designed, the following stage is to create a suitable control law that will enhance its attractiveness. The control law is structured as follows to meet this requirement:

(21)${\vartheta }_i={\vartheta }_{i,eq}+\mathsf{\Delta }{\vartheta }_i,$

${\vartheta }_{i,eq}$ represents the equivalent control components, whereas $\mathsf{\Delta }{\vartheta }_i$ represents the discontinuous control components used to achieve the intended sliding movement.

The equivalent control parts ${\vartheta }_{i,eq}$ are constructed by setting ${\dot{s}}_i=0$, as shown in the following:

(22)${\dot{s}}_i={\ddot{\tilde{x}}}_{1i}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-x_{2i}\right)+{\beta }_i {\tilde{x}}_{1i}={\ddot{x}}_{1i}^{*}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-{\dot{x}}_{1i}\right)+{\beta }_i {\tilde{x}}_{1i}-{\dot{x}}_{2i},$

By replacing (14) into (22), the equivalent control inputs may be computed as follows:

(23)${\vartheta }_{i,eq}={\displaystyle \frac{1}{g_i\left(X\right)}} \left({\ddot{x}}_{1i}^{*}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-{\dot{x}}_{1i}\right)-f_i\left(X\right)+{\beta }_i {\tilde{x}}_{1i}\right),$

Subsequently, a discontinuous control component is created to counteract disturbances and ensure the required sliding movement is maintained while also keeping the sliding variable on the intended sliding surface. This control is synthesized as follows:

(24)$\mathsf{\Delta }{\vartheta }_i={\displaystyle \frac{1}{g_i\left(X\right)}}{k}_i sign\left(s_i\right),$

where $k_i$ are positive design parameters.

The global control laws are formed by combining the equivalent and discontinuous control components as follows:

(25)${\vartheta }_i={\displaystyle \frac{1}{g_i\left(X\right)}} \left({\ddot{x}}_{1i}^{*}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-{\dot{x}}_{1i}\right)-f_i\left(X\right)+{\beta }_i {\tilde{x}}_{1i}+k_i sign\left(s_i\right)\right),$

Theorem 1.

Assume nonlinear system in (14) with limited external disturbances. By using the global control laws in (25) and the designated sliding surfaces in (17), the intended sliding movement may be attained and sustained on the designated sliding surfaces, irrespective of disturbances. This is accomplished by selecting discontinuous control gains in (24) as follows: $k_i\ge D_d+{\eta }_i$.

Proof of Theorem 1.

Let us define the following Lyapunov candidate functions as follows:

(26) $V_1={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^6{s}_i^2},$

The temporal derivative of (26) is given as follows:

(27)${\dot{V}}_1={\displaystyle \sum _{i=1}^6{s}_i {\dot{s}}_i},$

Replacing (25) into (27) yields the following:

(28)$\begin{array}{l}{\dot{V}}_1={\displaystyle \sum _{i=1}^6\left[s_i\left({\ddot{x}}_{1i}^{*}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-{\dot{x}}_{1i}\right)+{\beta }_i {\tilde{x}}_{1i}-{\ddot{x}}_{1i}^{*}-f_i\left(X\right)-{\beta }_i {\tilde{x}}_{1i}-\right.\right.} \\ \left.\left.{\lambda }_i \left({\dot{x}}_{1i}^{*}-{\dot{x}}_{1i}\right)+f_i\left(X\right)-k_i sign\left(s_i\right)-d_i\right)\right]\\ ={\displaystyle \sum _{i=1}^6\left[s_i\left(-k_i sign\left(s_i\right)-d_i\right)\right]}\\ \le {\displaystyle \sum _{i=1}^6\left[\left(-\left[D_d+{\eta }_i\right]\left|s_i\right|- d_i s_i \right)\right]} <0,\end{array}$

Hence, by selecting a small positive value of ${\eta }_i$, the suggested global control laws in (25) ensure the stability of the system even in the face of external disturbances. □

Remark 1.

Sliding mode control may mitigate disturbance effects, but only if the discontinuous control gains $k_i$ are larger than the maximum disturbance. However, control chattering may occur if the discontinuous control gain is raised to ensure system stability as disturbances get stronger. In addition, the amount of chattering is directly proportional to the values of $k_i$.

The implementation of the control laws in (25) is ineffective due to the need for a precise model. An adaptive control technique using type-2 fuzzy systems is offered as a solution to these kinds of issues. The proposed methodology is based on the use of online estimation to determine the local nonlinearities of each subsystem. These nonlinearities are represented by the functions $g_i\left(X\right)$ and $f_i\left(X\right)$, for $i=1,\dots ,6$ are estimated using adaptive type-2 fuzzy systems and local state variables specific to each subsystem.

4. Background of the Type-2 Fuzzy Logic Inference System

Figure 4 illustrates the configuration of a type-2 fuzzy logic system. The system has resemblance to a type-1 fuzzy logic system. One notable distinction is in the inclusion of a type-2 fuzzy set, which necessitates the use of a type-reducer to transform the output of the fuzzy inference engine into a type-1 fuzzy set, referred to as a type-reduced set. The set with reduced types is subjected to defuzzification in order to achieve a crisp result. In this study, we focus on the analysis of an interval type-2 fuzzy logic system that employs singleton input fuzzification. The process of fuzzification is referred to as singleton fuzzification when the input fuzzy set consists of a single point with a degree of membership equal to 1. The use of interval type-2 fuzzy logic systems proves to be straightforward and efficient in streamlining the computing procedure for type reduction.

Suppose we have a type-2 Takagi–Sugeno–Kang fuzzy logic system with n inputs, $x=\left[x_1,\dots ,x_n\right]\in {\Re }^n$, and a single output, $y\in \Re$. In order to build the fuzzy rules, we suppose that the type-2 fuzzy system has N rules, where the Lth rule has the following form:

(29)$R^L: if x_1 is {\tilde{F}}_1^L and\dots and x_n is {\tilde{F}}_n^L Then y^L=C^L, L=1,\dots N,$

The antecedent linguistic phrases, denoted as ${\tilde{F}}_1^L$, ${\tilde{F}}_2^L$,…${\tilde{F}}_n^L$, are represented by interval type-2 Gaussian fuzzy sets, as seen in Figure 5. The output of the Lth rule, $R^L$, is denoted as $y^L$, whereas the consequent parameter, $C^L$, is an interval type-1 set.

In Figure 5, the uncertainty range of each membership function (MF) may be shown as a bounded interval using the upper MF ${\overline{\mu }}_{F_p^L}\left(x_p\right)$ and the lower MF ${\underset{\_}{\mu }}_{F_p^L}\left(x_p\right)$, where

(30)$\left\{\begin{array}{l}{\overline{\mu }}_{{\tilde{F}}_p^L}\left(x_p\right)=\exp \left\{-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{x_p-m_p}{{\sigma }_p}}\right)}^2\right\}\\ {\underset{\_}{\mu }}_{{\tilde{F}}_p^L}\left(x_p\right)=0.8 {\overline{\mu }}_{{\tilde{F}}_p^L}\left(x_p\right)\end{array}\right.,$

In this context, ${\sigma }_p$ and $m_p$ represent, respectively, the mean and standard deviation of the Gaussian primary MF of the type-2 fuzzy system ${\tilde{F}}_p^L$ in the Lth rule.

