Nonlinear Extended State Observer Based Prescribed Performance Control for Quadrotor UAV with Attitude and Input Saturation Constraints

Abstract

In this paper, a prescribed performance control scheme of the quadrotor unmanned aerial vehicle (UAV) under attitude and input saturation constraints is introduced. According to the underactuated feature, the quadrotor UAV system can be decomposed into an underactuated subsystem and a fully actuated subsystem. With the feedback linearization technique, a single nonlinear extended state observer (ESO) is proposed, and multiple observations are utilized to estimate both matched and unmatched disturbances, which not only can obtain a uniform convergence, but also reduces the complexity of the observer’s parameter adjustment. To improve system stability, an input saturation algorithm for each single rotor is introduced to modify the final control output. In addition, the limited attitude for the quadrotor UAV is also considered as a saturation constraint in the control scheme with a compensation auxiliary system. On this basis, dynamic surface control (DSC) with prescribed performance is adopted to guarantee the bounded convergence and steady-state error. All state errors of the closed-loop system are proven to be uniformly bounded using the Lyapunov theory, and the simulation results are given to demonstrate the stability, effectiveness, and superiority of the proposed control strategies at last.

1. Introduction

Unmanned aerial vehicles (UAVs) have attracted significant attention from research institutes, companies and industries over the last decade, owing to their wide applications in environment mapping, safety monitoring, disaster preventive management, service delivery, etc. [1]. The quadrotor UAV is the most widely used vertical take-off and landing aircraft with flexible maneuverability.

For the quadrotor UAV, the highly nonlinear underactuated system has led to essential challenges in designing effective flight controller with robust stability and high performance. Many linear techniques have been designed to reach a full control of the quadrotor UAV, such as the cascade PID control [2,3] and LQR control [4]. However, for such a highly nonlinear system, the linearized control methods have great limitations for the analysis of system robustness and anti-interference ability. Therefore, to overcome the limitations of the linear control methods, nonlinear control methods, such as feedback linearization [5,6,7], backstepping approach [8,9], sliding mode control (SMC) [10,11,12], observer-based robust control [13,14,15] or some adaptive control approaches [16,17,18,19] have been investigated to achieve good stability and tracking performance. Additionally, research works, such as refs. [20,21,22], also introduced the prescribed performance method to manipulate the convergence rate and the steady-state error inside the prescribed bounds. Among the aforementioned methods, observer-based backstepping approaches have been widely adopted by researchers and demonstrate great advantages in the control of the quadrotor UAV system [9,23]. However, there is a significant drawback in that the backstepping control is sensitive to the initial error due to the repeated differentiation of the virtual controls [24]. To solve this limitation, the dynamic surface control (DSC) [25,26] is represented in this work and integrated with the prescribed performance technique to guarantee the bounded convergence and steady-state errors.

Integrated with the microelectronic mechanical system (MEMS), positioning technologies and fusion algorithms, the real-time translational and rotational states of the quadrotor UAV can be obtained for state and attitude estimation [27,28,29]. Various state or disturbance observers were integrated with controllers to enhance stability and performance robustness [30]. For instance, in ref. [31], Liu et al. designed two novel finite-time disturbance observers for position dynamics and attitude dynamics separately. The selection of control gains is improved to be mildly greater than the observation error instead of disturbances, which is more practical for implementation, but the system states are directly obtained from measurements. To observe both the states and disturbances, in [32], Xi et al. proposed an adaptive sliding mode disturbances observer for a robot manipulator system. The backstepping-based auxiliary system with error feedback is used as a system state observer. Compared to the disturbance observer, the extended state observer (ESO) [33], with the attractive advantages of having a concise structure and the ability to estimate both the states of the dynamics and the lumped disturbances via the system outputs, is widely studied [22,34,35]. In ref. [15], Zhang et al. applied six conventional ESOs to estimate the disturbances exist on quadrotor UAV system. Liu et al. [36] designed a finite-time ESO for attitude tracking of a quadrotor UAV using angular rates as the observer feedback. In the work of [37], Niu et al. applied the finite-time ESO to estimate the disturbance for the terminal sliding mode surface. Although the above-mentioned schemes have achieved the observation purpose, all these ESOs are only effective on integral-chain systems with matched disturbance using linear observer gains and a single measurement. The quadrotor UAV dynamics is usually decomposed into an underactuated subsystem and a fully actuated (UF) subsystem [38], where the former is a fourth-order pure-feedback system with both matched and unmatched disturbances. To compensate for the matched or unmatched disturbance from output, Chen et al. proposed two generalized approaches on unmatched disturbances situation based on ESO [39] and nonlinear disturbance observer [40]; the unmatched disturbance is effectively attenuated from output via a disturbance compensation gain. However, the estimation of both matched and unmatched disturbances with only one observer is still challenging, especially for the quadrotor UAV system. Consequently, in this work, we proposed a nonlinear ESO to approximate both the matched and unmatched disturbance estimation problems of the quadrotor UAV system.

