An adaptive sliding mode actuator fault tolerant control scheme for octorotor system

Abstract

In this article, an adaptive sliding mode control is used in the framework of fault tolerant control to mitigate the effects of actuator faults without requiring the actuator health information. Since unmanned aerial vehicles are being used in multiple fields such as military, surveillance, media, agriculture, communication and trading sector, therefore it is of vital importance to overcome the effects of actuator faults that can decline system performance and can even lead to some serious accidents. The proposed adaptive sliding mode control approach can handle actuator faults directly without requiring any faults information and adaptively adjusts controller gains to maintain acceptable level of performance. To validate the effectiveness of the proposed adaptive fault tolerant control scheme, it has been tested in simulations using non-linear Benchmark model of Octorotor system and its performance is compared with the optimal LQR control approach.

Introduction

Research on unmanned aerial vehicles (UAV) is increasing day by day due to improvement and development of different control laws. Nowadays, these UAVs are used in different fields like agriculture, commercial, education and also in media sector. ¹ Due to the fact that the applications relating to UAVs are increasing rapidly, the need of stable operation of UAVs is on rise.

The fundamental problem with the safe operation of vehicles with wingspan smaller than 1 m is reliable stabilization, robustness to unpredictable changes in the environment and resilience to noisy data from small sensor systems. Autonomous operation of aerial vehicles relies upon on-board stabilization and trajectory tracking capabilities, and significant effort has to be carried on to make sure that these systems are able to achieve stable flight in the presence of actuator faults/failures. These problems are compounded at smaller scale, since as the vehicle is more susceptible to environmental effects (wind, temperature).

Octorotors are the special class of UAVs which have eight fixed pitch propellers where the desired movement is achieved by varying the speed of rotors. Due to its structure, it is capable of vertical takeoff and landing. Numerous control methods have been proposed for an octorotor system, for both regulation and trajectory tracking. The goal is to find a control strategy which allows the controlled states of an octorotor to converge to an arbitrary set of time-varying reference signals. Many previous works, for example, by Sadeghi et al., ² Benzaid et al., ³ Argentim et al., ⁴ and Khatoon et al., ⁵ have demonstrated that it is possible to control the octorotor using linear control techniques by linearizing the dynamics around an operating point, usually chosen to be at the hover state. However, a wider flight envelope and better performance can be achieved by using non-linear control techniques. Within these non-linear methods, for example, back-stepping, ⁶ sliding mode control (SMC) ^7,8 and feedback linearization ⁹ have been demonstrated to be effective for UAV control.

In the past, the control methodologies such as PID controller and Kalman filter ^10,11 were very successful in industry with wide range of applications. However, control methods that received good attention in last couple of years are robust and adaptive control. ^{12 –14} These controllers are motivated by the need for the safety purpose of different critical systems such as aircraft, chemical plants and nuclear power plants, which require such control systems that can deal with various changes in the dynamics of the plant. Robust control technique such as SMC has been extensively used in recent years. ^{15 –18} SMC technique is considered as a powerful tool to design controller for uncertain systems. SMC provides the robustness against the matched uncertainty and external disturbance. In Yang and Kim, ¹⁹ SMC technique was applied to non-holonomic wheeled mobile robots to achieve asymptotic stabilization to the desired trajectory. The robustness of SMC scheme was validated experimentally. SMC technique was also applied to unicycle-like robotic in Savkin and Wang ²⁰ to avoid collision with moving obstacles. Moreover, the authors in Sarfraz et al. ²¹ proposed integral SMC (ISMC) to provide the robust stabilization control of non-holonomic underwater systems. The unknown term through input transformation was computed using adaptive technique. Some recent results on SMC techniques have been applied to control the autopilot system of the ship ²² and attitude tracking control of quadrotor system in Promkajin and Parnichkun. ²³

A robust fault tolerant scheme was proposed in Chakravarty and Mahanta ¹⁴ for non-linear systems affected by actuator faults. The controller implemented employed an adaptive second-order sliding mode scheme integrated with the back-stepping technique, retaining the benefits of both the techniques. The proposed scheme ensured robustness towards linearly parametrized mismatched uncertainties and also ensured that the proposed controller yields enhanced transient performance with no chattering in the control input in comparison with a dedicated back-stepping or sliding mode controller. Neural networks approach has also been used in the literature. As discussed in Amezquita et al., ¹² an adaptive dynamic surface control scheme for a class of MIMO non-linear systems with actuator failures and uncertainties was presented. In the proposed control scheme, the dynamic changes and disturbances induced by actuator failures were detected and isolated by means of radial basis function neural networks, which also compensated system uncertainties that arise from the mismatch between nominal model and real plant. In Fan et al., ¹³ the adaptive control methodology was discussed for a class of uncertain systems in case of stochastic actuator failures. In this scheme, some knowledge has been established on basis of Markovian variables and then back-stepping control technique was employed for the adaptive compensation of the actuator failures. The authors in Hongzeng et al. ²⁴ have considered the combination of SMC with control allocation for the fault tolerance of a UAV system, whereas in Chakravarty and Mahanta, ²⁵ a fault tolerant control (FTC) scheme was proposed to improve the transient behaviour of an aircraft after actuator faults. In Semprun et al., ²⁶ an actuator FTC scheme was considered for a class of non-linear feedback linearizable systems.