The inference engine maps the crisp inputs to interval type-2 fuzzy output sets by combining the fuzzy rules. It takes the input and the rules’ antecedents into account to determine the firing interval for each rule. Then, the subsequent fuzzy sets are trained using these firing levels. The Lth rule’s firing interval, $\left[{\underset{\_}{f}}^L,{\overline{f}}^L\right]$, is a type-1 interval set that is defined by the points ${\underset{\_}{f}}^L$ and ${\overline{f}}^L$, which are located on the left and right sides of the set, respectively, such that

(31)${\underset{\_}{f}}^L={\underset{\_}{\mu }}_{{\tilde{F}}_1^L}\left(x_1\right)\times {\underset{\_}{\mu }}_{{\tilde{F}}_2^L}\left(x_2\right)\times \dots \times {\underset{\_}{\mu }}_{{\tilde{F}}_n^L}\left(x_n\right),$

(32)${\overline{f}}^L={\overline{\mu }}_{{\tilde{F}}_1^L}\left(x_1\right)\times {\overline{\mu }}_{{\tilde{F}}_2^L}\left(x_2\right)\times \dots \times {\overline{\mu }}_{{\tilde{F}}_n^L}\left(x_n\right),$

${\underset{\_}{\mu }}_{{\tilde{F}}_p^L}\left(x_p\right)$ and ${\overline{\mu }}_{{\tilde{F}}_p^L}\left(x_p\right)$ denote the membership values of the lower and upper membership functions of the crisp input $x_p$ to the type-2 fuzzy system ${\tilde{F}}_p^L$ in the Lth rule.

The final result may be stated to be the same as [66], when type-2 Takagi–Sugeno–Kang rules are applied with interval type-2 fuzzy systems for the antecedents and interval type-1 fuzzy systems for the consequent sets.

(33)$Y=\left[y_l y_r\right]={\displaystyle \underset{{\theta }^1}{\int }\dots }{\displaystyle \underset{{\theta }^M}{\int }\dots {\displaystyle \underset{f^1}{\int }\dots {\displaystyle \underset{f^M}{\int }\times \frac{1}{\frac{{\displaystyle \sum _{L=1}^N{f}^L{\theta }^L}}{{\displaystyle \sum _{L=1}^N{f}^L}}}}}},$

Considering that the output Y is a type-1 interval set, it is sufficient to calculate its two endpoints, $y_l$ and $y_r$, using an expansion of fuzzy basis functions in the following manner:

(34)$y_l={\displaystyle \frac{{\displaystyle \sum _{L=1}^N{f}_l^L{\theta }^L}}{{\displaystyle \sum _{L=1}^N{f}_l^L}}}={\displaystyle \sum _{L=1}^N{W}_l^L\left(x\right) {\theta }^L}=W_l^T\left(x\right)\theta ,$

(35)$y_r={\displaystyle \frac{{\displaystyle \sum _{L=1}^N{f}_r^L{\theta }^L}}{{\displaystyle \sum _{L=1}^N{f}_r^L}}}={\displaystyle \sum _{L=1}^N{W}_r^L\left(x\right) {\theta }^L}=W_r^T\left(x\right)\theta ,$

where $f_l^L$ and $f_r^L$ are the firing strength membership grades that contribute to $y_l$ and $y_r$, respectively. Parameters ${\theta }^i$ represent type-1 centroids of consequent sets. Note that the centroid of a type-1 set $C_L$ with $Q$ discretized points is written as follows:

(36)$y_c={\displaystyle \frac{{\displaystyle \sum _{L=1}^Q{y}_L {\mu }_{C^L}\left(y_L\right)}}{{\displaystyle \sum _{L=1}^Q{\mu }_{C^L}\left(y_L\right)}}},$

The fuzzy basis functions can be obtained as follows:

(37)$W_l^L\left(x\right)={\displaystyle \frac{f_l^L}{{\displaystyle \sum _{L=1}^M{f}_l^L}}} \mathrm{and} W_l^L\left(x\right)={\displaystyle \frac{f_r^L}{{\displaystyle \sum _{L=1}^M{f}_r^L}}},$

Equation (37) denotes that the components of the left and right fuzzy basis functions vectors are determined as follows:

(38)$W_l^T\left(x\right)=\left[W_l^1\left(x\right),\dots ,W_l^N\left(x\right)\right],$

(39)$W_r^T\left(x\right)=\left[W_r^1\left(x\right),\dots ,W_r^N\left(x\right)\right],$

(40)$\theta =\left[{\theta }^1,\dots ,{\theta }^N\right],$

To calculate Y, it is necessary to calculate $y_l$ and $y_r$. This may be accomplished by implementing the iterative Algorithms 1 and 2 provided in Ref. [67]. Here, a concise description of the computational process is given. Firstly, we calculate the right-most point $y_r$.

For the sake of simplicity, let us suppose that the parameters ${\theta }^i$ are organized in ascending sequence, meaning that ${\theta }^1\le {\theta }^2\le \dots {\theta }^N$.

Algorithm 1. Karnik–Mendel algorithm for $y_r$ [67]

1. Calculate $y_r$ in (35) by first assigning a starting value $f_r^L={\displaystyle \frac{{\underset{\_}{f}}^L+{\overline{f}}^L}{2}}$ for $L=1,2,\dots N$where ${\underset{\_}{f}}^L$ and ${\overline{f}}^L$ have been computed using (31) and (32), respectively, and let ${y^{\prime }}_r=y_r$.2. Determine the value of $k$ (where $1\le k\le N-1$) such that ${\theta }^k\le {y^{\prime }}_r\le {\theta }^{k+1}$.3. Calculate $y_r$ in (35) where $f_r^L={\underset{\_}{f}}^L$ for $L\le k$ and $f_l^L={\overline{f}}^L$ for $L>k$; then, set ${y^{″}}_r=y_r$.4. If ${y^{″}}_r\ne {y^{\prime }}_r$, proceed to step 5. If ${y^{″}}_r={y^{\prime }}_r$, establish thereafter $y_r={y^{″}}_r$ and proceed to step 6. 5. Set ${y^{\prime }}_r={y^{″}}_r$ and proceed to step 2.6. End.

Algorithm 2. Karnik–Mendel algorithm for $y_l$ [67]

1. Calculate $y_l$ in (34) by first assigning a starting value $f_l^L={\displaystyle \frac{{\underset{\_}{f}}^L+{\overline{f}}^L}{2}}$ for $L=1,2,\dots N$ where ${\underset{\_}{f}}^L$ and ${\overline{f}}^L$ have been computed using (31) and (32), respectively, and let ${y^{\prime }}_l=y_l$.2. Determine the value of $k^{\prime }$ (where $1\le k^{\prime }\le N-1$) such that ${\theta }^{k^{\prime }}\le {y^{\prime }}_l\le {\theta }^{k^{\prime }+1}$.3. Calculate $y_l$ in (34) where $f_l^L={\overline{f}}^L$ for $L\le k^{\prime }$ and $f_l^L={\underset{\_}{f}}^L$ for $L>k^{\prime }$; then, set ${y^{″}}_l=y_l$.4. If ${y^{″}}_l\ne {y^{\prime }}_l$, proceed to step 5. If ${y^{″}}_l={y^{\prime }}_l$, establish thereafter $y_l={y^{″}}_l$ and proceed to step 6.5. Set ${y^{\prime }}_l={y^{″}}_l$ and proceed to step 2.6. End.

