Sliding mode control for quadrotor with disturbance observer

Abstract

In this article, a sliding mode control scheme is proposed for a quadrotor in the presence of an exogenous disturbance. A nonlinear sliding mode surface is constructed based on the estimate output of a disturbance observer to reject the effect of the unknown disturbance in the quadrotor. The desired control performance is achieved by bringing the state from unstable state to stable ones. To show the effectiveness of the developed control scheme, simulation results are provided for illustration of the designed controller based on disturbance observer.

Introduction

Some significant researches have been conducted on different applications in the area of automatic control for several decades. But recently, the focus has been shifted toward an unmanned aerial vehicle (UAV), quadrotor. Unlike other fixed-wings UAVs, quadrotor is a rotary-wings UAV and has a feature of vertical take-off and landing (VTOL) which gives a necessary benefit of using quadrotor over other UAVs. ¹ It is used widely in commercial and military operations including traffic monitoring, photography, search and rescue, journalism, surveillance, sports, photo shoot, and many more. ² The quadrotor is a rotary-wings, four-rotor setup with 6 degrees of freedom (DOF). It has four control inputs and six outputs, that is, roll (φ), pitch (θ), and yaw (ψ) in attitude while the direction along x, y and z in position. ³ Since the number of inputs is less than the number of outputs, quadrotor can be categorized among the under-actuated nonlinear systems. Hence, it opens vast areas for researchers to conduct research on highly nonlinear under-actuated system model of quadrotor as well as the automatic control methods of the quadrotor.

In early days, the focus of research was on developing an effective and accurate quadrotor model. In Bouabdallah and Siegwart, ⁴ a dynamic model of quadrotor was presented. Following development of the model, research goals were oriented to develop control synthesis for hovering, tracking, and robustness of the quadrotor using linear control techniques such as proportional–integral–derivative (PID) and linear quadratic regulator (LQR) method. ⁵ The knowledge of proportional–derivative (PD) control was utilized for altitude control of quadrotor in Erginer and Altug. ¹ PID control algorithms have been discussed in previous works.^6–9 Although the satisfactory control performances were achieved using these techniques, there are some drawbacks of drift from initial conditions and tuning of controller.^10–13 Therefore, it is mandatory to investigate nonlinear control synthesis for quadrotor.

Nonlinear control synthesis for quadrotor has been investigated extensively by researchers. Due to the ability of robustness, disturbance rejection, and achieving control performance asymptotically, sliding mode control (SMC) gained notable significance.^14–16 Super twisting SMC algorithm for quadrotor was proposed in Jayakrishnan. ¹⁷ A novel adaptive control scheme was presented in Xu et al. ¹⁸ In Yang et al., ¹⁹ variable gain control is presented. Backstepping control and nonlinear dynamic inversion (NDI) algorithms were investigated in previous works.^20–22 Other nonlinear control methods are discussed in Zhao et al. ²³ and Zu et al. ²⁴ Backstepping integrated with the SMC for a miniature quadrotor was constructed and presented in Madani and Benallegue. ²⁵ Self-repairing control of quadrotor using the technique of adaptive SMC was proposed in Yang et al. ²⁶ However, constructing SMC control algorithm leads to high frequency fluctuation in controller, that is, chattering. Therefore, an area is available to conduct research on chattering removing as well as considering the effects of uncertainty and disturbances in the model.

It is known that disturbance can either be measurable or unmeasurable. For the former case, feedforward strategy is followed, while for the latter one, disturbance estimation methodology is followed. Furthermore, it can be said that the uncertainties and unmodelled dynamics are included in the disturbance and divided into two kinds, that is, matched and mismatched. Matched disturbance and uncertainties appear in the same channel of input ²⁷ while mismatched uncertainties appear in different channels. ²⁸ Haptic identification is discussed in Yang et al. ²⁹ Extensive researches have been conducted to develop disturbance observer (DO) and disturbance observer–based controls (DOBCs) in previous works.^30–36 DO-based backstepping was investigated in Tripathi et al. ³⁷ Parameter uncertainties in quadrotor were considered using SMC in Xiong and Zhang. ³⁸ DO integrated for a class of uncertain systems using SMC was presented in Chen and Chen. ³⁹ Adaptive-based SMC for quadrotor without knowledge of the bound on uncertainties was investigated in Li et al. ⁴⁰ In Chen and Sam, ⁴¹ adaptive neural output feedback control schemes are utilized based on DO. Neural network–based control for vibrating systems is discussed in Zhao et al. ⁴² A research has been conducted to show that the SMC was insensitive to the matched uncertainties in Choi, ⁴³ while the use of SMC for mismatched uncertainties was addressed in Chan et al. ⁴⁴ However, while developing SMC controllers, the problem of singularity should be faced. ⁴⁵ To overcome this problem, terminal sliding mode control (TSMC) and nonsingular terminal sliding mode control (NSTSMC) have been proposed in Zak, ⁴⁶ Venkataraman and Gulati^47,48 and Yu and colleagues,^49–59 respectively. However, quadrotor is a flying object and is always under disturbance of different kinds. Therefore, in this article, it is assumed that the quadrotor is experiencing the effect of harmonic disturbance represented by the exogenous system for which a DO is constructed in Chen. ³³ However, TSMC/NSTSMC cannot be integrated with enhanced exogenous DO because the former is finite-time stable while the latter is asymptotic stable. Moreover, a research has been conducted for using continuous NSTSMC with finite-time disturbance observer (FTDO) in Yang et al. ⁵⁰ Therefore, to the author’s knowledge, there exists a void to be filled by conducting research for the control of quadrotor using SMC with enhanced exogenous DO providing a baseline for this research work.

The motivation of this article is to propose and construct a SMC with a criterion of removing singularity as well as work with enhanced exogenous DO, asymptotically. The proposed sliding mode surface is similar in structure with sliding mode surface of NSTSMC. In this dynamic sliding mode surface, the estimated disturbance is given to a matched states having input to remove the disturbance from the mismatched state with no input.

The layout of this article is organized as follows. In section “Model of quadrotor,” a model of quadrotor is presented. In section “Control design based on disturbance estimation,” a modified continuous nonsingular TSMC with DO is constructed followed by the proof of controller as well as stability analysis. In section “SMC with DO for quadrotor,” the proposed controller is utilized for automatic control of quadrotor. Simulation results and conclusions are discussed in section “Simulation results” and section “Conclusion,” respectively.

Model of quadrotor

Quadrotor is an UAV with four rotors directed upward. These four rotors are at equal distance from each other making a shape of a square as shown in Figure 1. ³

Figure 1.

Schematic of quadrotor.

In Figure 1, T₁, T₂, T₃, and T₄ represent the corresponding thrusts of each rotor, and θ and φ represent roll and pitch angle, respectively. Moreover, dynamics of quadrotor are derived in both body frame and inertial frame shown in Figure 2. ³ The body frame is fixed with a quadrotor, describing the orientation of the quadrotor with the axes of z in positive upward direction while x and y in left and right directions, respectively. ³

Figure 2.

Frame of reference of quadrotor.