FTC schemes considering ISMC have been recently proposed for state feedback and output feedback in the literatures. ^{27 –29} All these techniques require FDI scheme, wherein FDI scheme was used to estimate the effectiveness levels of the actuators because this information was explicitly used. Recently in Zhong et al., ³⁰ an overview of the tilt rotor UAV was presented where the concept and some typical platforms while focusing on various control techniques is discussed.

In the proposed solution of this article, a FTC scheme is introduced without requiring any fault information and can effectively tolerate actuator faults. To cope with the sudden changes in system dynamics due to actuator faults, an adaptive SMC approach is used, to make sure that regulation of the sliding mode is not disturbed. The proposed adaptation law adjusts the controller gains by taking into account the magnitude of the actuator faults. The proposed adaptive FTC scheme is tested using different fault scenarios and is compared with the optimal LQR controller to show its effectiveness.

In this article as in Bouabdallah and Siegwart, ⁶ the following assumptions are made:

The structure is rigid and symmetric.

The centre of gravity lies at the origin of the body axis reference frame.

The inertia matrix is diagonal.

The propellers are rigid.

Actuator lag is considered negligible.

The thrust is proportional to the square of the speed of the rotor.

The drag is proportional to the square of the speed of the rotor.

The article is structured as follows: First, the non-linear dynamic model of an octorotor is presented and then afterwards the second section will comprise detailed design of different adaptive SMC laws specifically: altitude control law, x-y translation and attitude control laws. In the third section, simulation results are shown to validate the proposed FTC scheme. The fourth section provides the concluding remarks.

Description of non-linear dynamical model

The non-linear state space model of an octorotor system in the presence of actuator faults/failures can be represented as

$\dot{x}=f(t,x)+g(t,x)W(t)\tau (t)$

1

where

$f(t,x)=\left[\begin{array}{l}{x}_b\hfill \\ y_b\hfill \\ z_b\hfill \\ p+q\text{ sin}(\varphi )\text{tan}(\theta )+r\text{ cos}(\varphi )\text{tan}(\theta )\hfill \\ q\text{ cos}(\varphi )-r\text{sin}(\theta )\hfill \\ q\text{ sin}(\varphi ){\text{cos}}^{-1}(\theta )+r\text{ cos}(\varphi ){\text{cos}}^{-1}(\theta )\hfill \\ 0\hfill \\ 0\hfill \\ g\hfill \\ qr\frac{I_y-I_z}{I_x}+\frac{J_r}{I_x}q\Omega \hfill \\ pr\frac{I_z-I_x}{I_y}-\frac{J_r}{I_y}p\Omega \hfill \\ pq\frac{I_x-I_y}{I_z}\hfill \end{array}\right]$

and

$g(t,x)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{1}{m}\text{cos}\varphi \text{sin}\theta \text{cos}\psi +\text{sin}\varphi \text{sin}\psi & 0& 0& 0\\ \frac{1}{m}\text{cos}\varphi \text{sin}\theta \text{sin}\psi -\text{sin}\varphi \text{cos}\psi & 0& 0& 0\\ -\frac{1}{m}\text{cos}(\varphi )\text{cos}(\theta )& 0& 0& 0\\ 0& \frac{1}{I_x}& 0& 0\\ 0& 0& \frac{1}{I_y}& 0\\ 0& 0& 0& \frac{1}{I_z}\end{array}\right]$

The diagonal weighting matrix $W(t)$ with entries $w_i\mathrm{,...},w_m$ represent the effectiveness level of actuators. The diagonal entries $w_i=1$ and $w_i=0$ show that the concerned $i$ th actuator is healthy or completely failed (failure), whereas entry $0<w_i<1$ indicates that the $i$ th actuator has some level of actuator fault. In equation (1), the state vector is

$x(t)=[x_b y_b z_b \varphi \theta \psi {\dot{x}}_b {\dot{y}}_b {\dot{z}}_b p q r{]}^{\text{T}}$

2

where the states $x_b$ , $y_b$ and $z_b$ represent the position of the octorotor with respect to body fixed frame and $p$ , $q$ and $r$ represent the angular velocities with respect to the body fixed frame and $\varphi$ , $\theta$ , $\psi$ are respectively the roll angle, pitch angle and yaw angle. The term $\Omega$ is termed as disturbance present in the system which depends on the rotor speeds and represents the overall residual propeller speed. The expression for $\Omega$ is

$\Omega =-{\Omega }_1-{\Omega }_2+{\Omega }_3+{\Omega }_4-{\Omega }_5-{\Omega }_6+{\Omega }_7+{\Omega }_8$

The inputs of the system are

$\tau (t)=[{\tau }_1 {\tau }_2 {\tau }_3 {\tau }_4{]}^{\text{T}}$

3

where ${\tau }_1$ denotes the roll torque, ${\tau }_2$ pitch torque, ${\tau }_3$ yaw torque and ${\tau }_4$ denotes the total thrust.

In order to achieve desired x-y translation, the control variables ${\tau }_{xx}$ and ${\tau }_{yy}$ can be considered as virtual commands, where

${\tau }_{xx}=\text{cos}\varphi \text{sin}\theta \text{cos}\psi +\text{sin}\varphi \text{sin}\psi$

4

${\tau }_{yy}=\text{cos}\varphi \text{sin}\theta \text{sin}\psi -\text{sin}\varphi \text{cos}\psi$

5

The desired expressions for ${\varphi }_d$ and ${\theta }_d$ can be obtained from these virtual commands specifically

${\varphi }_d={\text{sin}}^{-1}(\text{sin}\psi {\tau }_{xx}-\text{cos}\psi {\tau }_{yy})$

6

${\theta }_d={\text{sin}}^{-1}\frac{\text{cos}\psi {\tau }_{xx}+\text{sin}\psi {\tau }_{yy}}{\text{cos}{\varphi }_d}$

7

The parametric values for the model are taken from the study by Weidong et al. ⁸ and are given in Table 1.