The average of $y_l$ and $y_r$ gives the type-2 fuzzy system’s crisp output. Therefore, the defuzzified crisp output is obtained as follows:

(41)$y={\displaystyle \frac{W_l^T\left(x\right)+W_r^T\left(x\right)}{2}}\theta =W^T\left(x\right)\theta ,$

5. Type-2 Fuzzy Adaptive Integral Sliding Mode Control Synthesis and Stability Analysis

The matrix K in (3) will not be an identity matrix due to the presence of actuator defects. Given the equation $\tilde{g}\left(X\right)A=g\left(X\right)A\left(I-K\right)$, it is possible to rephrase the system described in (14) as [63]:

(42)$\left\{\begin{array}{l}{\dot{x}}_{1i}=x_{2i}\\ {\dot{x}}_{2i}=f_i\left(X\right)+{\widehat{g}}_i\left(X\right) {\vartheta }_i+d_i\end{array}\right.,$

where ${\widehat{g}}_i\left(X\right)=g\left(X\right)-\tilde{g}\left(X\right)$.

The approximated value must be used instead of $g\left(X\right)$ to compensate for actuator faults. Since model uncertainties exist in the system, the approximated value ${\widehat{f}}_i\left(X\right)$ must be used to create the control law [63].

The suggested control technique utilizes four adaptive type-2 fuzzy inference systems to approximate the functions $f_i\left(X\right)$ and $g_i\left(X\right)$ separately. The approximation is done in the following manner:

(43)$f_i\left(X\right)=W_i^T \left(x_{ai}\right){\theta }_i^{*}+{\epsilon }_{f_i},$

(44)$g_i\left(X\right)=V_i^T \left(x_{ai}\right){\mathsf{\Gamma }}_i^{*}+{\epsilon }_{g_i},$

where ${\theta }_i^{*}$ and ${\mathsf{\Gamma }}_i^{*}$ are the ideal parameters, $W_i^T \left(x_{ai}\right)$ and $V_i^T \left(x_{ai}\right)$ are the mean basis functions obtained using type-2 fuzzy logic system where $x_{ai}={\left[x_{1i} x_{2i}\right]}^T$. Each basis function is computed as the average of the respective left and right basis functions. The adaptive type-2 fuzzy inference systems are considered to have limited approximation errors, denoted as follows:

$\left|{\epsilon }_{f_i}\right|\le {\epsilon }_{\alpha f}$ and $\left|{\epsilon }_{g_i}\right|\le {\epsilon }_{\alpha g}$. ${\epsilon }_{\alpha f}$ and ${\epsilon }_{\alpha g}$ are unknown positive parameters.

The selected input vectors for the used type-2 fuzzy systems are determined as follows:

(45)$\left\{\begin{array}{l}{x}_{a1}={\left[x_{11} x_{21}\right]}^T={\left[z \dot{z}\right]}^T, x_{a2}={\left[x_{12} x_{22}\right]}^T={\left[x \dot{x}\right]}^T\\ x_{a3}={\left[x_{13} x_{23}\right]}^T={\left[y \dot{y}\right]}^T, x_{a4}={\left[x_{14} x_{24}\right]}^T={\left[\phi \dot{\phi }\right]}^T\\ x_{a5}={\left[x_{15} x_{25}\right]}^T={\left[\theta \dot{\theta }\right]}^T, x_{a6}={\left[x_{16} x_{26}\right]}^T={\left[\psi \dot{\psi }\right]}^T\end{array}\right.,$

The conventional sliding mode controller causes chattering by oscillating its outputs at high frequencies. Unfortunately, this phenomenon may stimulate the system’s high-frequency dynamics. To avoid chattering, a type-1 fuzzy logic control (T1FLC) $\mathsf{\Delta } u_i$ approximates the discontinuous control.

(46)$\mathsf{\Delta } u_i=\mathrm{T}1\mathrm{FLC} \left(s_i\right),$

where $\mathsf{\Delta } u_i$ is the T1FLC output produced by sliding surfaces variables $s_i$, for $i=1,2,\dots 6$.

Figure 6 illustrates the fuzzy membership functions of the sets representing the input variables $s_i$ and the output discontinuous control $\mathsf{\Delta } u_i$.

In Figure 6, triangular membership functions are selected for all the membership functions of the fuzzy input variable. The labels assigned to the fuzzy variable surfaces $s_i$ are negative big (NB), negative small (NS), zero (ZE), positive small (PS), and positive big (PB). The discontinues control variables $\mathsf{\Delta } u_i$ depicted in Figure 6 are assigned fuzzy labels, which are membership functions that are split into five levels. These levels are represented by a set of linguistic variables: negative big (NB), negative small (NS), zero (ZE), and positive small (PS) and positive big (PB). Table 1 displays the rules basis, with five rules [68].

(47)$\begin{array}{cc}\mathsf{\Delta } u_i& =\mathrm{T}1\mathrm{FLC} \left(s_i\right)\hfill \\ & ={\displaystyle \frac{{\alpha }_{i 1} {\rho }_i+{\alpha }_{i 2} {\omega }_i+{\alpha }_{i 3} {\vartheta }_i-{\alpha }_{i 4} {\omega }_i-{\alpha }_{i 5} {\rho }_i}{{\displaystyle \sum _{p=1}^5{\alpha }_{i p}}}} i=1,\dots ,6\hfill \\ & ={\alpha }_{i 1} {\rho }_i+{\alpha }_{i 2} {\omega }_i+{\alpha }_{i 3} {\vartheta }_i-{\alpha }_{i 4} {\omega }_i-{\alpha }_{i 5} {\rho }_i\hfill \end{array}$

where $0\le {\alpha }_{i 1} \le 1$, $0\le {\alpha }_{i 2} \le 1$, $0\le {\alpha }_{i 4} \le 1$ and $0\le {\alpha }_{i 5} \le 1$ for $i=1,\dots ,6$ are the membership degree of rules 1, 2, 3, 4, and 5 presented in Table 1. ${\rho }_i$, $-{\rho }_i$, ${\vartheta }_i=0$, ${\omega }_i$ and $- {\omega }_i$ are the center of output membership functions $\mathsf{\Delta } u_i$ for $i=1,\dots ,6$. The triangle membership functions determine the validity of the relation ${\displaystyle \sum _{p=1}^5{\alpha }_{i p}} =1$ for $i=1,\dots ,6$. Further analysis of the fuzzy discontinues control $\mathsf{\Delta } u_i$ may be done using the following six criteria [68]. It is shown in Figure 6 that for each value of $s_i$, only one of these six circumstances will occur.

Condition 1.

Only rule 1 is enabled $\left(s_i>{\delta }_i, {\alpha }_{i 1} =1, {\alpha }_{i 2}= {\alpha }_{i 3}={\alpha }_{i 4}={\alpha }_{i 5}=0\right)$.