The linear position of the system is represented by ξ = [x, y, z]^T and angular position is represented by η = [φ, θ, ψ]^T. Both of the positions are derived in inertial frame with x, y, z axes for linear position while φ, θ, and ψ for angular position. To transform the parameters from body frame to inertial frame, a rotational matrix, R, is given as ³

$$R=\left[\begin{array}{ccc}{C}_{\psi }{C}_{\theta }\hfill & C_{\psi }{S}_{\theta }{S}_{\varphi }\hfill & C_{\psi }{S}_{\theta }{C}_{\varphi }+S_{\psi }{S}_{\varphi }\hfill \\ \multicolumn{1}{c}{S_{\psi }{C}_{\theta }}& S_{\psi }{S}_{\theta }{S}_{\varphi }+C_{\psi }{C}_{\theta }\hfill & S_{\psi }{S}_{\theta }{C}_{\varphi }-C_{\psi }{S}_{\varphi }\hfill \\ \multicolumn{1}{c}{-S_{\theta }}& C_{\theta }{S}_{\varphi }\hfill & C_{\theta }{C}_{\varphi }\hfill \end{array}\right]$$

(1)

where C_φ = cos(φ), S_θ = sin(θ), and so on. It should be noted that to convert parameters from inertial frame to body frame, R⁻¹ is needed. Because the rotation matrix R is orthogonal matrix, it can be written that R⁻¹ = R^T. The model of quadrotor can be written in the following state space form

$$\stackrel{·}{X}=f(X,U)$$

(2)

where X and U are representing state vectors and inputs, respectively, written as follows

$$X={\left[\begin{array}{cccccccccccc}\varphi \hfill & \stackrel{·}{\varphi }\hfill & \theta \hfill & \stackrel{·}{\theta }\hfill & \psi \hfill & \stackrel{·}{\psi }\hfill & z\hfill & \stackrel{·}{z}\hfill & x\hfill & \stackrel{·}{x}\hfill & y\hfill & \stackrel{·}{y}\hfill \end{array}\right]}^T$$

(3)

and

$$U={\left[\begin{array}{cccc}{U}_1\hfill & U_2\hfill & U_3\hfill & U_4\hfill \end{array}\right]}^T$$

(4)

The control inputs are given by Bouabdallah and Siegwart ⁴

$$\begin{array}{c}{U}_1=k({\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2+{\mathrm{\Omega }}_3^2+{\mathrm{\Omega }}_4^2)\hfill \\ U_2=k(-{\mathrm{\Omega }}_2^2+{\mathrm{\Omega }}_4^2)\hfill \\ U_3=k({\mathrm{\Omega }}_1^2-{\mathrm{\Omega }}_3^2)\hfill \\ U_4=b(-{\mathrm{\Omega }}_1^2+{\mathrm{\Omega }}_2^2-{\mathrm{\Omega }}_3^2+{\mathrm{\Omega }}_4^2)\hfill \end{array}$$

(5)

where k and b are thrust and drag coefficients, respectively. Ω₁, Ω₂, Ω₃, and Ω₄ represent angular velocity of each rotor. The nonlinear dynamic system of quadrotor can be written as follows ³

$$f(X,U)=\left[\begin{array}{c}\stackrel{·}{\varphi }\hfill \\ \multicolumn{1}{c}{\frac{\stackrel{·}{\varphi }\stackrel{·}{\psi }(I_{\mathrm{yy}}-I_{\mathrm{zz}})}{I_{\mathrm{xx}}}+\stackrel{·}{\theta }\left(\frac{J_r}{I_{\mathrm{xx}}}\right){\mathrm{\Omega }}_r+\left(\frac{l}{I_{\mathrm{xx}}}\right)U_2}\\ \multicolumn{1}{c}{\stackrel{·}{\theta }}\\ \multicolumn{1}{c}{\frac{\stackrel{·}{\varphi }\stackrel{·}{\psi }(I_{\mathrm{zz}}-I_{\mathrm{xx}})}{I_{\mathrm{yy}}}-\stackrel{·}{\varphi }\left(\frac{J_r}{I_{\mathrm{yy}}}\right){\mathrm{\Omega }}_r+\left(\frac{l}{I_{\mathrm{yy}}}\right)U_3}\\ \multicolumn{1}{c}{\stackrel{·}{\psi }}\\ \multicolumn{1}{c}{\frac{\stackrel{·}{\theta }\stackrel{·}{\varphi }(I_{\mathrm{xx}}-I_{\mathrm{yy}})}{I_{\mathrm{zz}}}+\left(\frac{1}{I_{\mathrm{zz}}}\right)U_4}\\ \multicolumn{1}{c}{\stackrel{·}{z}}\\ \multicolumn{1}{c}{g-\left(\cos \varphi \cos \theta \right)\frac{1}{m}{U}_1}\\ \multicolumn{1}{c}{\stackrel{·}{x}}\\ \multicolumn{1}{c}{(\cos \varphi \sin \theta \cos \psi +\sin \varphi \sin \psi )\frac{1}{m}{U}_1}\\ \multicolumn{1}{c}{\stackrel{·}{y}}\\ \multicolumn{1}{c}{(\cos \varphi \sin \theta \sin \psi -\sin \varphi \cos \psi )\frac{1}{m}{U}_1}\end{array}\right]$$

(6)

where m is mass, g is gravity, l is distance, J_r is rotor inertia. I_xx, I_yy, and I_zz are airframe inertia of roll, pitch, and yaw, respectively. Angular rotation is represented by Ω _r . Figure 3 illustrates that the system of angles is independent of the translation system but translation system is dependent on the angles.

Figure 3.

Block diagram of translational and rotational subsystem.

Let us redefine the variables for simplicity as x₁ = φ, $x_2=\stackrel{·}{\phi }$ , x₃ = θ, $x_4=\stackrel{·}{\theta }$ , x₅ = ψ, $x_6=\stackrel{·}{\psi }$ , x₇ = z, $x_8=\stackrel{·}{z}$ , x₉ = x, $x_{10}=\stackrel{·}{x}$ , x₁₁ = y, and $x_{12}=\stackrel{·}{y}$ . Then, considering the external disturbance, the model of quadrotor can be written as follows

$$f(X,U)=\left[\begin{array}{c}{x}_2+d_1\hfill \\ \multicolumn{1}{c}{x_4{x}_6{a}_1+x_4{\Omega }_r{a}_2+b_1{U}_2}\\ \multicolumn{1}{c}{x_4+d_2}\\ \multicolumn{1}{c}{x_2{x}_6{a}_3-x_2{\Omega }_r{a}_4+b_2{U}_3}\\ \multicolumn{1}{c}{x_6+d_3}\\ \multicolumn{1}{c}{x_2{x}_4{a}_5+b_3{U}_4}\\ \multicolumn{1}{c}{x_8+d_4}\\ \multicolumn{1}{c}{g-\frac{U_1}{m}\left(\cos x_1\cos x_3\right)}\\ \multicolumn{1}{c}{x_{10}}\\ \multicolumn{1}{c}{\frac{U_1}{m}(\sin x_1\sin x_5+\cos x_1\sin x_3\cos x_5)}\\ \multicolumn{1}{c}{x_{12}}\\ \multicolumn{1}{c}{\frac{U_1}{m}(\sin x_1\cos x_5+\cos x_1\sin x_3\sin x_5)}\end{array}\right]$$