Table 1.

System parameters. ²⁰

Parameter	Value	Unit
G	9.81	m/s2
M	1.64	Kg
$I_x$	0.044	Kg·m2
$I_y$	0.044	Kg·m2
$I_z$	0.088	Kg·m2
D	0.3 × ${10}^{-6}$	N·m·s2
L	0.4	m
$J_r$	90 × 10−6	Kg·10−6
B	10 × 10−6	N·m2

Controller design

This section explains the design procedure of an adaptive sliding mode controller. The design process is simplified by assuming that perturbation over hover state are small, therefore it is assumed that $(\dot{\varphi },\dot{\theta },\dot{\psi })\approx (p,q,r)$ .

Design of an adaptive sliding mode controller

The controller design is based on the nominal fault free system which means that while designing the controller, it will be assumed that $W(t)=I$ in (1). The basic idea of the sliding mode controller is to define a sliding surface and enforce the system trajectories to remain on this sliding surface by using a discontinuous control law. ³¹ A typical unit vector control can be defined as

$v_n(t)=-k(t,x)\frac{s(t)}{||s(t)||}\text{\hspace{1em}\hspace{0.05em}for\hspace{0.05em}\hspace{1em}}s(t)\ne 0$

8

where $s(t)$ is the switching function (which is defined in the sequel) and $k(t,x)>0$ is a modulation gain. In this article, the idea is to make the modulation gain adaptive, ³² so that the feedback gain can adapt the sudden changes in the system dynamics due to possible occurrence of actuator faults or in the presence of external disturbances.

In the design process, the controller for translational motion will provide desired roll and pitch angles depending on the desired values of x and y. These angles will be provided to the attitude controller which will compute roll torque ${\tau }_1$ , pitch torque ${\tau }_2$ and yaw torque ${\tau }_3$ .

Altitude controller

An altitude controller provides the overall thrust ${\tau }_4$ needed by an octorotor. The vertical motion of an octorotor can be described by the differential equation as

$\ddot{z}=-g+\text{cos}\varphi \text{cos}\theta \frac{{\tau }_4}{m}$

9

Consider the tracking error between state $z$ and desired state $z_d$ as

$e_1=z-z_d$

10

Define a sliding surface as

$s_1(t)={\dot{e}}_1(t)+{\beta }_1{e}_1(t)\text{\hspace{1em}\hspace{0.05em}where\hspace{0.05em}\hspace{1em}}{\beta }_1>0$

11

By taking the time derivative of equation (11) and putting values of $e_1$ and equation (9) it becomes

${\dot{s}}_1=-g+\text{cos}\varphi \text{cos}\theta \frac{{\tau }_4}{m}-{\ddot{z}}_d+{\beta }_1(\dot{z}-{\dot{z}}_d)$

12

In order to find the control law such that the motion along the surface $s_1(t)$ is stable, choose a Lyapunov candidate function as

$V_1=\frac{1}{2}{s}_1^{\text{T}}{s}_1$

13

By taking the time derivative of $V_1$ along the trajectories it becomes

${\dot{V}}_1=s_1^{\text{T}}{\dot{s}}_1$

14

Consider the control law ${\tau }_4$ in the form

${\tau }_4={\tau }_{l_4}+{\tau }_{n_4}$

15

where ${\tau }_{l_4}$ is

${\tau }_{l_4}=\frac{m}{\text{cos}\varphi \text{cos}\theta }[g+{\ddot{z}}_d-\beta (\dot{z}-{\dot{z}}_d)-{\mu }_1{s}_1]$

16

and ${\tau }_{n_4}$ is

${\tau }_{n_4}(t)=-\frac{m}{\text{cos}\varphi \text{cos}\theta }{k}_1(t,x)\frac{s_1(t)}{||s_1(t)||}\text{\hspace{0.05em} for \hspace{0.05em}}{s}_1\ne 0$

17

where $k_1(t,x)>0$ is the adaptive modulation gain (AMG) and ${\mu }_1>0$ . In the presence of actuator faults, the state trajectory deviates from the sliding surface, therefore a choice of an AMG (to adjust the feedback gain accordingly) is given by

$k_1(t,x)=\parallel {\tau }_{l_4}\parallel {\overline{k}}_1(t,x)+{\eta }_1$

18

where ${\eta }_1>0$ is a positive scalar and the adaptation gain ${\overline{k}}_1(t,x)$ evolves according to

${\dot{\overline{k}}}_1(t,x)=-{\zeta }_1{\overline{k}}_1(t,x)+{\zeta }_2\parallel {\tau }_{l_4}\parallel \parallel s_1\parallel$

19

where ${\zeta }_1$ and ${\zeta }_2$ are design parameters and their values are given in Table 2. By putting value of ${\tau }_4$ in equation (12) which yields

${\dot{s}}_1=-k_1(t,x)\frac{s_1}{\parallel s_1\parallel }-{\mu }_1{s}_1$

20

$=-(\parallel {\tau }_{l_4}\parallel {\overline{k}}_1(t,x)+{\eta }_1)\frac{s_1}{\parallel s_1\parallel }-{\mu }_1{s}_1$

21

Table 2.