(48) $\mathsf{\Delta } u_i=\mathrm{T}1\mathrm{FLC} \left(s_i\right)= {\rho }_i i=1,\dots ,6,$

Condition 2.

Rule 1 and 2 are enabled concurrently $\left({\chi }_i<s_i\le {\delta }_i, 0<{\alpha }_{i 1}, {\alpha }_{i 2} \le 1, {\alpha }_{i 3}={\alpha }_{i 4}={\alpha }_{i 5} =0\right)$.

(49) $\mathsf{\Delta } u_i=\mathrm{T}1\mathrm{FLC} \left(s_i\right)={\alpha }_{i 1} {\rho }_i+{\alpha }_{i 2} {\omega }_i i=1,\dots ,6,$

Condition 3.

Rule 2 and 3 are enabled concurrently $\left(0<s_i\le {\chi }_i, 0<{\alpha }_{i 2}, {\alpha }_{i 3} \le 1, {\alpha }_{i 1} ={\alpha }_{i 4}={\alpha }_{i 5} =0\right)$.

(50) $\mathsf{\Delta } u_i=\mathrm{T}1\mathrm{FLC} \left(s_i\right)={\alpha }_{i 2} {\omega }_i i=1,\dots ,6,$

Condition 4.

Rule 3 and 4 are enabled concurrently $\left(-{\chi }_i<s_i\le 0, 0<{\alpha }_{i 3}, {\alpha }_{i 4} \le 1, {\alpha }_{i 1} ={\alpha }_{i 2}={\alpha }_{i 5} =0\right)$.

(51) $\mathsf{\Delta } u_i=\mathrm{T}1\mathrm{FLC} \left(s_i\right)=- {\alpha }_{i 4} {\omega }_i i=1,\dots ,6$

Condition 5.

Rule 4 and 5 are enabled concurrently $\left(- {\delta }_i<s_i\le -{\chi }_i, 0<{\alpha }_{i 4} , {\alpha }_{i 5} \le 1, {\alpha }_{i 1} ={\alpha }_{i 2}={\alpha }_{i 3} =0\right)$.

(52) $\mathsf{\Delta } u_i=\mathrm{T}1\mathrm{FLC} \left(s_i\right)=- {\alpha }_{i 4} {\omega }_i -{\alpha }_{i 5} {\rho }_i i=1,\dots ,6$

Condition 6.

Only rule 5 is enabled $\left(s_i<- {\delta }_i, {\alpha }_{i 5} =1, {\alpha }_{i 1}= {\alpha }_{i 2}={\alpha }_{i 3}={\alpha }_{i 4}=0\right)$.

(53) $\mathsf{\Delta } u_i=\mathrm{T}1\mathrm{FLC} \left(s_i\right)=- {\rho }_i i=1,\dots ,6$

Based on the four potential conditions presented in Equations (48)–(53)

(54)$\mathsf{\Delta } u_i=\mathrm{T}1\mathrm{FLC} \left(s_i\right)={\rho }_i\left({\alpha }_{i 1}-{\alpha }_{i 5}\right)+ {\omega }_i\left({\alpha }_{i 2}-{\alpha }_{i 4}\right) \mathrm{for} i=1,\dots ,6$

and it can be seen that

(55)$s_i \mathsf{\Delta } u_i=s_i \mathrm{T}1\mathrm{FLC} \left(s_i\right)=\left({\rho }_i\left|{\alpha }_{i 1}-{\alpha }_{i 5}\right|+ {\omega }_i\left|{\alpha }_{i 2}-{\alpha }_{i 4}\right|\right) \left|s_i\right| \mathrm{for} i=1,\dots ,6$

The adaptive type-2 fuzzy inference system is primarily used to approximate the nonlinear functions $f_i\left(X\right)$ and $g_i\left(X\right)$. These functions are denoted as follows:

(56)${\widehat{f}}_i\left(X\right)=W_i^T \left(x_{ai}\right){\theta }_i \mathrm{for} i=1,\dots ,6$

(57)${\widehat{g}}_i\left(X\right)=V_i^T \left(x_{ai}\right){\mathsf{\Gamma }}_i i=1,\dots ,6$

where ${\theta }_i$ and ${\mathsf{\Gamma }}_i$ are the adjusted vector parameters.

By using Equations (43), (44), (56), and (57), it can be shown that

(58)${\widehat{f}}_i\left(X\right)-f_i\left(X\right)=- W_i^T\left(x_{ai}\right){\tilde{\theta }}_i-{\epsilon }_{f_i}$

(59)${\widehat{g}}_i\left(X\right)-g_i\left(X\right)=- V_i^T\left(x_{ai}\right){\tilde{\mathsf{\Gamma }}}_i-{\epsilon }_{g_i}$

where ${\tilde{\theta }}_i={\theta }_i^{*}-{\theta }_i$ and ${\tilde{\mathsf{\Gamma }}}_i={\mathsf{\Gamma }}_i^{*}-{\mathsf{\Gamma }}_i$ are the parametric errors.

The feedback control law is ultimately formulated as follows:

(60)${\vartheta }_i={\displaystyle \frac{1}{{\widehat{g}}_i\left(X\right)}} \left({\ddot{x}}_{1i}^{*}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-{\dot{x}}_{1i}\right)-{\widehat{f}}_i\left(X\right)+{\beta }_i {\tilde{x}}_{1i}+\mathsf{\Delta } u_i\right) \mathrm{for} i=1,\dots ,6$

The following rules define the adjustment of type-2 fuzzy system parameters:

(61)$\left\{\begin{array}{l}{\dot{\theta }}_i=- a_{1i} s_i W_i\left(x_{ai}\right)\\ {\dot{\mathsf{\Gamma }}}_i=- a_{2i} s_i V_i\left(x_{ai}\right) {\vartheta }_i\end{array}\right.$

where $a_{1i}$ and $a_{2i}$ are positive tuning parameters.

Theorem 2.

Assume nonlinear system in (14) that includes actuator defects and external disturbances. By utilizing the control laws in (60) and updating them with the suggested adaptation laws in (61), the intended sliding movement may be achieved and sustained despite the occurrence of external disturbances and actuator faults. The discontinuous control given by the type-1 fuzzy inference system satisfied ${\rho }_i\left|{\alpha }_{i 1}-{\alpha }_{i 5}\right|+ {\omega }_i\left|{\alpha }_{i 2}-{\alpha }_{i 4}\right|\ge D_d+\left|{\epsilon }_{f_i}+ {\epsilon }_{g_i} {\vartheta }_i\right|+{\eta }_i$      for      $i=1,\dots ,6$.

Proof of Theorem 2.