The input saturation caused by limited propeller thrust forces often constrains the performance severely, and even dominates the stability of the quadrotor UAV [41].Thus, the input constraints are practically important issues for control problems, and extensive research works have considered stabilizing a nonlinear system with input saturation [42,43,44]. In ref. [41], Wang et al. introduced the hyperbolic function and Nussbaum function to designed priori-bounded control inputs for the trajectory tracking control of quadrotor UAV. Liu et al. in [23] applied two input saturation functions for position dynamics and attitude dynamics separately with two auxiliary control systems for compensation of the saturation effects. However, all these researchers only focused on the synthetic torque and force control input, but for the quadrotor UAV system, the limited force generated from each single rotor presents a more practical constraint rather than the synthetic torque and force control input. Thus, an input saturation algorithm on each single rotor of quadrotor UAV is proposed and an auxiliary system is designed in this work to compensate for the saturation effects.

A quadrotor UAV mounted with precise inspection instrument could fit the requirements for conducting healthy monitoring in limited spaces, such as tunnel safety patrolling, as shown in Figure 1. Highly aggressive maneuvers are not allowed, and the body inclination angle $\phi$ has to be within safety ranges. The large inclination angle of the quadrotor UAV will also cause insufficient lift force and further affect the system stability and performance. Thus, the attitude constraints turned out to be an important issue. Nevertheless, the input saturation for the position dynamics were considered in refs. [23,41], which can be seen as a bound for calculating the desired attitude angles, but did not directly consider the attitude constraints in designing the trajectory tracking controller for the quadrotor UAV. To realize the attitude constraint for quadrotor UAV, a saturation function is added to the inclination angle of the body with an auxiliary system.

Motivated by the above observations, the input saturation and prescribe performance control were taken into account in most existing research works. It is of practical significance to consider the attitude constraint for the control of the quadrotor UAV. Compared to other control schemes, a backstepping-typed DSC approach is more suitable for implementing the attitude constraint with a compensation auxiliary system. In this work, we focus on the prescribed performance control issue for the quadrotor UAV in the presence on both input saturation and attitude constraint, where the uncertainties and disturbances are approximated and attenuated through a novel nonlinear ESO base DSC. The main contributions of this paper are as follows:

1.

The control scheme is developed by the DSC technique with two auxiliary systems designed for attitude and input saturation constraints. Additionally, the prescribed performance method provides a more intuitive way to adjust the tracking speed and steady-state error.

2.

Considering the limitations of existing ESOs, two nonlinear ESOs are developed for approximating the pure-feedback subsystems of quadrotor UAV. Under such scheme, only one ESO is utilized for each quadrotor UF subsystem to estimate both the matched and unmatched disturbances with multiple state observations. Thus, a uniform convergence speed can be obtained, and the complexity of the observer’s parameter adjustment are reduced compared to conventional ESO designs.

3.

To improve the control stability of quadrotor UAV, the input saturation constraint is modified to exert on the thrust force generated by each rotor rather than the synthetic torque and force control inputs. Furthermore, the attitude constraint is firstly taken into account for stabilizing a quadrotor UAV. The constraint is realized by a saturation function with an auxiliary system as compensation to keep the inclination of quadrotor UAV within safety region.

The rest of this paper is organized as follows. In Section 2, problem formulation and preliminaries are illustrated for a typical configuration of quadrotor UAV. Section 3 introduces and proves a nonlinear ESO with multi-measurement feedback and varying-observer gain. The main results, which are the design of proposed controller, with stability analysis of the closed-loop system including the ESO system, are provided in Section 4. Then, several cases are simulated to validate the effectiveness of the proposed method in Section 5. The conclusion is given in Section 6.