(7)

where d₁, d₂, d₃, and d₄ represent the mismatched disturbances in roll, pitch, yaw, and altitude axis of quadcopter. a₁, a₂, a₃, a₄, a₅, b₁, b₂, and b₃ are constants given in equation (8). ⁴ The constants in equation (7) are written as follows

$$\begin{array}{c}{a}_1=\frac{I_{\mathrm{yy}}-I_{\mathrm{zz}}}{I_{\mathrm{xx}}}\hfill \\ a_2=\frac{J_r}{I_{\mathrm{xx}}}\hfill \\ a_3=\frac{I_{\mathrm{zz}}-I_{\mathrm{xx}}}{I_{\mathrm{yy}}}\hfill \\ a_4=\frac{J_r}{I_{\mathrm{yy}}}\hfill \\ a_5=\frac{I_{\mathrm{xx}}-I_{\mathrm{yy}}}{I_{\mathrm{zz}}}\hfill \\ b_1=\frac{l}{I_{\mathrm{xx}}}\hfill \\ b_2=\frac{l}{I_{\mathrm{yy}}}\hfill \\ b_3=\frac{1}{I_{\mathrm{zz}}}\hfill \end{array}$$

(8)

The control objective is to develop a control algorithm based on nonsingular TSMC with capabilities of disturbance rejection after integrating it with DO. However, to simplify the control design, only the attitude and height control are considered in this article. The attitude and altitude system of a quadrotor can be written as follows

$$f(X,U)=\left[\begin{array}{c}{x}_2+d_1\hfill \\ \multicolumn{1}{c}{x_4{x}_6{a}_1+x_4{\mathrm{\Omega }}_r{a}_2+b_1{U}_2}\\ \multicolumn{1}{c}{x_4+d_2}\\ \multicolumn{1}{c}{x_2{x}_6{a}_3-x_2{\mathrm{\Omega }}_r{a}_4+b_2{U}_3}\\ \multicolumn{1}{c}{x_6+d_3}\\ \multicolumn{1}{c}{x_2{x}_4{a}_5+b_3{U}_4}\\ \multicolumn{1}{c}{x_8+d_4}\\ \multicolumn{1}{c}{g-\frac{U_1}{m}\left(\cos x_1\cos x_3\right)}\end{array}\right]$$

(9)

For designing the control algorithm based on SMC, we need following Lemmas and Assumptions.

Lemma 1

An affine nonlinear system can be written in a generalized form as follows ³²

$$\begin{array}{c}\stackrel{·}{x}=f\left(x\right)+g_1\left(x\right)u+g_2\left(x\right)d\hfill \\ y=h\left(x\right)\hfill \end{array}$$

(10)

where x ∈ R ⁿ , u ∈ R ^m , d ∈ R ^l , and y ∈ R ^s are the vectors of state, input, disturbance, and output, respectively. The disturbance d is a harmonic disturbance with known frequency and unknown amplitude, generated using an exogenous system give below ³⁰

$$\begin{array}{c}\stackrel{·}{\xi }=A\xi \hfill \\ \multicolumn{1}{c}{d=C\xi }\end{array}$$

(11)

where ξ ∈ R^s is a state of exogenous system, A represents a square matrix, and C represents a row matrix with dimensions of s × s and 1 × s, respectively.

According to Chen, ³³ DO proposed for rejection of disturbance generated from exogenous system (11) is designed as follows

$$\begin{array}{c}\stackrel{·}{z}=[A-l\left(x\right)g_2\left(x\right)C]z+\mathrm{Ap}\left(x\right)-l\left(x\right)\hfill \\ [g_2\left(x\right)\mathrm{Cp}\left(x\right)+f\left(x\right)+g_1\left(x\right)u]\hfill \\ \hat{\xi }=z+p\left(x\right)\hfill \\ \hat{d}=C\hat{\xi }\hfill \end{array}$$

(12)

where z is auxiliary variable, u is control input, $\hat{\xi }$ is the estimation of ξ, and $\hat{d}$ is the disturbance estimation. The variable p(x) and l(x) are calculated as ³³

$$l\left(x\right)=\frac{\partial p\left(x\right)}{\partial x}$$

(13)

and

$$p\left(x\right)={\mathrm{KL}}_f^{r-1}h\left(x\right)$$

(14)

where K is a gain vector, K = [k₁, k₂, …, k_n] ^T , to be determined, L_f represents a Lie derivative of function f(x), and r is relative degree. It has been illustrated in Chen ³³ that the estimation of harmonic disturbance, $\hat{d}$ , yielded by the DO in equation (12) approaches disturbance, d, globally exponentially and satisfies the following inequality

$${\stackrel{·}{V}}_0\left(e_{\xi }\right)=-\delta e_{\xi }^T{e}_{\xi }$$

(15)

where $e_{\xi }=\hat{\xi }-\xi$ , is the disturbance estimate error. δ > 0 is a constant to ensure Lyapunov stability. Furthermore, from equation (15), we know that e_ξ satisfies that||e_ξ|| < ε₀ with ε₀ > 0.

Lemma 2

A first-order sliding mode differentiator is designed as ⁵²

$$\begin{array}{c}{\stackrel{·}{\rho }}_0={\zeta }_0=-{\epsilon }_0|{\rho }_0-f\left(t\right){|}^{\frac{1}{2}}\mathrm{sign}({\rho }_0-f\left(t\right))+{\rho }_1\hfill \\ {\stackrel{·}{\rho }}_1=-{\epsilon }_1\mathrm{sign}({\rho }_1-{\zeta }_0)\hfill \end{array}$$

(16)

where ρ₀, ρ₁, and ζ₀ are representing the states, 0 and 1, are the parameters to be designed for the differentiator and f(t) is a known function. Then, $\stackrel{·}{f}(t)$ , the differential term can be approximated in finite time by ζ₀ given that the initial conditions (ρ₀ − f(t₀)) and (ζ₀ − f(t₀)) are bounded. We can assume that|ε₀ − f(t)| < ε₁ with ε₁ > 0.

Lemma 3

If a positive definite function V (t) satisfies the inequality presented below ⁵³

$$\stackrel{·}{V}\left(t\right)\le -\alpha V^{\eta },\forall t\ge t_0,V\left(t\right)\ge 0$$

(17)

where the constant α and η are taken such that α > 0 while 0 < η < 1. Then, for any given initial condition of t₀, V (t) satisfies the following inequality

$$V^{1-\eta }\left(t\right)\le V^{1-\eta }\left(t_0\right)-\alpha (1-\eta )(t-t_0),t_0\le t\le t_r$$

(18)

and also satisfies

$$V\left(t\right)=0,t\ge t_r$$

(19)

where t_r is given as follows

$$t_r=t_0+\frac{V^{1-\eta }\left(t_0\right)}{\alpha (1-\eta )}$$

(20)

Assumption 1

It is assumed that the disturbance in the system is bounded by a positive constant d*. But for the bound, it is not necessary to be known, d* = sup|d(t)| for t > 0.

Assumption 2

It is assumed that the roll angle, φ and pitch angle, θ lie in the interval $[-\pi /2,\pi /2]$ .