Let us define the following Lyapunov candidate functions as:

(62) $V_2={\displaystyle \sum _{i=1}^6\left(\frac{1}{2}{s}_i^2+\frac{1}{ 2 a_{1i}}{\tilde{\theta }}_i^T {\tilde{\theta }}_i+\frac{1}{ 2 a_{2i}}{\tilde{\mathsf{\Gamma }}}_i^T {\tilde{\mathsf{\Gamma }}}_i\right)}$

The temporal derivative of the sliding surfaces in (17) may be calculated as follows:

(63)${\dot{s}}_i={\ddot{x}}_{1i}^{*}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-x_{2i}\right)+{\beta }_i {\tilde{x}}_{1i}-f_i\left(X\right)-g_i\left(X\right) {\vartheta }_i-d_i$

By performing addition and subtraction operations on the variables ${\widehat{g}}_i\left(X\right){\vartheta }_i$ in Equation (63), we obtain the following result:

(64)${\dot{s}}_i={\ddot{x}}_{1i}^{*}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-x_{2i}\right)+{\beta }_i {\tilde{x}}_{1i}-f_i\left(X\right)-{\widehat{g}}_i\left(X\right) {\vartheta }_i+\left\{{\widehat{g}}_i\left(X\right)-g_i\left(X\right)\right\} {\vartheta }_i-d_i$

By replacing the control laws ${\vartheta }_i$ in (60) into (64), we obtain the following:

${\dot{s}}_i={\ddot{x}}_{1i}^{*}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-x_{2i}\right)+{\beta }_i {\tilde{x}}_{1i}-f_i\left(X\right)- \left({\ddot{x}}_{1i}^{*}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-{\dot{x}}_{1i}\right)-{\widehat{f}}_i\left(X\right)+{\beta }_i {\tilde{x}}_{1i}+\mathsf{\Delta } u_i\right)+$

(65)$\begin{array}{l}\left\{{\widehat{g}}_i\left(X\right)-g_i\left(X\right)\right\} {\vartheta }_i-d_i\\ =\left[{\widehat{f}}_i\left(X\right)-f_i\left(X\right)\right]-\mathsf{\Delta } u_i+\left\{{\widehat{g}}_i\left(X\right)-g_i\left(X\right)\right\} {\vartheta }_i-d_i\end{array}$

Replacing $\mathsf{\Delta } u_i$ in (54) into (65) yields the following:

(66)${\dot{s}}_i=\left[{\widehat{f}}_i\left(X\right)-f_i\left(X\right)\right]-{\rho }_i\left({\alpha }_{i 1}-{\alpha }_{i 5}\right)- {\omega }_i\left({\alpha }_{i 2}-{\alpha }_{i 4}\right) +\left\{{\widehat{g}}_i\left(X\right)-g_i\left(X\right)\right\} {\vartheta }_i-d_i$

Substituting (58) and (59) into (66) yields the following:

(67)${\dot{s}}_i=\left[- W_i^T\left(x_{ai}\right){\tilde{\theta }}_i-{\epsilon }_{f_i}\right]-{\rho }_i\left({\alpha }_{i 1}-{\alpha }_{i 5}\right)- {\omega }_i\left({\alpha }_{i 2}-{\alpha }_{i 4}\right) +\left\{- V_i^T\left(x_{ai}\right){\tilde{\mathsf{\Gamma }}}_i-{\epsilon }_{g_i}\right\} {\vartheta }_i-d_i$

Then, the temporal derivative of the chosen Lyapunov candidate functions in (62) can be expressed as follows:

(68)${\dot{V}}_2={\displaystyle \sum _{i=1}^6\left(s_i {\dot{s}}_i +\frac{1}{a_{1i}}{\tilde{\theta }}_i^T {\dot{\tilde{\theta }}}_i+\frac{1}{ a_{2i}}{\tilde{\mathsf{\Gamma }}}_i^T {\dot{\tilde{\mathsf{\Gamma }}}}_i\right)}$

By substituting (67) into (68), we obtain the following:

(69)$\begin{array}{l}{\dot{V}}_2={\displaystyle \sum _{i=1}^6{s}_i \left(\left[- W_i^T\left(x_{ai}\right){\tilde{\theta }}_i-{\epsilon }_{f_i}\right]-{\rho }_i\left({\alpha }_{i 1}-{\alpha }_{i 5}\right)- {\omega }_i\left({\alpha }_{i 2}-{\alpha }_{i 4}\right) +\left\{- V_i^T\left(x_{ai}\right){\tilde{\mathsf{\Gamma }}}_i-{\epsilon }_{g_i}\right\} {\vartheta }_i-d_i\right) -}\\ {\displaystyle \frac{1}{a_{1i}}}{\tilde{\theta }}_i^T {\dot{\theta }}_i-{\displaystyle \frac{1}{ a_{2i}}}{\tilde{\mathsf{\Gamma }}}_i^T {\dot{\mathsf{\Gamma }}}_i\\ ={\displaystyle \sum _{i=1}^6{\tilde{\theta }}_i^T\left(- s_i W_i\left(x_{ai}\right)-\frac{1}{a_{1i}}{\dot{\theta }}_i\right)+{\tilde{\mathsf{\Gamma }}}_i^T \left(- s_i V_i\left(x_{ai}\right) {\vartheta }_i-\frac{1}{a_{2m}}{\dot{\mathsf{\Gamma }}}_i\right)+}\\ s_i\left(-{\epsilon }_{f_i}-{\rho }_i\left({\alpha }_{i 1}-{\alpha }_{i 5}\right)- {\omega }_i\left({\alpha }_{i 2}-{\alpha }_{i 4}\right)- {\epsilon }_{g_i}{\vartheta }_i-d_i\right)\end{array}$

By replacing (61) in (69), we obtain the following:

(70)${\dot{V}}_2={\displaystyle \sum _{i=1}^6{s}_i\left(-{\rho }_i\left({\alpha }_{i 1}-{\alpha }_{i 5}\right)- {\omega }_i\left({\alpha }_{i 2}-{\alpha }_{i 4}\right)-{\epsilon }_{f_i}- {\epsilon }_{g_i}{\vartheta }_i-d_i\right)}$

Using (55), (70) is rewritten as follows:

(71)${\dot{V}}_2={\displaystyle \sum _{i=1}^6-\left({\rho }_i\left|{\alpha }_{i 1}-{\alpha }_{i 5}\right|+ {\omega }_i\left|{\alpha }_{i 2}-{\alpha }_{i 4}\right|\right) \left|s_i\right|-\left({\epsilon }_{f_i}+ {\epsilon }_{g_i}{\vartheta }_i+d_i\right)} s_i$

Ultimately, it may be determined that ${\dot{V}}_2<0$ by satisfying the inequality ${\rho }_i\left|{\alpha }_{i 1}-{\alpha }_{i 5}\right|+ {\omega }_i\left|{\alpha }_{i 2}-{\alpha }_{i 4}\right|\ge D_d+\left|{\epsilon }_{f_i}+ {\epsilon }_{g_i} {\vartheta }_i\right|+{\eta }_i$. As a result, even in the event of actuator failures and external disturbances, the suggested control rule may preserve system performance. □

6. Disturbance Observer Based on Type-2 Fuzzy Adaptive Integral Sliding Mode Control Strategy

This section introduces the creation of a nonlinear disturbance observer into the previously established adaptive integral sliding mode control to adjust for actuator defects, model errors, and external disturbances.

Assumption 5.