2. Problem Formulation and Preliminaries

The dynamics model of a typically configured quadrotor UAV, shown in Figure 2, is well established in many works in the literature [45]. The position and attitude vectors are represented by $\xi \triangleq {\left[x,y,z\right]}^T$ and $\eta \triangleq {\left[\varphi ,\theta ,\psi \right]}^T$, separately. The three orientation angles are referred to as roll $\left(-\frac{\pi }{2}<\varphi <\frac{\pi }{2}\right)$, pitch $\left(-\frac{\pi }{2}<\theta <\frac{\pi }{2}\right)$ and yaw $\left(-\pi \le \psi <\pi \right)$. The translational velocity vector $v\triangleq {\left[v_x,x_y,v_z\right]}^T$ is given in an earth-fixed coordinate, and the angular velocity vector $\omega \triangleq {\left[p,q,r\right]}^T$ is defined according to the body-fixed coordinate. Then, the dynamics of the quadrotor UAV are described as follows:

$$\left\{\begin{array}{cc}\hfill \dot{\xi }=& v\hfill \\ \hfill \dot{v}=& m^{-1}{R}_t{u}_{\xi }-g{e}_3+d_{\xi }\hfill \\ \hfill \dot{\eta }=& R_r\omega \hfill \\ \hfill \dot{\omega }=& J^{-1}{u}_{\eta }+C_{\eta }+d_{\eta }\hfill \end{array}\right.$$

(1)

where $R_t$ and $R_r$ are the translation velocity matrix and rotation velocity matrix [45], m and $J=\mathrm{diag}\left(J_x,J_y,J_z\right)$ are the body mass and inertia of the quadrotor UAV, g is the gravity constant, $d_{\xi }={\left[d_x,d_y,d_z\right]}^T$ and $d_{\eta }={\left[d_{\varphi },d_{\theta },d_{\psi }\right]}^T$ represent the lumped variables of force disturbances and torque disturbances [46], $C_{\eta }=-\omega \times J\omega$ denotes the body gyroscopic effect term, and $u_{\xi }={\left[0,0,{\tau }_f\right]}^T$ and $u_{\eta }={\left[{\tau }_p,{\tau }_q,{\tau }_r\right]}^T$ are the force and torque with respect to the body-fixed coordinate, respectively.

2.1. Problem Formulation

In order to implement control strategies for quadrotor UAV, the UF decomposition is adopted with state vectors $x_1={\left[x,y\right]}^T$, $x_2={\left[v_x,v_y\right]}^T$, $x_3={\left[\varphi ,\theta \right]}^T$, $x_4={\left[p,q\right]}^T$, $x_5={\left[\psi ,z\right]}^T$ and $x_6={\left[r,v_z\right]}^T$. Accordingly, the underactuated subsystem is defined as

$$\left\{\begin{array}{cc}\hfill {\dot{x}}_1& =x_2\hfill \\ \hfill {\dot{x}}_2& =g_{1}u\left({\phi }_1\right)+d_1\hfill \\ \hfill {\dot{x}}_3& =g_2{x}_4+d_2\hfill \\ \hfill {\dot{x}}_4& =g_3{\phi }_2+d_3\hfill \end{array}\right.$$

(2)

and a fully actuated subsystem is defined as

$$\left\{\begin{array}{cc}\hfill {\dot{x}}_5& =g_4{x}_6+d_4\hfill \\ \hfill {\dot{x}}_6& =g_5{\phi }_3+d_5\hfill \end{array}\right.$$

(3)

where the parameters $g_i,i=1\dots 5$ are

$$\begin{array}{cc}\hfill g_1& =\frac{{\tau }_f}{m}\left[\begin{array}{cc}{S}_{\psi }& C_{\psi }\\ -C_{\psi }& S_{\psi }\end{array}\right],g_2=\left[\begin{array}{cc}1& S_{\varphi }{T}_{\theta }\\ 0& C_{\varphi }\end{array}\right],g_3=\left[\begin{array}{cc}\frac{1}{J_x}& 0\\ 0& \frac{1}{J_y}\end{array}\right],\hfill \\ \hfill g_4& =\left[\begin{array}{cc}{C}_{\varphi }S{e}_{\theta }& 0\\ 0& 1\end{array}\right],g_5=\left[\begin{array}{cc}\frac{1}{J_z}& 0\\ 0& \frac{C_{\varphi }{C}_{\theta }}{m}\end{array}\right],\hfill \end{array}$$

and the lumped disturbances $d_i,i=1\dots 5$ are

$$\begin{array}{cc}\hfill d_1& =\left[\begin{array}{c}-d_x\\ -d_y\end{array}\right],d_2=\left[\begin{array}{c}{C}_{\varphi }{T}_{\theta }r\\ -S_{\varphi }r\end{array}\right],d_3=\left[\begin{array}{c}\frac{J_y-J_z}{J_x}qr-d_{\varphi }\\ \frac{J_z-J_x}{J_y}pr-d_{\theta }\end{array}\right],\hfill \\ \hfill d_4& =\left[\begin{array}{c}{S}_{\varphi }S{e}_{\theta }q\\ 0\end{array}\right],d_5=\left[\begin{array}{c}\frac{J_x-J_y}{J_z}pq-d_{\psi }\\ -g-d_z\end{array}\right],\hfill \end{array}$$