Control design based on disturbance estimation

To design the controller, we first start by assuming f(x) = [x_i + 1a(x)] ^T , g₁(x) = [0 b(x)] ^T , g₂(x) = 1, d(t) = [d 0] ^T , and h(x) = x₁; therefore, nonlinear system with mismatched disturbance can be derived from equation (10) as follows

$$\begin{array}{c}{\stackrel{·}{x}}_i=x_{i+1}+d\left(t\right)\hfill \\ {\stackrel{·}{x}}_n=a\left(x\right)+b\left(x\right)u\hfill \\ y=x_1\hfill \end{array}$$

(21)

where n represents the order of the system while i is a positive integer such that i = 1, 2, 3, …, n − 1. The states are represented by x_i, where i = 1, 2, 3, …, n and y is the output. Disturbance and input is represented by d(t) and u, respectively. a(x) is the function of x_i and it represents states of the system containing x = [x₁, x₂, …, x_n] ^T . b(x) is a nonzero function of x. Steps needed for designing a controller incorporated with DO are as follows:

1. Design a controller for the required parameters of stability and achieve specifications of needed performance with the supposition that disturbance is measurable.

2. Design a DO for estimation of disturbance.

3. Integrate DO with controller by replacing the term of disturbance in controller with estimated disturbance.

Control design of SMC

From equation (21), the second-order system with mismatched disturbance can be written as follows

$$\begin{array}{c}{\stackrel{·}{x}}_1=x_2+d\left(t\right)\hfill \\ {\stackrel{·}{x}}_2=a\left(x\right)+b\left(x\right)u\hfill \\ y=x_1\hfill \end{array}$$

(22)

In this article, TSMC synthesis is considered. But, while constructing a conventional terminal sliding mode controller, a problem of singularity is faced. In order to overcome the singularity problem, in Feng et al., ⁴⁹ a sliding mode surface is proposed as $s=x_1+(1/\beta )x_2^{p/q}$ . Based on this sliding mode surface, a modified dynamic nonsingular sliding mode surface is proposed for the second-order system (22) as follows

$$s=x_1+\frac{1}{\beta }(x_2+\hat{d}{)}^{\frac{p}{q}}$$

(23)

where β, p, and q are defined such that β > 0, p and q are positive odd integers designed such that 1 < p/q < 2. While $\hat{d}$ represents the disturbance estimation. Equation (23) can be written in standard form as follows

$$s={\tilde{x}}_1+\frac{1}{\beta }{\tilde{x}}_2^{\frac{p}{q}}$$

(24)

where

$${\tilde{x}}_1=x_1$$

(25)

$${\tilde{x}}_2=x_2+\hat{d}$$

(26)

For a nonlinear system in equation (22), the proposed DO-based SMC is constructed as follows

$$u=-b^{-1}\left(x\right)[\frac{1}{\rho }{\tilde{x}}_2^{2-\frac{p}{q}}+a\left(x\right)+{\zeta }_{\hat{d}}+\mathrm{Lsgn}\left(s\right)]$$

(27)

where L > 0 is the designing constant, ${\zeta }_{\hat{d}}$ is the approximation of disturbance estimation derivative obtained from using the first-order differentiator in equation (16), and the constant ρ is written as follows

$$\rho =\frac{p}{q\beta }$$

(28)

For the initial conditions as well as with the design of L such that L > 0, the control proposed in equation (27) will converge the states to zero in finite time. After constructing the SMC, it is needed to ensure the stability of the controller. To proceed, take derivative of sliding surface presented in equation (23) as

$$\stackrel{·}{s}={\stackrel{·}{x}}_1+\frac{p}{q\beta }(x_2+\hat{d}{)}^{\frac{p}{q}-1}({\stackrel{·}{x}}_2+\stackrel{·}{\hat{d}})$$

(29)

Invoking equation (22), (26), and (28) yields

$$\stackrel{·}{s}=(x_2+d)+\rho {\tilde{x}}_2^{\frac{p}{q}-1}(a\left(x\right)+b\left(x\right)u+\widehat{\stackrel{·}{d}})$$

(30)

The derivative of estimated DO is obtained using differentiator presented in Lemma 2; however, an error is incurred during approximation. Therefore, the derivative is written as $\hat{d}={\zeta }_{\hat{d}}+{\epsilon }_{\hat{d}}$ . Hence, equation (30) can be written as follows

$$\begin{array}{c}\begin{array}{c}\stackrel{·}{s}=(x_2+d)+\rho {\tilde{x}}_2^{\frac{p}{q}-1}(a\left(x\right)+b\left(x\right)u+{\zeta }_{\hat{d}}+{\epsilon }_{\hat{d}})\hfill \end{array}\hfill \end{array}$$

(31)

Invoking equation (27) yields

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{s}}& =(x_2+d)+\rho {\tilde{x}}_2^{\frac{p}{q}-1}\hfill \\ \multicolumn{1}{c}{}& (a\left(x\right)+b\left(x\right)(-b^{-1}\left(x\right)[\frac{1}{\rho }{\tilde{x}}_2^{2-\frac{p}{q}}+a\left(x\right)+{\zeta }_d+\mathrm{Lsgn}\left(s\right)])+{\zeta }_{\hat{d}}+{\epsilon }_{\hat{d}})\hfill \\ \multicolumn{1}{c}{}& =(x_2+d)+\rho {\tilde{x}}_2^{\frac{p}{q}-1}(-\frac{1}{\rho }{\tilde{x}}_2^{2-\frac{p}{q}}-\mathrm{Lsgn}\left(s\right)+{\epsilon }_{\hat{d}})\hfill \\ \multicolumn{1}{c}{}& =(x_2+d)-(x_2+\hat{d})-\rho (x_2+\hat{d}{)}^{\frac{p}{q}-1}(\mathrm{Lsgn}\left(s\right)-{\epsilon }_{\hat{d}})\hfill \\ \multicolumn{1}{c}{}& =(d-\hat{d})-\rho (x_2+\hat{d}{)}^{\frac{p}{q}-1}(\mathrm{Lsgn}\left(s\right)-{\epsilon }_{\hat{d}})\hfill \\ \multicolumn{1}{c}{}& =\tilde{d}-\rho (x_2+\hat{d}{)}^{\frac{p}{q}-1}(\mathrm{Lsgn}\left(s\right)-{\epsilon }_{\hat{d}})\hfill \\ \multicolumn{1}{c}{}& =\tilde{d}-\rho {\tilde{x}}_2^{\frac{p}{q}-1}(\mathrm{Lsgn}\left(s\right)-{\epsilon }_{\hat{d}})\hfill \end{array}$$

(32)

where $\tilde{d}=e_d=d-\hat{d}$ . From Lemma 1, $\tilde{d}<{\epsilon }_0$ and from Lemma 2, it is known that ${\epsilon }_{\hat{d}}<{\epsilon }_1$ with ε₁ > 0. Now, define a Lyapunov candidate as follows

$$V_1\left(t\right)=\frac{1}{2}{s}^2$$

(33)

Taking time derivative yields

$${\stackrel{·}{V}}_1\left(t\right)=s\stackrel{·}{s}$$

(34)