The disturbance’s time derivative in (14) is bounded and fulfills $\underset{t\to \infty }{\lim } {\dot{d}}_i\left(t\right)=0$. In order to approximate the value of the unknown disturbances $d_i$, a nonlinear disturbance observer is formulated as per Refs. [12,69]:

(72) $\left\{\begin{array}{l}{\dot{\mu }}_i=-{\kappa }_i {\mu }_i-{\kappa }_i\left({\kappa }_i x_{2i}+f_i\left(X\right)+g_i\left(X\right){\vartheta }_i\right)\\ {\widehat{d}}_i={\mu }_i+{\kappa }_i x_{2i}\end{array}\right. \mathrm{for} i=1,\dots ,6$

The variables ${\mu }_i$ represent the internal state of the nonlinear observer, while ${\kappa }_i$ denotes the gain of the nonlinear disturbance observer.

The disturbances estimation errors can be described as the difference between the actual disturbances and the estimated disturbances, expressed as follows:

(73)${\tilde{d}}_i=d_i-{\widehat{d}}_i$

Subsequently, by combining (14), (72), and (73), the time derivative of the disturbance estimation error can be described as follows:

(74)$\begin{array}{cc}{\dot{\tilde{d}}}_i& ={\dot{d}}_i-{\dot{\widehat{d}}}_i\hfill \\ & =-{\dot{\mu }}_i-{\kappa }_i {\dot{x}}_{2i}\\ & ={\kappa }_i \left({\widehat{d}}_i -{\kappa }_i x_{2i}\right)+{\kappa }_i\left({\kappa }_i x_{2i} +f_i\left(X\right)+g_i\left(X\right){\vartheta }_i\right)-\\ & {\kappa }_i\left(f_i\left(X\right)+g_i\left(X\right){\vartheta }_i+d_i\right)\\ & =-{\kappa }_i {\tilde{d}}_i\end{array}$

By using the adaptive sliding mode control law obtained in (60), which does not take into account the unknown disturbances in order to produce the continuous control components, the composite control laws may be generated by including the estimated disturbances ${\widehat{d}}_i$, expressed as follows:

(75)${\vartheta }_i={\displaystyle \frac{1}{{\widehat{g}}_i\left(X\right)}} \left({\ddot{x}}_{1i}^{*}+{\lambda }_i \left({\dot{x}}_{1i}^{*}-{\dot{x}}_{1i}\right)-{\widehat{f}}_i\left(X\right)+{\beta }_i {\tilde{x}}_{1i}-{\widehat{d}}_i+\mathsf{\Delta } u_i\right) \mathrm{for} i=1,\dots ,6$

Theorem 3.

Consider a nonlinear system (27), including actuator defects and unknown disturbances. Using the feedback control laws in (75) and disturbance observer (72) adjusted by (56), (57), and (61), the desired sliding movement is maintained despite external disturbances and actuator faults. The discontinuous control given by the type-1 fuzzy inference system satisfied ${\rho }_i\left|{\alpha }_{i 1}-{\alpha }_{i 5}\right|+ {\omega }_i\left|{\alpha }_{i 2}-{\alpha }_{i 4}\right|\ge E_d+\left|{\epsilon }_{f_i}+ {\epsilon }_{g_i} {\vartheta }_i\right|+{\eta }_i$ for $i=1,\dots ,6$.

Proof of Theorem 3.

Let us define the following Lyapunov candidate functions as

(76) $V_3={\displaystyle \sum _{i=1}^6\left(\frac{1}{2}{s}_i^2+\frac{1}{2}{\tilde{\alpha }}_i^2+\frac{1}{ 2 a_{1i}}{\tilde{\theta }}_i^T {\tilde{\theta }}_i+\frac{1}{ 2 a_{2i}}{\tilde{\mathsf{\Gamma }}}_i^T {\tilde{\mathsf{\Gamma }}}_i\right)} \mathrm{for} i=1,\dots ,6$

Then, the temporal derivative of the chosen Lyapunov candidate functions in (76) can be expressed as follows:

(77)$$\begin{gathered} {\dot{V}}_3={\displaystyle \sum _{i=1}^6 \left(s_i {\dot{s}}_i +{\tilde{d}}_i {\dot{\tilde{d}}}_i +\frac{1}{a_{1i}}{\tilde{\theta }}_i^T {\dot{\tilde{\theta }}}_i+\frac{1}{ a_{2i}}{\dot{\tilde{\mathsf{\Gamma }}}}_i^T {\tilde{\mathsf{\Gamma }}}_i\right)} \\ ={\displaystyle \sum _{i=1}^6{s}_i \left(\left[- W_i^T\left(x_{ai}\right){\tilde{\theta }}_i-{\epsilon }_{f_i}\right]-{\rho }_i\left({\alpha }_{i 1}-{\alpha }_{i 5}\right)- {\omega }_i\left({\alpha }_{i 2}-{\alpha }_{i 4}\right)+\right. } \\ \left.\left\{ -V_m^T\left(x_{ai}\right){\tilde{\mathsf{\Gamma }}}_i -{\epsilon }_{g_i}\right\} {\vartheta }_i-{\tilde{d}}_i\right)-{\displaystyle \frac{1}{a_{1i}}}{\tilde{\theta }}_i^T {\dot{\theta }}_i-{\displaystyle \frac{1}{ a_{2i}}}{\tilde{\mathsf{\Gamma }}}_i^T {\dot{\mathsf{\Gamma }}}_i -{\kappa }_i {\tilde{d}}_i^2 \\ ={\displaystyle \sum _{i=1}^6{\tilde{\theta }}_i^T\left(- s_i W_i\left(x_{ai}\right)-\frac{1}{a_{1i}}{\dot{\theta }}_i\right)+{\tilde{\mathsf{\Gamma }}}_i \left(- s_i V_i\left(x_{ai}\right) {\vartheta }_i-\frac{1}{a_{2i}}{\dot{\mathsf{\Gamma }}}_i \right)+} \\ {s}_i\left(-{\epsilon }_{f_i}-{\rho }_i\left({\alpha }_{i 1}-{\alpha }_{i 5}\right)- {\omega }_i\left({\alpha }_{i 2}-{\alpha }_{i 4}\right)- {\epsilon }_{g_i}{\vartheta }_i-{\tilde{d}}_i \right)-{\kappa }_i {\tilde{d}}_i^2 \end{gathered}$$

By replacing (61) in (77), we obtain the following:

(78)$$\begin{gathered} {\dot{V}}_3={\displaystyle \sum _{i=1}^6{s}_i\left(-{\epsilon }_{f_i}-{\rho }_i\left({\alpha }_{i 1}-{\alpha }_{i 5}\right)- {\omega }_i\left({\alpha }_{i 2}-{\alpha }_{i 4}\right)- {\epsilon }_{g_i}{\vartheta }_i-{\tilde{d}}_i\right)-{\kappa }_i {\tilde{d}}_i^2} \\ ={\displaystyle \sum _{i=1}^6-\left({\rho }_i\left|{\alpha }_{i 1}-{\alpha }_{i 5}\right|+ {\omega }_i\left|{\alpha }_{i 2}-{\alpha }_{i 4}\right|\right) \left|s_i\right|-\left({\epsilon }_{f_i}+ {\epsilon }_{g_i}{\vartheta }_i+d_i\right)} s_i-{\kappa }_i {\tilde{d}}_i^2 \end{gathered}$$