Design parameters for adaptive SMC.

Parameter	Value	Parameter	Value
${\beta }_1$	6.5	${\mu }_3$	0
${\beta }_2$	1	${\mu }_4$	0.0025
${\beta }_3$	1	${\mu }_5$	0.0025
${\beta }_4$	3.5	${\mu }_6$	0.00025
${\beta }_5$	3.5	${\zeta }_1$	200
${\beta }_6$	2.5	${\zeta }_2$	100
${\mu }_1$	8	$\eta$	0.9
${\mu }_2$	0	${\epsilon }_0$	0.003

SMC: sliding mode control.

Therefore equation (14) becomes

$\begin{array}{c}{\dot{V}}_1=s_1^{\text{T}}((-\parallel {\tau }_{l_4}\parallel {\overline{k}}_1(t,x)-{\eta }_1)\frac{s_1}{\parallel s_1\parallel }-{\mu }_1{s}_1)\\ \le (-\parallel {\tau }_{l_4}\parallel {\overline{k}}_1(t,x)-{\eta }_1)\parallel s_1||-{\mu }_1\parallel s_1{\parallel }^2\end{array}$

22

It is clear that ${\dot{V}}_1\le 0$ ( ${\dot{V}}_1$ is semi negative definite), therefore the control input ${\tau }_4$ ensures that the motion along the sliding surface $s_1(t)$ is stable.

x-y translation

The x-position of an octorotor can be described by the following differential equation as

$\ddot{x}={\tau }_x\frac{{\tau }_4}{m}$

23

The tracking error between state $x$ and desired state $x_d$ can be defined as

$e_2=x-x_d$

24

Define a sliding surface as

$s_2(t)={\dot{e}}_2(t)+{\beta }_2{e}_2(t)\text{\hspace{1em}\hspace{0.05em}where\hspace{0.05em}\hspace{1em}}{\beta }_2>0$

25

By taking the time derivative of (25) and putting values of $e_2$ and (23) which yields

${\dot{s}}_2={\tau }_x\left(\frac{{\tau }_4}{m}\right)-{\ddot{x}}_d+{\beta }_2(\dot{x}-{\dot{x}}_d)$

26

To show that the motion along the surface $s_2(t)$ is stable, choose a Lyapunov candidate function as

$V_2=\frac{1}{2}{s}_2^{\text{T}}{s}_2$

27

By taking the time derivative of $V_2$ along the trajectories it becomes

${\dot{V}}_2=s_2^{\text{T}}{\dot{s}}_2$

28

Consider the control law for $x$ -translation in the form

${\tau }_x={\tau }_{l_x}+{\tau }_{n_x}$

29

where ${\tau }_{l_x}$ is

${\tau }_{l_x}=\frac{m}{{\tau }_4}[{\ddot{x}}_d-{\beta }_2(\dot{x}-{\dot{x}}_d)-{\mu }_2{s}_2]$

30

and ${\tau }_{n_x}$ is

${\tau }_{n_x}(t)=-\frac{m}{{\tau }_4}{k}_2(t,x)\frac{s_2(t)}{\parallel s_2(t)\parallel }\text{\hspace{1em}\hspace{0.05em}for \hspace{0.05em}}{s}_2\ne 0$

31

where $k_2(t,x)$ is the AMG and ${\mu }_2>0$ . To adapt the dynamical changes in case of actuator fault, a choice of an AMG $k_2(t,x)>0$ is given by

$k_2(t,x)=||{\tau }_{l_x}||{\overline{k}}_2(t,x)+{\eta }_2$

32

where ${\eta }_2>0$ is a positive scalar. The positive adaptation gain ${\overline{k}}_2(t,x)$ evolves according to

${\dot{\overline{k}}}_2(t,x)=-{\zeta }_1{\overline{k}}_2(t,x)+{\zeta }_2\parallel {\tau }_{l_x}\parallel \parallel s_2\parallel$

33

By putting value of ${\tau }_x$ in equation (26) gives

${\dot{s}}_2=-(||{\tau }_{l_x}||{\overline{k}}_2(t,x)+{\eta }_2)\frac{s_2}{\parallel s_2\parallel }-{\mu }_2{s}_2$

34

Therefore equation (28) becomes

$\begin{array}{c}{\dot{V}}_2=s_2^{\text{T}}(-(||{\tau }_{l_x}||{\overline{k}}_2(t,x)+{\eta }_2)\frac{s_2}{\parallel s_2\parallel }-{\mu }_2{s}_2)\\ \le -||{\tau }_{l_x}||{\overline{k}}_2(t,x)\parallel s_2\parallel -{\eta }_2\parallel s_2\parallel -{\mu }_2\parallel s_2{\parallel }^2\end{array}$

35

It is clear that ${\dot{V}}_2\le 0$ , hence the control law given in (29) is attractive for the sliding surface given in (25). Following the similar procedure adopted for the $x$ -translation, the control law for the $y$ -translation motion ${\tau }_y$ can be designed.