Invoking equation (32) yields

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{V}}_1\left(t\right)}& =s[\tilde{d}-\rho {\tilde{x}}_2^{\frac{p}{q}-1}(\mathrm{Lsgn}\left(s\right)-{\epsilon }_d)]\hfill \\ \multicolumn{1}{c}{}& \le \left|s\right|{\epsilon }_0-L\rho {\tilde{x}}_2^{\frac{p}{q}-1}\left|s\right|+\rho {\tilde{x}}_2^{\frac{p}{q}-1}{\epsilon }_1\left|s\right|\hfill \\ \multicolumn{1}{c}{}& \le -\left|s\right|[L\rho -\rho {\epsilon }_1]{\tilde{x}}_2^{\frac{p}{q}-1}+\left|s\right|{\epsilon }_0\hfill \\ \multicolumn{1}{c}{}& \le -[L^{*}{\tilde{x}}_2^{\frac{p}{q}-1}-{\epsilon }_0]\left|s\right|\hfill \\ \multicolumn{1}{c}{}& \le -[L^{*}{\tilde{x}}_2^{\frac{p}{q}-1}-\Gamma ]\left|s\right|\hfill \end{array}$$

(35)

where L* = Lρ − ρε₁ and Γ > ε₀. Now, a constant can be designed for Lyapunov stability such that $L^{**}>L^{*}\rho {\tilde{x}}_2^{q^{p\_-1}}-\Gamma$ . Moreover, following the proof in Yang et al., ⁵⁴ by designing sufficiently large gain constant L*, the states will be attracted toward the sliding surface. It is known that p and q are odd constants with the condition of 1 < p/q < 2, then we have ${\tilde{x}}_2^{\frac{p}{q}}-1>0$ for nonzero x₂. Assuming $H(x_2,\hat{d})=L\rho {\tilde{x}}_2^{\frac{p}{q}}-\Gamma$ yields

$${\stackrel{·}{V}}_1\left(t\right)=s\stackrel{·}{s}\le -H(x_2,d)\left|s\right|$$

(36)

Therefore, Lyapunov stability is ensured for the case of x₂ ≠ 0. The states of the nonlinear system can reach sliding surface s = 0 is reached asymptotically. This can be proved during the second case when x₂ = 0. Invoking equation (27), equation (22) yields

$${\stackrel{·}{x}}_2=-\frac{1}{\rho }{\tilde{x}}_2^{2-\frac{p}{q}}-{\zeta }_{\hat{d}}-\mathrm{Lsgn}\left(s\right)$$

(37)

Substituting x₂ = 0 in ${\tilde{x}}_2=x_2+\hat{d}=\hat{d}$ yields

$${\stackrel{·}{x}}_2=-\frac{1}{\rho }{\hat{d}}^{2-\frac{p}{q}}-{\zeta }_{\hat{d}}-\mathrm{Lsgn}\left(s\right)$$

(38)

$${\stackrel{·}{x}}_2=-(\frac{1}{\rho }{\hat{d}}^{2-\frac{p}{q}}+{\zeta }_{\hat{d}}+\mathrm{Lsgn}\left(s\right))$$

(39)

Using the knowledge in Yang et al., ⁵⁴ let us define a bound D as $D\ge (1/\rho ){\hat{d}}^{2-\frac{\rho }{q}}+{\zeta }_{\hat{d}}$ Invoking yields

$${\stackrel{·}{x}}_2\le -(D+\mathrm{Lsgn}\left(s\right))$$

(40)

Furthermore, following the procedure of proof in Feng et al., ⁴⁹ for sufficiently large L > 0, it is derived that ${\stackrel{·}{x}}_2\le -L$ and ${\stackrel{·}{x}}_2\ge L$ for s > 0 and s < 0, respectively. It means that x₂ = 0 is not an attractor. It can also be said that for a region nearby x₂ = 0, that is,|x₂| < δ, the aforementioned condition still exists, that is, for s > 0 and s < 0, it can be said that x₂ ≤ −L and x₂ ≥ L, respectively. Therefore, the crossing for the trajectory for s > 0 is from x₂ = δ to x₂ = −δ and vice versa for s < 0. For the region outside the vicinity of x₂ > δ, it can be concluded from equation (35) that the switching surface s > 0 can be reached asymptotically. Once the switching surface is reached, it can be concluded that the equilibrium point can be reached in finite time because the structure of sliding surface chosen in equation (24) is similar to the standard sliding surface.

Stability analysis of DO-based SMC

In this section, stability analysis of DO-based SMC is presented. Let us define a Lyapunov candidate function as follows

$$\begin{array}{c}\multicolumn{1}{c}{V_T\left(t\right)=V_1\left(t\right)+V_0\left(t\right)}\\ \multicolumn{1}{c}{=\frac{1}{2}{s}^2+V_0\left(t\right)}\end{array}$$

(41)

V ₁(t) and V₀(t) are suggested in equation (33) and Lemma 1, respectively. The former Lyapunov candidate is utilized for the stability of SMC controller, while the latter is for stability of DO. Taking time derivative of equation (41) followed by invoking equations (35) and (15) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_T\left(t\right)\le -(L^{*}{\tilde{x}}_2^{\frac{p}{q}-1}\left|s\right|-\Gamma )-\delta e_{\xi }^T{e}_{\xi }}\\ \multicolumn{1}{c}{\le -(L^{*}{\tilde{x}}_2^{\frac{p}{q}-1}\left|s\right|-\Gamma +\delta e_{\xi }^T{e}_{\xi })}\end{array}$$

(42)

The stability for the functions involved in equation (42) is presented in their respective sections, that is, the Lyapunov stability for DO is proved in Chen, ³³ Lemma 1. While the stability of SMC is derived and shown in equations (35) and (40). Once the sliding mode surface is approached, we have s = 0, invoking the system dynamics in (23) yields

$$s=x_1+\frac{1}{\beta }(x_2+\hat{d}{)}^{\frac{p}{q}}=x_1+\frac{1}{\beta }{\stackrel{·}{x}}_1=0$$

(43)

Now, it can be derived that

$${\stackrel{·}{x}}_1=-\beta x_1$$

(44)

where β > 0.

Furthermore, from equation (35), it can be stated that $s\stackrel{·}{s}<0$ . Moreover, for a Lyapunov candidate $V_T(t)=V_1(t)+V_0(t)$ , it has been shown that ${\stackrel{·}{V}}_T(t)\le -(L*{\tilde{x}}_2^{q-1}-\Gamma +\delta e_{\xi }^T{e}_{\xi })$ and it can be stated that |s|^α ≤ 1 + |s|, where 0 < α < 1. Therefore, it can be concluded that for the chosen Lyapunov candidate function and the aforementioned condition on s, the states x₁ and ${\tilde{x}}_2=x_2+\hat{d}$ will not escape the manifold. Hence, it can be concluded that by designing appropriate gain constants, DO-based SMC is asymptotically stable. This completes the proof.

SMC with DO for quadrotor

UAV, quadrotor, is highly nonlinear under-actuated system. The model of quadrotor needs the controller for roll, pitch, yaw, and altitude simultaneously. The first controlling part of quadrotor is roll control based on DO which is presented as follows.

DO-based roll control

From a state space model of quadrotor in equation (7), the second-order subsystem of roll model is written where as follows

$$\begin{array}{c}{\stackrel{·}{x}}_1=x_2+d_1\hfill \\ {\stackrel{·}{x}}_2=x_4{x}_6{a}_1+x_4{\mathrm{\Omega }}_r{a}_2+b_1{U}_2\hfill \end{array}$$

(45)

a ₁ and a₂ are constants given in equation (7), and Ω _r is quadrotor parameter.