Ultimately, it may be determined that ${\dot{V}}_3<0$ by satisfying the inequality ${\rho }_i\left|{\alpha }_{i 1}-{\alpha }_{i 5}\right|+ {\omega }_i\left|{\alpha }_{i 2}-{\alpha }_{i 4}\right|\ge E_d+\left|{\epsilon }_{f_i}+ {\epsilon }_{g_i} {\vartheta }_i\right|+{\eta }_i$, where $E_d$ is the upper bound of the disturbance estimation errors. As a result, even in the event of actuator failures and external disturbances, the suggested control law (75) may preserve system performance. The block diagram of the proposed control algorithm is presented in Figure 7. The suggested type-2 fuzzy adaptive integral sliding mode control approach may create suitable control signals to concurrently address actuator faults and model uncertainties without necessitating previous knowledge of fault information and uncertainty limits. To alleviate conservativeness, actuator faults, model uncertainties, and external disturbances are individually addressed. Actuator faults and model uncertainties are mitigated through the proposed type-2 fuzzy adaptive schemes, while external disturbances are countered by the implementation of a nonlinear disturbance observer. A type-1 fuzzy logic system is used to develop a fuzzy hitting control rule to prevent the development of chattering problems. □

7. Simulation Results and Discussions

This section presents a series of simulated case studies and tests to evaluate the efficacy and performance of the proposed type-2 fuzzy adaptive integral sliding mode control PT2FAISMC technique for 4Y octocopter aircraft. The objective is to assess its capability in handling actuator defects and external disturbances. The physical parameters of the 4Y octocopter aircraft under study are shown in Table 2.

In order to provide a fair comparison and showcase the benefits of the proposed control strategy, we will also examine the performance of a conventional integral sliding mode control (CISMC). The control parameters are carefully selected to ensure optimal performance, as depicted in Table 3. In addition, to optimize the computation time of the suggested control method, just three type-2 membership functions are used for each input variable in the fuzzy systems being studied.

To verify the efficacy of the suggested control technique, two scenarios are presented.

Scenario 1:

Step trajectory tracking for the 4Y octocopter aircraft in the presence of actuator faults and external disturbances.

In Scenario 1, simulations of the 4Y octocopter aircraft in a faulty situation have been carried out, where 30% loss of control effectiveness faults are injected to actuator#5, actuator#6, actuator#7, and actuator#8 at t = 50 s. Subsequently, a similar fault is intentionally introduced to actuator#1, actuator#2, actuator#3, and actuator#4 at t = 75 s. Furthermore, the equations of the 4Y octocopter aircraft model take into consideration the external disturbances. When the 4Y octocopter aircraft is flying outside, interruptions can occur due to wind gusts. Consequently, the disturbing expressions are considered at time t = 60 s as sinusoidal waves of different frequencies, given as [70]:

(79)$\left\{\begin{array}{l}{d}_1\left(t\right)=0.5 \cos \left(0.7 t\right)+0.7 \cos \left(0.3 t\right)\\ d_2\left(t\right)=- 0.08 \cos \left(0.1013 t-3.043\right)+0.1 \sin \left(0.448 t-13.464\right)+0.05\sin \left(1.5708 t-15 \pi \right)\\ d_3\left(t\right)=0.05 \cos \left(0.4 t\right)+0.05 \cos \left(0.7 t\right)\\ d_4\left(t\right)=0.5 \cos \left(0.4 t\right)+1 \\ d_5\left(t\right)=0.5 \sin \left(0.5 t\right)+1 \\ d_6\left(t\right)=0.5 \sin \left(0.7 t\right)+1 \end{array}\right.$

The trajectory tracking of the 4Y octocopter aircraft is shown in Figure 8 and Figure 9, when there are numerous actuator problems and external disturbances. The PT2FAISMC is able to adequately handle these faults such that the tracking performance remains unchanged, and the tracking errors eventually converge to zero. In reality, it is not possible to provide appropriate trajectory tracking for the provided reference command using CISMC in the event of actuator faults and external disturbances. In addition, the CISMC tracks the reference command with considerable oscillations, as presented in Figure 8, and it has the weakest tracking performance once faults arise. Due to the fact that yaw and height motion are affected by eight actuators, when faults occur in the actuators, (x, y, z, ψ) positions are more affected than (φ, θ). Also, as shown in Figure 10, the CISMC requires greater control efforts than the PT2FAISMC to keep the system tracking performance stable, since the discontinuous control gain increases. Chattering from the controls may potentially cause the system to become unstable by triggering unwanted dynamics. The global trajectory in 3D is depicted in Figure 11.

Scenario 2:

Sine trajectory tracking for the UAV and manipulator in the presence of parametric uncertainty, actuator faults, and external disturbances.

In Scenario 2, a sinusoidal desired trajectory is considered as [70]: $x_d=0.5 \cos \left({\displaystyle \frac{\pi t}{20}}\right)$, $y_d=0.5 \sin \left({\displaystyle \frac{\pi t}{20}}\right)$, $z_d=3-2\cos \left({\displaystyle \frac{\pi t}{20}}\right)$, and the actuator faults in the 4Y octocopter aircraft are injected to actuator#5, actuator#6, actuator#7, and actuator#8 at t = 15 s. Subsequently, a similar fault is intentionally introduced to actuator#1, actuator#2, actuator#3, and actuator#4 at t = 20 s. To evaluate the effectiveness of the proposed control method in dealing with external disturbances, the unsettling expressions in (79) has been added at t = 25 s.

In this simulation test, additional unknown aspects are considered to verify the efficacy of the suggested control technique. The tracking performance of the 4Y octocopter aircraft in faulty situation is shown in Figure 12 and Figure 13. It is remarked that the PT2FAISMC is capable of effectively preserving system performance in the presence of multiple actuator faults and external disturbances. This is achieved through the implementation of a designed disturbance observer and adaptive control schemes. Additionally, the PT2FAISMC is able to suppress control chattering, as depicted in Figure 14. However, when actuator defects occur, the CISMC is unable to ensure the successful tracking of the desired reference command owing to their focus on accommodating external disturbances. Figure 14 displays the control inputs of the actuators using the two techniques. The excessive use of discontinuous control approaches to handle actuator defects and external disturbances results in the unwanted occurrence of control chattering in the CISMC. Moreover, the introduction of additional uncertain elements into the system exacerbates the oscillation of the system output via CISMC. Figure 15 illustrates the global path in 3D.

A comparative analysis is presented in Table 4 to enhance visibility and demonstrate the efficacy of the suggested control approach relative to other relevant studies.

In Ref. [76], the authors formulated a stable type-1 fuzzy adaptive control law for a category of MIMO nonlinear systems to enhance the controller’s robustness against parametric uncertainties and significant uncertain nonlinearities, subsequently applying this methodology to the quadrotor aircraft.

Type-1 fuzzy systems are used to approximate the model of the controlled system. To mitigate the impact of inevitable reconstruction mistakes, the authors include a sliding term into the control rule. The approximation theory and the Lyapunov approach are used in tandem to develop the fuzzy adaptive control law in the first phase and to demonstrate the convergence of the tracking error and the boundedness of the adaptive parameters and all plant signals in the subsequent phase.