Define a sliding surface as

$s_3(t)={\dot{e}}_3(t)+{\beta }_3{e}_3(t)\text{\hspace{1em}\hspace{0.05em}where\hspace{0.05em}\hspace{1em}}{\beta }_3>0$

36

where the tracking error between state $y$ and desired state $y_d$ can be defined as

$e_3=y-y_d$

37

In a similar fashion, the control law for the $y$ -translation motion ${\tau }_y$ cab be written as

${\tau }_y={\tau }_{l_y}+{\tau }_{n_y}$

38

where ${\tau }_{l_y}$ is

${\tau }_{l_y}=\frac{m}{{\tau }_4}[{\ddot{y}}_d-{\beta }_3(\dot{y}-{\dot{y}}_d)-{\mu }_3{s}_3]$

39

and ${\tau }_{n_y}$ is

${\tau }_{n_y}(t)=-\frac{m}{{\tau }_4}{k}_3(t,x)\frac{s_3(t)}{\parallel s_3(t)\parallel }\text{\hspace{1em}\hspace{0.05em}for \hspace{0.05em}}{s}_3\ne 0$

40

where ${\mu }_3>0$ and $k_3(t,x)>0$ is an adaptive modulation function given by

$k_3(t,x)=\parallel {\tau }_{l_y}\parallel {\overline{k}}_3(t,x)+{\eta }_3$

41

where ${\eta }_3$ is a positive scalar. The positive adaptation gain ${\overline{k}}_3(t,x)$ evolves according to

${\dot{\overline{k}}}_3(t,x)=-{\zeta }_1{\overline{k}}_3(t,x)+{\zeta }_2\parallel {\tau }_{l_y}\parallel \parallel s_3\parallel$

42

From x-y translation, the desired ${\varphi }_d$ and ${\theta }_d$ are calculated by putting values of ${\tau }_x$ and ${\tau }_y$ in equations (6) and (7) in place of ${\tau }_{xx}$ and ${\tau }_{yy}$ .

Attitude controller

Desired roll ${\varphi }_d$ and pitch ${\theta }_d$ angles from the x-y translational controller and yaw angle are fed into attitude controller to calculate the required control efforts. The roll dynamics of an octorotor can be described by the following differential equation as

$\ddot{\varphi }=qr\frac{I_y-I_z}{I_x}+\frac{J_r}{I_x}q\Omega +\frac{{\tau }_1}{I_x}$

43

where $\Omega$ term is considered as a disturbance (which depends on the rotor speeds). During controller design, it is assumed that there is no disturbance in the system. The tracking error between state $\varphi$ and desired state ${\varphi }_d$ can be defined as

$e_4=\varphi -{\varphi }_d$

44

Define a sliding surface as

$s_4(t)={\dot{e}}_4(t)+{\beta }_4{e}_4(t)\text{\hspace{1em}\hspace{0.05em}where\hspace{0.05em}\hspace{1em}}{\beta }_4>0$

45

Taking the time derivative of (45) and putting values of $e_4$ and (43) it becomes

${\dot{s}}_4=qr\frac{I_y-I_z}{I_x}+\frac{J_r}{I_x}q+\frac{{\tau }_1}{I_x}-{\ddot{\varphi }}_d+{\beta }_4(\dot{\varphi }-{\dot{\varphi }}_d)$

46

The control law for the roll torque is defined as

${\tau }_1={\tau }_{l_1}+{\tau }_{n_1}$

47

where ${\tau }_{l_1}$ is

${\tau }_{l_1}=I_x(-qr\frac{I_y-I_z}{I_x}-\frac{J_r}{I_x}q+{\ddot{\varphi }}_d-{\beta }_4(\dot{\varphi }-{\dot{\varphi }}_d)-{\mu }_4{s}_4)$

48

and ${\tau }_{n_1}$ is

${\tau }_{n_1}(t)=-I_x{k}_4(t,x)\frac{s_4(t)}{||s_4(t)||}\text{\hspace{1em}\hspace{0.05em}for \hspace{0.05em}\hspace{0.05em}}{s}_4\ne 0$

49

where $k_4(t,x)>0$ is an adaptive modulation function given by

$k_4(t,x)=\parallel {\tau }_{l_1}\parallel {\overline{k}}_4(t,x)+{\eta }_4$

50

where ${\eta }_4$ is a positive scalar. The positive adaptation gain ${\overline{k}}_4(t,x)$ evolves according to

${\dot{\overline{k}}}_4(t,x)=-{\zeta }_1{\overline{k}}_4(t,x)+{\zeta }_2\parallel {\tau }_{l_1}\parallel \parallel s_4\parallel$

51

where the values of the design parameters ${\zeta }_1$ and ${\zeta }_2$ are given in Table 2. To show that the motion along the surface $s_4(t)$ is stable, choose a Lyapunov candidate function as

$V_4=\frac{1}{2}{s}_4^{\text{T}}{s}_4$

52

Taking the time derivative of $V_4$ along the trajectories it becomes

${\dot{V}}_4=s_4^{\text{T}}{\dot{s}}_4$

53

By substituting the value of ${\tau }_1$ in equation (46) which yields

${\dot{s}}_4=-(\parallel {\tau }_{l_1}\parallel {\overline{k}}_4(t,x)+{\eta }_4)\frac{s_4}{\parallel s_4\parallel }-{\mu }_4{s}_4$

54

Therefore equation (53) becomes

$\begin{array}{c}{\dot{V}}_4=s_4^{\text{T}}(-(\parallel {\tau }_{l_1}\parallel {\overline{k}}_4(t,x)+{\eta }_4)\frac{s_4}{\parallel s_4\parallel }-{\mu }_4{s}_4)\\ \le -||{\tau }_{l_1}||{\overline{k}}_4(t,x)\parallel s_4\parallel -{\eta }_4\parallel s_4\parallel -{\mu }_4\parallel s_2{\parallel }^2\end{array}$

55

It is clear that ${\dot{V}}_4\le 0$ , which shows that the control input ${\tau }_1$ is stabilizing the system.