and the control inputs ${\phi }_i,i=1,2,3$ are

$$\begin{array}{c}\hfill {\phi }_1=\left[\begin{array}{c}{S}_{\varphi }\\ C_{\varphi }{S}_{\theta }\end{array}\right],{\phi }_2=\left[\begin{array}{c}{\tau }_p\\ {\tau }_q\end{array}\right],{\phi }_3=\left[\begin{array}{c}{\tau }_r\\ {\tau }_f\end{array}\right],\end{array}$$

the symbols $S_{\left(·\right)}$, $C_{\left(·\right)}$, $T_{\left(·\right)}$ and $S{e}_{\left(·\right)}$ are the short form of trigonometric function $sin\left(·\right)$, $cos\left(·\right)$, $tan\left(·\right)$ and $sec\left(·\right)$.

Remark 1.

From (2) and (3), it can be seen that the subsystem does not satisfy the matching condition because translational and rotational dynamics both involve uncertainties and external disturbances, and the translational motions of quadrotor UAV are achieved through body rotational motions.

The rotor dynamic can be approximated by a first-order low-pass filter. However, for non-aggressive maneuvers, the fast rotor dynamics can be regarded as a conventional assumption, and the rotor dynamic effects can also be diminished by the high bandwidth of the controller [19,47]. Therefore, the rotor dynamics are ignored and the four virtual control variables $\left\{{\tau }_f,{\tau }_p,{\tau }_r,{\tau }_q\right\}$ are obtained from thrust force as

$${\left[{\tau }_p,{\tau }_q,{\tau }_r,{\tau }_f\right]}^T=\mathsf{\Phi }u\left({\phi }_4\right)$$

(4)

where

$$\mathsf{\Phi }=\left[\begin{array}{cccc}l& -l& -l& l\\ -l& -l& l& l\\ c& -c& c& -c\\ 1& 1& 1& 1\end{array}\right],{\phi }_4=\left[\begin{array}{c}{F}_1\\ F_2\\ F_3\\ F_4\end{array}\right]$$

(5)

and $u\left(·\right)$ represents the amount of control after the saturation constraints, $F_i,i=1\dots 4$ is the thrust force generated by $ith$ rotor, l is the arm length shown in Figure 2, and c is the torque-thrust coefficient.

Assumption 1.

All state vectors position, ξ, velocity, v, attitude angle, η, and angular rate, ω, of quadrotor UAV are available from measurement during flight.

Assumption 2.

The lumped disturbances are bounded to the first-order derivative $\parallel {\dot{d}}_i\parallel \le {\overline{d}}_i$ and have constant values in steady state ${lim}_{t\to \infty }{\dot{d}}_i\left(t\right)=0$ and ${lim}_{t\to \infty }{d}_i\left(t\right)=D_{i,\infty }$ where $i=1\dots 5$.

Assumption 3.

The orientation angle $\left\{\varphi ,\theta ,\psi \right\}$ are all within valid ranges as $\left(-\frac{\pi }{2}<\varphi <\frac{\pi }{2}\right)$, $\left(-\frac{\pi }{2}<\theta <\frac{\pi }{2}\right)$ and $\left(-\pi \le \psi <\pi \right)$.

Assumption 4.

The desired x-y-z position $\left\{x_d,y_d,z_d\right\}$ and yaw angle ${\psi }_d$ and their derivatives are assumed to be piecewise uniformly bounded.

2.2. Input and State Constraints

Due to the limited force generated by each rotor, the input saturation constraint is expressed using saturation function as follows:

$$u\left(F_i\right)=\left\{\begin{array}{cc}{F}_{max},\hfill & \mathrm{if}{F}_i>F_{max}\hfill \\ F_i,\hfill & \mathrm{if}{F}_{min}\le F_i\le F_{max}\hfill \\ F_{min},\hfill & \mathrm{if}{F}_i<F_{min}\hfill \end{array}\right.$$

(6)

where $F_i,i=1\dots 4$ are the thrust force generated by each rotor, and $F_{max}>F_{min}\in \left(0,\infty \right)$ are the maximum and minimum force values.