This approach’s rule base has several rules; hence, its execution requires considerable processing time, posing challenges for real-time implementation. In this methodology, the authors used for each fuzzy system has eight inputs, and each input has two fuzzy sets. Therefore, the rule number in each fuzzy system is 256. In the proposed approach, each type-2 fuzzy system has only two inputs, and each input contains three type-2 fuzzy sets. The rules base of the developed method comprises only 9 rules. Thus, this strategy necessitates reduced computational time for execution relative to the method given in Ref. [76].

8. Conclusions

In this work, an adaptive type-2 fuzzy sliding mode control strategy based on a disturbance observer is applied to the 4Y octocopter aircraft, targeting various actuator faults and external disturbances while reducing chattering phenomenon. Separate processes are employed to overcome actuator faults and external disturbances. Specifically, the suggested adaptive control strategies are used to accommodate actuator faults, while the developed disturbance observer is employed to reduce external disturbances. The discontinuous control strategy of sliding mode control is greatly decreased by using type-2 fuzzy adaptive control parameters in both continuous and discontinuous control sections, preventing chattering phenomena in the control signals. In addition, by combining the suggested disturbance observer with the established adaptive type-2 fuzzy sliding mode control, the discontinuous control gain of sliding mode control is decreased even more. This effectively reduces the chattering phenomena issue while still ensuring accurate tracking capability of the 4Y octocopter aircraft. Additionally, in order to eliminate the chattering phenomenon on the traditional integral sliding mode control, a fuzzy hitting control law is implemented using a type-1 fuzzy logic inference method.

The control technique described in this study has a much higher level of robustness when compared to earlier research, particularly in dealing with actuator faults, model uncertainties, and external disturbances. The efficacy and benefits of the proposed controller are confirmed and justified by numerical simulation tests. These simulations compare the performance of the controller with a conventional integral sliding mode controller. The 4Y octocopter aircraft is subjected to fluctuations in the varied levels of actuator failures and unknown external disturbances during testing.

Author Contributions

Methodology, S.Z. and A.D.; Software, S.Z.; Formal analysis, H.R. and M.F.B.; Investigation, H.M.; Resources, L.B. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. The 4Y octocopter aircraft configuration [16], (a) Real 4Y octocopter (b) 4Y octocopter configuration.

Figure 2. Controlling the motion of the 4Y octocopter aircraft utilizing virtual control.

Figure 3. Overview of the developed fault-tolerant control.

Figure 4. Architecture of a type-2 fuzzy logic system.

Figure 5. Interval type-2 Gaussian membership functions for the antecedent sets.

Figure 6. The membership functions of input variables [Forumla omitted. See PDF.] and output [Forumla omitted. See PDF.] for [Forumla omitted. See PDF.].

Figure 7. Block diagram of the proposed control algorithm.

Figure 8. (x, y, z) positions and yaw angle (ψ) outputs of the 4Y octocopter aircraft in the presence of actuator faults and external disturbances (Scenario 1)). (a) Evolution of x real vs. x desired (b) Evolution of x real vs. y desired (c) Evolution of z real vs. z desired (d) Evolution of ksi real vs. ksi desired.

Figure 9. Roll and pitch angles (φ, θ) of the 4Y octocopter aircraft in the presence of actuator faults and external disturbances (Scenario 1).(a) Evolution of phi angle (b) Evolution of theta angle.

Figure 10. The input forces of the 4Y octocopter aircraft (Scenario 1). (a) Evolution of input F1 (b) Evolution of input F2 (c) Evolution of input F3 (d) Evolution of input F4 (e) Evolution of input F5 (f) Evolution of input F6 (g) Evolution of input F7 (h) Evolution of input F8.

Figure 11. 3D position tracking result of the 4Y octocopter aircraft (Scenario 1).

Figure 12. 3D position tracking result of the 4Y octocopter aircraft (Scenario 1). (a) Evolution of x real vs. x desired (b) Evolution of x real vs. y desired (c) Evolution of z real vs. z desired (d) Evolution of ksi real vs. ksi desired.

Figure 13. Roll and pitch angles (φ, θ) of the 4Y octocopter aircraft in the presence of actuator faults and external disturbances (Scenario 2). (a) Evolution of phi angle (b) Evolution of th et a angle.

Figure 14. The input forces of the 4Y octocopter aircraft (Scenario 2). (a) Evolution of input F1 (b) Evolution of input F2 (c) Evolution of input F3 (d) Evolution of input F4.

Figure 15. 3D position tracking result of the 4Y octocopter aircraft (Scenario 2).

Table 1

Rules base [68].

	Rule 1	Rule 2	Rule 3	Rule 4	Rule 5
$s_i$	PB	PS	ZE	NS	NB
$\mathsf{\Delta }{u}_i$	PB	PS	ZE	NS	NB

Table 2

Physical parameters of the 4Y octocopter aircraft [16].

Parameters	Values
$m$	1.56 (Kg)
$K_p$	10 × 10−6 (N.s2)
$K_d$	0.3 × 10−6 (N.s2)
$I_x,I_y,I_z$	0.04 (Kg·m2), 0.04 (Kg·m2), 0.08 (Kg·m2)
$\alpha$	1.36 (rad)
$L$	0.158 (m)
$l_1$	0.25 (m)
$g$	9.81 (m/s2)
$K_{ftx}, K_{fty},K_{ftz}$	5.567 × 10−4 (N/m/s), 5.567 × 10−4 (N/m/s), 6.354 × 10−4 (N/m/s)
$K_{fax}, K_{fay},K_{faz}$	5.567 × 10−4 (N/rad/s), 5.567 × 10−4 (N/rad/s), 6.354 × 10−4 (N/rad/s)

Table 3

Control parameters.

Parameters	Values
${\lambda }_i$ $\mathrm{for} i=1,\dots ,6$	1
${\beta }_i$	1
$k_i$	1
${\eta }_i$	0.1
$a_{1 i}$	100
$a_{2 i}$	0.5
${\kappa }_i$	120
${\delta }_i$	1
${\chi }_i$	0.5
${\rho }_i$	1
${\omega }_i$	0.5

Table 4

Comparative study.

Our Control Algorithm	Other Control Algorithm	Corresponding Papers
No information on actuator defects is required; the controller is updating online to mitigate the effects of the faults.	Require information on actuator fault models.	[71,72]
The absence of a fault detection and isolation (FDI) module necessitates that the controller autonomously addresses actuator defects by adjusting itself to mitigate adverse effects. Moreover, the detection-induced time delay is circumvented.	The employment of a failure detection and isolation (FDI) module.	[73,74]
The closed-loop system exhibits global asymptotic stability, with the tracking error converging exponentially to the origin, attributable to the accurate approximation derived from type-2 fuzzy systems, while the residual terms from these type-2 fuzzy systems are managed by the included robust control term based on a type-1 fuzzy system.	The closed-loop system is uniformly ultimately bounded (UUB) stable, and the tracking error converges exponentially only to a compact set. This is due to the terms residue from approximation.	[72,75]
To alleviate conservativeness, actuator faults, model uncertainties, and external disturbances are individually addressed. Actuator faults and model uncertainties are mitigated through the proposed neural adaptive schemes, while external disturbances are countered by the design of a nonlinear disturbance observer.	The impacts of actuator faults, model errors, and external disturbances are combined regarded as a comprehensive perturbation, leading to a cautious design.	[60,61,62]

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