The pitch dynamics of an octorotor can be described by the following differential equation as

$\ddot{\theta }=pr\frac{I_z-I_x}{I_y}+\frac{J_r}{I_y}p\Omega +\frac{{\tau }_2}{I_y}$

56

The tracking error between state $\theta$ and desired state ${\theta }_d$ can be defined as

$e_5=\theta -{\theta }_d$

57

Define a sliding surface as

$s_5(t)={\dot{e}}_5(t)+{\beta }_5{e}_5(t)\text{\hspace{1em}\hspace{0.05em}where\hspace{0.05em}\hspace{1em}}{\beta }_5>0$

58

Taking the time derivative of (58) and putting values of $e_5$ and (56) it becomes

${\dot{s}}_5=pr\frac{I_z-I_x}{I_y}+\frac{J_r}{I_y}p\Omega +\frac{{\tau }_2}{I_y}-{\ddot{\theta }}_d+{\beta }_5(\dot{\theta }-{\dot{\theta }}_d)$

59

To show that the motion along the surface $s_5(t)$ is stable, choose a Lyapunov candidate function as

$V_5=\frac{1}{2}{s}_5^{\text{T}}{s}_5$

60

Taking the time derivative of $V_5$ along the trajectories it becomes

${\dot{V}}_5=s_5^{\text{T}}{\dot{s}}_5$

61

Consider the control law for the pitch torque ${\tau }_2$ in the form

${\tau }_2={\tau }_{l_2}+{\tau }_{n_2}$

62

where

${\tau }_{l_2}=I_y(pr\frac{I_z-I_x}{I_y}+\frac{J_r}{I_y}p+{\ddot{\theta }}_d-{\beta }_5(\dot{\theta }-{\dot{\theta }}_d)-{\mu }_5{s}_5)$

63

and ${\tau }_{n_2}$ is

${\tau }_{n_2}(t)=-I_y{k}_5(t,x)\frac{s_5(t)}{||s_5(t)||}\text{\hspace{1em}\hspace{0.05em}for \hspace{0.05em}}{s}_5\ne 0$

64

where $k_5(t,x)>0$ is an adaptive modulation function given by

$k_5(t,x)=\parallel {\tau }_{l_2}\parallel {\overline{k}}_5(t,x)+{\eta }_5$

65

where ${\eta }_5$ is a positive scalar. The positive adaptation gain ${\overline{k}}_5(t,x)$ evolves according to

${\dot{\overline{k}}}_5(t,x)=-{\zeta }_1{\overline{k}}_5(t,x)+{\zeta }_2\parallel {\tau }_{l_2}\parallel \parallel s_5\parallel$

66

By putting value of ${\tau }_2$ in equation (59) gives

${\dot{s}}_5=-(\parallel {\tau }_{l_2}\parallel {\overline{k}}_5(t,x)+{\eta }_5)\frac{s_5}{\parallel s_5\parallel }-{\mu }_5{s}_5$

67

Therefore equation (61) becomes

$\begin{array}{c}{\dot{V}}_5=s_5^{\text{T}}(-(\parallel {\tau }_{l_2}\parallel {\overline{k}}_5(t,x)+{\eta }_5)\frac{s_5}{\parallel s_5\parallel }-{\mu }_5{s}_5)\\ \le -\parallel {\tau }_{l_2}\parallel {\overline{k}}_5(t,x)\parallel s_5\parallel -{\eta }_5\parallel s_5\parallel -{\mu }_5\parallel s_5{\parallel }^2\end{array}$

68

It is clear that ${\dot{V}}_5\le 0$ , hence the control input ${\tau }_2$ is stabilizing the system.

The yaw dynamics of an octorotor can be described by the following differential equation

$\ddot{\psi }=pq\frac{I_x-I_y}{I_z}+\frac{{\tau }_3}{I_z}$

69

The tracking error between state $\psi$ and desired state ${\psi }_d$ can be defined as

$e_6=\psi -{\psi }_d$

70

Define a sliding surface as

$s_6(t)={\dot{e}}_6(t)+{\beta }_6{e}_6(t)\text{\hspace{1em}\hspace{0.05em}where\hspace{0.05em}\hspace{1em}}{\beta }_6>0$

71

Taking the time derivative of (71) and putting values of $e_6$ and (69) it becomes

${\dot{s}}_6=pq\frac{I_x-I_y}{I_z}+\frac{{\tau }_3}{I_z}-{\ddot{\psi }}_d+{\beta }_6(\dot{\psi }-{\dot{\psi }}_d)$

72

To show that the motion along the surface $s_6(t)$ is stable, choose a Lyapunov candidate function as

$V_6=\frac{1}{2}{s}_6^{\text{T}}{s}_6$

73

Taking the time derivative of $V_6$ along the trajectories it becomes

${\dot{V}}_6=s_6^{\text{T}}{\dot{s}}_6$

74

Consider the yaw torque ${\tau }_3$ in the form

${\tau }_3={\tau }_{l_3}+{\tau }_{n_3}$

75

where

${\tau }_{l_3}=I_z(-pq\frac{I_x-I_y}{I_z}+{\ddot{\psi }}_d-{\beta }_6(\dot{\psi }-{\dot{\psi }}_d)-{\mu }_6{s}_6)$