A generalized form of DO is presented in equation (12), which requires the calculations for p(x) and l(x). In the first step, it is needed to design a vector p(x). For roll control, using equation (13) and invoking equation (14), it is designed as follows

$$p\left(x\right)=K_1\frac{\partial \left(x_1\right)}{\partial x}\left[\begin{array}{c}{x}_2\hfill \\ \multicolumn{1}{c}{x_4{x}_6{a}_1+x_4{\mathrm{\Omega }}_r{a}_2}\end{array}\right]$$

(46)

$$p\left(x\right)=K_1\left[\begin{array}{cc}1\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{x}_2\hfill \\ \multicolumn{1}{c}{x_4{x}_6{a}_1+x_4{\mathrm{\Omega }}_r{a}_2}\end{array}\right]$$

(47)

$$p\left(x\right)=K_1{x}_2$$

(48)

where K₁ is a matrix with design constants k₁ and k₂,that is, K₁ = [k₁k₂]^T. The harmonic disturbance considered for the article is presented in equation (11). The matrices A₁ and C₁ cccrm as follows

$$\begin{array}{cc}{A}_1=\hfill & \left[\begin{array}{cc}0\hfill & a_{1_{12}}\hfill \\ \multicolumn{1}{c}{a_{1_{21}}}& 0\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{C_1=}& \left[\begin{array}{cc}{c}_{1_{11}}\hfill & 0\hfill \end{array}\right]\hfill \end{array}$$

(49)

where a₁₁₁ = a₁₂₂ = 0. a₁₁₂ and a₁₂₁ are constants such that a₁₁₂ > 0 and a₁₂₁ < 0, respectively. c₁₁₁ represents a positive constant. Now, invoking equations (49) and (48) in equation (12), the obtained DO is written as

$$\begin{array}{c}{\stackrel{·}{z}}_{\varphi }=\left[\begin{array}{c}{a}_{1_{12}}{z}_2+a_{1_{12}}{x}_2+a_{1_{12}}{k}_2-x_4{x}_6{a}_1-x_4{\mathrm{\Omega }}_r{a}_2-U_{\varphi }\hfill \\ \multicolumn{1}{c}{a_{1_{21}}{z}_1+a_{1_{21}}{x}_2+a_{1_{21}}{k}_1-x_4{x}_6{a}_1-x_4{\mathrm{\Omega }}_r{a}_2-U_{\varphi }}\end{array}\right]\hfill \\ {\hat{\xi }}_{\varphi }=z_{\varphi }+K_1{x}_2\hfill \\ {\hat{d}}_{\varphi }=C_1\hat{\xi }\hfill \end{array}$$

(50)

Resulting in ${\hat{d}}_{\phi }={\hat{\xi }}_1$ . It should be noted that, for simplicity, it is assumed U₂ = U_φ and ${\hat{d}}_2={\hat{d}}_{\phi }$ . Now for designing the controller based on SMC, a sliding surface is proposed based on equation (23) as follows

$$s_{\varphi }=x_1+\frac{1}{\beta }(x_2+d_{\varphi }{)}^{\frac{p}{q}}$$

(51)

Taking time derivative and invoking equation (45) yields

$${\stackrel{·}{s}}_{\varphi }={\tilde{x}}_2+{\rho }_1{\tilde{x}}_2^{\frac{p}{q}-1}[x_4{x}_6{a}_1+x_4{\mathrm{\Omega }}_r{a}_2+b_1{U}_2+\stackrel{·}{\hat{d}}]$$

(52)

Now setting $\stackrel{·}{s}$ equal to zero results in a roll controller, U₂ = U_φ derived as follows

$$U_{\varphi }=-\frac{1}{b_1}[\frac{1}{{\rho }_1}{\tilde{x}}_2^{2-\frac{p}{q}}+x_4{x}_6{a}_1+x_4{\mathrm{\Omega }}_r{a}_2+d_{\varphi }+L_1\mathrm{sgn}\left(s_{\varphi }\right)]$$

(53)

${\hat{d}}_{\phi }$ is an output from DO and $\hat{d}$ is derivative of estimated DO which is estimated using sliding mode differentiator presented in Lemma 2. Hence, the roll controller is derived as follows

$$U_{\varphi }=-\frac{1}{b_1}[\frac{1}{{\rho }_1}{\tilde{x}}_2^{2-\frac{p}{q}}+x_4{x}_6{a}_1+x_4{\mathrm{\Omega }}_r{a}_2+{\zeta }_{{\hat{d}}_{\varphi }}+L_1\mathrm{sgn}\left(s_{\varphi }\right)]$$

(54)

DO-based pitch control

The state space mode of pitch control is taken from the model of quadrotor in equation (7)

$$\begin{array}{c}{\stackrel{·}{x}}_3=x_4+d_2\hfill \\ {\stackrel{·}{x}}_4=x_2{x}_6{a}_3-x_2{\mathrm{\Omega }}_r{a}_4+b_2{U}_3\hfill \end{array}$$

(55)

For pitch control, it is assumed that A₂ = [0 a₂₁₂; a₂₂₁ 0] and C₂ = [c₂₁₁ 0]. Following similar procedure of roll control in the previous section, DO based on pitch control is derived as follows

$$\begin{array}{c}{\stackrel{·}{z}}_{\theta }=\left[\begin{array}{c}{a}_{2_{12}}{z}_1+a_{2_{12}}{x}_4+a_{2_{12}}{k}_3-x_4{x}_6{a}_3+x_2{\mathrm{\Omega }}_r{a}_4-b_2{U}_{\theta }\hfill \\ \multicolumn{1}{c}{a_{2_{21}}{z}_2+a_{2_{21}}{x}_4+a_{2_{21}}{k}_4-x_4{x}_6{a}_1-x_4{\mathrm{\Omega }}_r{a}_2-b_2{U}_{\varphi }}\end{array}\right]\hfill \\ {\hat{\xi }}_{\theta }=z_{\theta }+K_2{x}_4\hfill \\ {\hat{d}}_{\theta }=C_2\xi \hfill \end{array}$$

(56)

Resulting in ${\hat{d}}_{\theta }={\hat{\xi }}_3$ . For pitch control, observer gain constant K₂ consists of k₃ and k₄, that is, K₂ = [k₃k₄] ^T . Furthermore, the sliding surface proposed for pitch control is

$$s_{\theta }=x_3+\frac{1}{\beta }(x_4+{\hat{d}}_{\theta }{)}^{\frac{p}{q}}$$

(57)

Following the procedure presented in roll controller, the controller for pitch is derived as follows

$$U_{\theta }=-\frac{1}{b_2}[\frac{1}{{\rho }_2}{\tilde{x}}_4^{2-\frac{p}{q}}+x_2{x}_6{a}_3-x_2{\mathrm{\Omega }}_r{a}_4+{\zeta }_{{\hat{d}}_{\theta }}+L_2\mathrm{sgn}\left(s_{\theta }\right)]$$

(58)

where ${\varsigma }_{{\hat{d}}_{\theta }}$ is an output from first-order sliding mode differentiator representing the estimation of ${\hat{d}}_{\theta }$ .