76

and ${\tau }_{n_3}$ is

${\tau }_{n_3}(t)=-I_z{k}_6(t,x)\frac{s_6(t)}{\parallel s_6(t)\parallel }\text{\hspace{1em}\hspace{0.05em}for \hspace{0.05em}\hspace{0.05em}}{s}_6\ne 0$

77

where $k_6(t,x)>0$ is an adaptive modulation function given by

$k_6(t,x)=\parallel {\tau }_{l_3}\parallel {\overline{k}}_6(t,x)+{\eta }_6$

78

where ${\eta }_6$ is a positive scalar. The positive adaptation gain ${\overline{k}}_6(t,x)$ evolves according to

${\dot{\overline{k}}}_6(t,x)=-{\zeta }_1{\overline{k}}_6(t,x)+{\zeta }_2\parallel {\tau }_{l_3}\parallel \parallel s_6\parallel$

79

Finally by putting value of ${\tau }_3$ in equation (72) it becomes

${\dot{s}}_6=-(\parallel {\tau }_{l_3}\parallel {\overline{k}}_6(t,x)+{\eta }_6)\frac{s_6}{\parallel s_6\parallel }-{\mu }_6{s}_6$

80

Therefore equation (74) becomes

$\begin{array}{c}{\dot{V}}_6=s_6^{\text{T}}(-(\parallel {\tau }_{l_3}\parallel {\overline{k}}_6(t,x)+{\eta }_6)\frac{s_6}{\parallel s_6\parallel }-{\mu }_6{s}_6)\\ \le -||{\tau }_{l_3}||{\overline{k}}_6(t,x)\parallel s_6\parallel -{\eta }_6\parallel s_6\parallel -{\mu }_6\parallel s_6{\parallel }^2\end{array}$

81

It is clear that ${\dot{V}}_6\le 0$ , hence ${\dot{V}}_6$ is semi negative definite, that is, the control input ${\tau }_3$ is stabilizing the system.

The overall FTC strategy is shown in Figure 1.

Figure 1.

Schematic of overall FTC strategy. FTC: fault tolerant control.

Design of an optimal LQR Controller

In this section, an optimal LQR controller approach is adopted to design the altitude, x-y translational and attitude controllers. ³³ These controllers are then tested on the non-linear benchmark model of an octorotor. While designing the altitude controller, the state vector $x_z=\{z,\dot{z}{\}}^{\text{T}}$ and its desired state ${\left\{z_{\text{des}}\mathrm{,0}\right\}}^{\text{T}}$ is considered. While designing the altitude controller using LQR approach, the objective is to minimize the cost function $J_z={\displaystyle {\int }_0^{\infty }}(x_z^{\text{T}}{Q}_z{x}_z+{\tau }_{4_{\text{LQR}}}^{\text{T}}{R}_z{\tau }_{4_{\text{LQR}}})\text{d}t$ , where $Q_z$ and $R_z$ matrices are chosen as

$Q_z=\left[\begin{array}{ll}1\hfill & 0\hfill \\ 0\hfill & 1\hfill \end{array}\right]R_z=0.1$

The designed altitude control law is

${\tau }_{4_{\text{LQR}}}=(K_z(z-z_{\text{des}})+g)\frac{m}{\text{cos}\varphi \text{cos}\theta }$

82

where $K_z=[3.16234.0404]$ . For x-y translational controller, the state vector $xy=\{x,\dot{x},y,\dot{y}{\}}^{\text{T}}$ and its desired state $x{y}_{\text{des}}=\{x_{\text{des}}\mathrm{,0},y_{\text{des}}\mathrm{,0}\}$ is considered. In designing the x-y translational controller, it is required to minimize the cost function $J_{xy}={\displaystyle {\int }_0^{\infty }}(x{y}^{\text{T}}{Q}_{xy}xy+{\tau }_{x{y}_{\text{LQR}}}^{\text{T}}{R}_{xy}{\tau }_{x{y}_{\text{LQR}}})\text{d}t$ , where $Q_{xy}$ and $R_{xy}$ matrices are chosen as

$Q_{xy}=\left[\begin{array}{cccc}0.125& 0& 0& 0\\ 0& 0.25& 0& 0\\ 0& 0& 0.125& 0\\ 0& 0& 0& 0.25\end{array}\right]R_{xy}=\left[\begin{array}{ll}2\hfill & 0\hfill \\ 0\hfill & 2\hfill \end{array}\right]$

The designed control law for x-y translational motion is

${\tau }_{x{y}_{\text{LQR}}}=-K_{xy}(xy-x{y}_{\text{des}})$

83

where

$K_{xy}=\left[\begin{array}{cccc}0.2500& 0.4195& 0& 0\\ 0& 0& 0.2500& 0.4195\end{array}\right]$