DO-based yaw control

From equation (7), the state space model for yaw is

$$\begin{array}{c}{\stackrel{·}{x}}_5=x_6+d_3\hfill \\ {\stackrel{·}{x}}_6=x_2{x}_4{a}_5+b_3{U}_4\hfill \end{array}$$

(59)

For yaw control, it is assumed that A₃ = [0 $a_{3_{12}}$ ; $a_{3_{10}}$ ] and C₃ = [c₃₁₁ 0]. Therefore, DO for yaw control is calculated as

$$\begin{array}{c}{\stackrel{·}{z}}_{\psi }=\left[\begin{array}{c}{a}_{3_{12}}{z}_1+a_{3_{12}}{x}_6+a_{3_{12}}{k}_5-x_2{x}_4{a}_5-b_3{U}_{\psi }\hfill \\ \multicolumn{1}{c}{a_{3_{21}}{z}_2+a_{3_{21}}{x}_6+a_{3_{21}}{k}_6-x_2{x}_4{a}_5-b_3{U}_{\psi }}\end{array}\right]\hfill \\ {\xi }_{\psi }=z_{\psi }+K_3{x}_6\hfill \\ {\hat{d}}_{\psi }=C_3\xi \hfill \end{array}$$

(60)

Resulting in ${\hat{d}}_{\psi }={\hat{\xi }}_5$ . Observer gain K₃ consists of k₅ and k₆, that is, K₃ = [k₅ derive k₆]^T. Next step is to controller, sliding surface for yaw is proposed as follows

$$s_{\theta }=x_5+\frac{1}{\beta }(x_6+{\hat{d}}_{\psi }{)}^{\frac{p}{q}}$$

(61)

SMC for yaw controller for the proposed sliding surface is calculated as

$$U_{\psi }=-\frac{1}{b_3}[\frac{1}{{\rho }_3}{\tilde{x}}_6^{2-\frac{p}{q}}+x_2{x}_4{a}_5+{\zeta }_{d_{\psi }}+L_3\mathrm{sgn}\left(s_{\psi }\right)]$$

(62)

where ${\varsigma }_{{\hat{d}}_{\psi }}$ is an output from first-order sliding mode differentiator representing the estimation of ${\hat{d}}_{\psi }$ .

DO-based altitude control

From equation (7), the state space model for altitude is

$$\begin{array}{c}{\stackrel{·}{x}}_7=x_8+d_4\hfill \\ {\stackrel{·}{x}}_8=g-\frac{U_1}{m}\left(\cos x_1\cos x_3\right)\hfill \end{array}$$

(63)

For yaw control, it is assumed that A₄ = [0 a₄₁₂; a₄₂₁ 0] and C₄ = [c₄₁₁ 0]. Hence, the DO for altitude controller is derived as

$$\begin{array}{c}{\stackrel{·}{z}}_p=\left[\begin{array}{c}{a}_{4_{12}}{z}_2+a_{4_{12}}{x}_8+a_{4_{12}}{k}_8+g-\frac{U_p}{m}\cos \left(x_1\right)\cos \left(x_3\right)\hfill \\ \multicolumn{1}{c}{a_{4_{21}}{z}_1+a_{4_{21}}{x}_7+a_{4_{21}}{k}_7+g-\frac{U_p}{m}\cos \left(x_1\right)\cos \left(x_3\right)}\end{array}\right]\hfill \\ {\xi }_p=z+K_4{x}_8\hfill \\ {\hat{d}}_p=C_4\xi \hfill \end{array}$$

(64)

Resulting in ${\hat{d}}_p={\hat{\xi }}_7$ . Observer gain K₄ for altitude consists of k₇ and k₈, that is, K₄ = [k₇k₈] ^T . Next step is to design controller for altitude. A sliding surface for altitude is taken as

$$s_p=x_7+\frac{1}{\beta }(x_8+{\hat{d}}_p{)}^{\frac{p}{q}}$$

(65)

SMC for altitude is given as follows

$$U_p=-\frac{m}{\cos x_1\cos x_3}[\frac{1}{{\rho }_4}{\tilde{x}}_8^{2-\frac{p}{q}}-g+{\zeta }_{d_p}+L_4\mathrm{sgn}\left(s_4\right)]$$

(66)

where ${\varsigma }_{{\hat{d}}_p}$ is an output from first-order sliding mode differentiator representing the estimation of ${\hat{d}}_p$ .

Stability analysis of quadrotor

Now, for stability analysis of complete quadrotor altogether, let us define a Lyapunov function as follows

$$V=V_{\varphi }+V_{\theta }+V_{\psi }+V_p$$

(67)

where V_φ, V_θ, V_ψ, and V_p represent Lyapunov functions for roll, pitch, yaw, and altitude, respectively. Assuming and invoking the Lyapunov function of roll, pitch, yaw, and height yields

$$\begin{array}{c}V=\frac{1}{2}(s_{\varphi }^2+s_{\theta }^2+s_{\psi }^2+s_p^2)\hfill \end{array}$$

(68)

Taking derivative of equation (68)

$$\begin{array}{c}\stackrel{·}{V}=s_{\varphi }{\stackrel{·}{s}}_{\varphi }+s_{\theta }{\stackrel{·}{s}}_{\theta }+s_{\psi }{\stackrel{·}{s}}_{\psi }+s_p{\stackrel{·}{s}}_p\hfill \end{array}$$

(69)

Utilizing equations (35) and (42), it can be written

$$\begin{array}{c}{\stackrel{·}{V}}_{\varphi }=s_{\varphi }{\stackrel{·}{s}}_{\varphi }=-((L_1^{*}{\tilde{x}}_2^{\frac{p}{q}-1}-{\Gamma }_1)\left|s_{\varphi }\right|+{\delta }_{\varphi }{e}_{{\xi }_{\varphi }}^T{e}_{{\xi }_{\varphi }})\hfill \\ {\stackrel{·}{V}}_{\theta }=s_{\theta }{\stackrel{·}{s}}_{\theta }=-((L_2^{*}{\tilde{x}}_4^{\frac{p}{q}-1}-{\Gamma }_2)\left|s_{\theta }\right|+{\delta }_{\theta }{e}_{{\xi }_{\theta }}^T{e}_{{\xi }_{\theta }})\hfill \\ {\stackrel{·}{V}}_{\psi }=s_{\psi }{\stackrel{·}{s}}_{\psi }=-((L_3^{*}{\tilde{x}}_6^{\frac{p}{q}-1}-{\Gamma }_3)\left|s_{\psi }\right|+{\delta }_{\psi }{e}_{{\xi }_{\psi }}^T{e}_{{\xi }_{\psi }})\hfill \\ {\stackrel{·}{V}}_p=s_p{\stackrel{·}{s}}_p=-((L_4^{*}{\tilde{x}}_8^{\frac{p}{q}-1}-{\Gamma }_4)\left|s_p\right|+{\delta }_p{e}_{{\xi }_p}^T{e}_{{\xi }_p})\hfill \end{array}$$

(70)

where $L_1^{*}=L_1{\rho }_1-{\rho }_1{\epsilon }_1$ , $L_2^{*}=L_2{\rho }_2-{\rho }_2{\epsilon }_2$ , $L_3^{*}=L_3{\rho }_3-{\rho }_3{\epsilon }_3$ , and $L_4^{*}=L_4{\rho }_4-{\rho }_4{\epsilon }_4$ . It should be noted that, ρ₁, ρ₂, ρ₃, and ρ₄ represent the sliding mode surface parameters derived using equation (28). And ε₁, ε₂, ε₃, and ε₄ represent the bound on the disturbance as stated in Lemma 2. Furthermore, from Lemma 1, δ_φ, δ_θ, δ_ψ, and δ_p represent positive real constants. Now, invoking in equation (69) yields