Based on the control law in (83), the desired values of ${\varphi }_d$ and ${\theta }_d$ can be computed using the expressions given in (6) and (7) which will be subsequently used in the attitude controller. For designing the attitude controller, the state vector $x_a=\{\varphi ,\dot{\varphi },\theta ,\dot{\theta },\psi ,\dot{\psi }\}$ and its desired state vector $x_{a_{\text{des}}}=\{\varphi \mathrm{,0},\theta \mathrm{,0},\psi \mathrm{,0}\}$ is considered. In designing the attitude controller, it is required to minimize the cost function $J_{x_a}={\displaystyle {\int }_0^{\infty }}(x_a{ }^{\text{T}}{Q}_{x_a}{x}_a+u^{\text{T}}{R}_{x_a}u)\text{d}t$ , where the weighting matrices $Q_{x_a}$ and $R_{x_a}$ are chosen as $Q_{x_a}=\text{diag}\left\{\mathrm{7.5,1,7.5,1,35,5}\right\}$ and $R_a=\text{diag}\{0.05,0.05,0.025\}$ . The designed attitude controller to generate the required roll toque, pitch torque and yaw torque is

$u=-Ka(x_a-x_{a_{\text{des}}})$

84

where

$K_a=\left[\begin{array}{cccccc}12.2474& 4.5911& 0& 0& 0& 0\\ 0& 0& 12.2474& 4.5911& 0& 0\\ 0& 0& 0& 0& 37.4166& 14.3731\end{array}\right]$

Simulation results

Simulation results presented in this article are based on a non-linear octorotor model taken from Victor et al. ³⁴ The efficiency of the proposed adaptive FTC approach is tested by introducing actuator faults on each rotor. In this section, three fault scenarios are considered:

10 $\%$ reduction in all the rotors effectiveness at 20 s;

25 $\%$ reduction in all the rotors effectiveness at 20 s; and

50 $\%$ reduction in all the rotors effectiveness at 20 s.

In Figure 2, 3D manoeuvering of an octorotor is shown for both fault free and faulty condition, where fault free manoeuvering results using the proposed adaptive SMC approach are shown in Figure 2(a). The starting point of the flight is [0,0,5] and after going through manoeuver it comes back at the same point. Similar manoeuver was considered after introducing 50 $\%$ reduction in actuators effectiveness level and no visible difference was seen as shown in Figure 2(b) as compared to nominal scenario. It reflects the effectiveness of the proposed FTC scheme as the AMG automatically adjusts itself to cope with the change in actuators effectiveness.

Figure 2.

3D manoeuvering of octorotor: (a) fault free and (b) 50% fault.

Figure 3.

x, y, z and psi states during faulty condition: (a) x-position for 10%, 25% and 50% reduction in actuators effectiveness; (b) y-position for 10%, 25% and 50% reduction in actuators effectiveness; (c) yaw angle for 10%, 25% and 50% reduction in actuators effectiveness; and (d) altitude-position for 10%, 25% and 50% reduction in actuators effectiveness.

In Figure 3, $x$ , $y$ , $z$ and $\psi$ states tracking performance is shown during different fault scenarios (reduction in effectiveness) in which each fault has occurred at 20 s and remained for the rest of the time. In $x$ -position, introducing different level of faults do not show visible difference from the nominal condition. Yaw angle and $y$ -position plots during all fault scenarios are also similar to that of fault free condition. In Figure 3(d), during $z$ -position tracking, there is a little overshoot for different level of faults according to fault magnitude.

In Figure 4, AMGs are given for altitude and attitude controllers. In Figure 4(a), AMG of T1 for all the three scenarios are shown and when fault occurs, it gives overshoot and after that adjusts itself and then maintains itself according to the magnitude of fault. In Figure 4(b) and (c), AMG of T2 and T3 are shown, respectively, for all the three scenarios and when fault occurs, it gives sudden overshoot and then adjusts itself according to the magnitude of fault. In Figure 4(c), AMG of T4 is shown, it has overshoot at 20 s but after that it maintains its gain according to the magnitude of fault.

Figure 4.

Modulation gains during faulty conditions in respective control laws: (a) modulation gain $(k_1)$ associated with ${\tau }_4$ for 10%, 25% and 50% reduction in actuators effectiveness; (b) modulation gain $(k_4)$ associated with ${\tau }_1$ for 10%, 25% and 50% reduction in actuators effectiveness; (c) modulation gain $(k_5)$ associated with ${\tau }_2$ for 10%, 25% and 50% reduction in actuators effectiveness; and (d) modulation gain $(k_6)$ associated with ${\tau }_3$ for 10%, 25% and 50% reduction in actuators effectiveness.

Figure 5.

3D manoeuvering of octorotor: (a) x-position for nominal and 50% reduction in actuators effectiveness; (b) y-position for nominal and 50% reduction in actuators effectiveness; and (c) altitude-position for nominal and 50% reduction in actuators effectiveness.

In Figure 5, an optimal LQR controller approach is adopted to demonstrate the tracking performance of $x$ , $y$ , $z$ states considering no fault and in the presence of 50 $\%$ reduction in actuators effectiveness. It is quite clear that after 20 s (when the actuators effectiveness reduces to 50 $\%$ ), the tracking performance of altitude-position degrades significantly, whereas in Figure 2 using the proposed adaptive FTC approach, the tracking performance does not degrade and tries to maintain the performance close to the nominal.

Conclusion

In this article, an adaptive FTC scheme is proposed which uses adaptive features of SMC in the framework of FTC. Incorporation of SMC in the passive FTC framework maintains an acceptable level of closed loop performance and has the ability to adjust feedback gains and adaptively tolerate the actuator faults in octorotor system. A range of actuator fault scenarios are considered in simulations, for validation purposes on the non-linear octorotor benchmark model. The simulations result show the effectiveness of the proposed scheme. The simulation results on a octorotor benchmark show fast convergence and demonstrate excellent FTC features of the scheme while comparing with the optimal LQR approach.