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}=-((L_1^{*}{\tilde{x}}_2^{\frac{p}{q}-1}-{\Gamma }_1)\left|s_{\varphi }\right|+{\delta }_{\varphi }{e}_{{\xi }_{\varphi }}^T{e}_{{\xi }_{\varphi }})-((L_2^{*}{\tilde{x}}_4^{\frac{p}{q}-1}-{\Gamma }_2)\left|s_{\theta }\right|+{\delta }_{\theta }{e}_{{\xi }_{\theta }}^T{e}_{{\xi }_{\theta }})}\\ \multicolumn{1}{c}{-((L_3^{*}{\tilde{x}}_6^{\frac{p}{q}-1}-{\Gamma }_3)\left|s_{\psi }\right|+{\delta }_{\psi }{e}_{{\xi }_{\psi }}^T{e}_{{\xi }_{\psi }})-((L_4^{*}{\tilde{x}}_8^{\frac{p}{q}-1}-{\Gamma }_4)\left|s_p\right|+{\delta }_p{e}_{{\xi }_p}^T{e}_{{\xi }_p})}\end{array}$$

(71)

Simplifying yields

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}=-((L_1^{*}{\tilde{x}}_2^{\frac{p}{q}-1}-{\Gamma }_1)\left|s_{\varphi }\right|+{\delta }_{\varphi }{e}_{{\xi }_{\varphi }}^T{e}_{{\xi }_{\varphi }})-((L_2^{*}{\tilde{x}}_4^{\frac{p}{q}-1}-{\Gamma }_2)\left|s_{\theta }\right|+{\delta }_{\theta }{e}_{{\xi }_{\theta }}^T{e}_{{\xi }_{\theta }})}\\ \multicolumn{1}{c}{-((L_3^{*}{\tilde{x}}_6^{\frac{p}{q}-1}-{\Gamma }_3)\left|s_{\psi }\right|+{\delta }_{\psi }{e}_{{\xi }_{\psi }}^T{e}_{{\xi }_{\psi }})-((L_4^{*}{\tilde{x}}_8^{\frac{p}{q}-1}-{\Gamma }_4)\left|s_p\right|+{\delta }_p{e}_{{\xi }_p}^T{e}_{{\xi }_p})}\end{array}$$

(72)

Hence, following the similar procedure of proof presented for a general nonlinear system, using Lemma 1 and the proofs in Feng et al. ⁴⁹ and Yang et al., ⁵⁴ the Lyapunov stability is satisfied. This completes the proof.

Simulation results

The parameters used for quadrotor are shown in Table 1. All the parameters used are converted to SI International system to have same units.

Table 1.

Quadrotor parameters.

Parameter	Symbol	Value	Unit
Gravity	g	9.81	m/s2
Mass	m	0.468	kg
Distance	l	0.225	M
Thrust coefficient	k	2.980 × 10−6
Drag coefficient	b	1.140 × 10−7
Rotor inertia	IM	3.357 × 10−5	kg m2
Airframe inertia of roll	Ixx	4.856 × 10−3	kg m2
Airframe inertia of pitch	Iyy	4.856 × 10−3	kg m2
Airframe inertia of yaw	Izz	8.801 × 10−3	kg m2
Drag force coefficient in roll direction	Ax	0.25	kg/s
Drag force coefficient in pitch direction	Ay	0.25	kg/s
Drag force coefficient in yaw direction	Az	0.25	kg/s

For simulation study, the harmonic disturbance considered for quadrotor is given below

$$\begin{array}{c}d\left(t\right)=\frac{1}{\tau }\sin ({\omega }_{j}t+1)\hfill \end{array}$$

(73)

where j = 1, 2, 3, 4. τ decides the amplitude of the disturbance and ω_j represents natural frequency of the disturbance. However, an exogenous disturbance has known frequency but unknown amplitude. ³⁰ Therefore, in simulation study, the amplitude of disturbance is varied in each channel while keeping frequency constant. Matrices A and C required in equation (11) are derived from equation (73) as follows

$$\begin{array}{c}{A}_j=\left[\begin{array}{cc}0\hfill & {\omega }_j\hfill \\ \multicolumn{1}{c}{-{\omega }_j}& 0\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{C=\left[\begin{array}{c}10\hfill \end{array}\right]}\end{array}$$

(74)

It should be noted down that matrices A and C are utilized for the derivation of enhanced exogenous DO only. During performing simulations, the time-dependent function of disturbance given in equation (73) is used. Furthermore, for designing the controller p > 0 and q > 0 are odd integers chooses such that 1 < p/q < 2 shown in Table 2. Using these design parameters, controller, and DO, the quadrotor model is simulated in MATLAB, and the following results are obtained for roll, pitch, yaw, and height. The simulation results show the three stabilized angles and their controllers output as well as the disturbance estimation for each of the parameter. The initial conditions for the angles prior to the presence of disturbance are assumed to be zero, that is, φ(0) = 0, θ(0) = 0, and ψ(0) = 0. For height, it is assumed that the quadrotor is initially in the air at the height of 15 m above the ground level, that is, z(0) = 15.

Table 2.

SMC and DO parameters.

Parameters	Roll	Pitch	Yaw	Altitude
p	7	7	7	13
q	5	5	5	7
β	15	20	15	5
Switching gain, L	180	1	280	215
Disturbance amplitude, τ	4	2	1	2
DO constant, k1	10	10	10	10
DO constant, k2	10	10	10	10

SMC: sliding mode control; DO: disturbance observer.

Simulation results of roll

Simulation results of roll are shown in Figures 4–6.

Figure 4.

Output of roll angle.

Figure 5.

Output of roll controller.

Figure 6.

Output of disturbance observer for roll angle.

Simulation results of pitch

Simulation results of pitch are shown in Figures 7–9.

Figure 7.

Output of pitch angle.

Figure 8.

Output of pitch controller.

Figure 9.

Output of disturbance observer for pitch angle.

Simulation results of yaw

Simulation results of yaw are shown in Figures 10–12.

Figure 10.

Output of yaw angle.

Figure 11.

Output of yaw controller.

Figure 12.

Output of disturbance observer for yaw angle.

Simulation results of altitude/height

Simulation results of altitude/height are shown in Figures 13–15.

Figure 13.

Altitude/height.

Figure 14.

Output of altitude controller.

Figure 15.

Output of disturbance observer for altitude/height.

Conclusion

In this research work, a modified SMC is proposed for achieving control performance of UAV, quadrotor. However, to simplify the control design, only the roll, pitch, yaw, and height control are considered. In the models of each system, mismatched exogenous disturbance has been considered for which DO is designed to estimate the disturbance. During the DO-based SMC design, two problems are faced. First, a derivative of DO is present in controller for which the states are not available. Second, singularity in the control design. The first problem is tackled by utilizing the first-order sliding mode differentiator, and for the second problem, the sliding mode surface is modified to avoid the singularity problem. After constructing the control scheme, the simulations are performed in MATLAB/Simulink tool for controller as well as DO showing that the proposed controller is effective, stable, and the performance of the system has been achieved asymptotically. Moreover, the presented research work is centered across the stabilization of quadrotor; therefore, this work can be expanded for tracking control of quadrotor.