The attitude constraint is designed with respect to the inclination angle of the quadrotor UAV, thus the saturation function for control variables ${\phi }_1$ in (2) is

$$u\left({\phi }_1\right)=\left\{\begin{array}{cc}\frac{M_{\eta }}{\parallel {\phi }_1\parallel }{\phi }_1,\hfill & \mathrm{if}\parallel {\phi }_1\parallel >M_{\eta }\hfill \\ {\phi }_1,\hfill & \mathrm{if}\parallel {\phi }_1\parallel \le M_{\eta }\hfill \end{array}\right.$$

(7)

where $M_{\eta }$ is the maximum allowed inclination value of the quadrotor UAV with respect to the horizontal level.

2.3. Prescribe Performance

Prescribe performance is achieved by ensuring the tracking error evolves within the following predefined region [20]:

$$\begin{array}{c}\hfill -{\underline{\delta }}_i{\rho }_i\left(t\right)<e_i\left(t\right)<{\overline{\delta }}_i{\rho }_i\left(t\right),\forall t\ge 0\end{array}$$

(8)

where $e_i,i=x,y,z,\psi$ are the errors, ${\underline{\delta }}_i$ and ${\overline{\delta }}_i$ are positive constants and ${\rho }_i\left(t\right)$ is a performance function which satisfies smooth, strictly positive and decaying bounded with ${\lim }_{t\to \infty }{\rho }_i\left(t\right)={\rho }_{i\infty }>0$. Thus, the performance function is defined as

$$\begin{array}{c}\hfill {\rho }_i\left(t\right)=\left({\rho }_{i0}-{\rho }_{i\infty }\right)e^{-l_{i}t}+{\rho }_{i\infty },\forall t\ge 0\end{array}$$

(9)

where ${\rho }_{i0}$, ${\rho }_{i\infty }$ and $l_i,i=x,y,z,\psi$ are positive constants and the initial condition $-{\underline{\delta }}_i{\rho }_i\left(0\right)<e_i\left(0\right)<{\overline{\delta }}_i{\rho }_i\left(0\right)$ are satisfied. Then, transforming the original constrained tracking error behavior into an equivalent unconstrained one as

$$\begin{array}{c}\hfill {\alpha }_i={\mathsf{\Phi }}_i\left(\frac{e_i}{{\rho }_i\left(t\right)}\right)\end{array}$$

(10)

where ${\mathsf{\Phi }}_i\left(·\right):\left(-{\underline{\delta }}_i,{\overline{\delta }}_i\right)\mapsto \left(-\infty ,+\infty \right)$ is a strictly increasing smooth function and chosen [21] as

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_i\left(\frac{e_i}{{\rho }_i\left(t\right)}\right)=\frac{1}{2}\ln \frac{{\underline{\delta }}_i+\frac{e_i}{{\rho }_i\left(t\right)}}{{\overline{\delta }}_i-\frac{e_i}{{\rho }_i\left(t\right)}}.\end{array}$$

(11)

Accordingly, the transformed error dynamics is derived as

$$\begin{array}{c}\hfill {\dot{\alpha }}_i={\gamma }_i\left({\dot{e}}_i-\frac{{\dot{\rho }}_i\left(t\right)}{{\rho }_i\left(t\right)}{e}_i\right)\end{array}$$

(12)

with,

$$\begin{array}{c}\hfill {\gamma }_i=\frac{1}{2{\rho }_i\left(t\right)}\left(\frac{1}{{\underline{\delta }}_i+\frac{e_i}{{\rho }_i\left(t\right)}}+\frac{1}{{\overline{\delta }}_i-\frac{e_i}{{\rho }_i\left(t\right)}}\right).\end{array}$$

(13)

Define $|{\gamma }_i{|}_{min}={\gamma }_{in}$, from (8), one can conclude that

$$\begin{array}{c}\hfill 0<{\gamma }_{in}=\frac{2}{{\rho }_{i0}\left({\underline{\delta }}_i+{\overline{\delta }}_i\right)}\le {\gamma }_i,-l_i<\frac{{\dot{\rho }}_i\left(t\right)}{{\rho }_i\left(t\right)}<0\end{array}$$

(14)

which will be used in the following proof of stability.

Lemma 1

([48]). Consider the original constrained error $e_i\left(t\right)$ and transformed error ${\alpha }_i\left(t\right)$ with error transformation defined in (10). If ${\alpha }_i\left(t\right)$ is bounded, $e_i\left(t\right)$ in (8) is satisfied for $\forall t\ge 0$.

For clarity, the following two notations are defined for this paper:

(1)

I without a superscript represents a $I^{2\times 2}$ identity matrix;

(2)

0 represents a zero matrix with proper dimensions;

(3)

$\mathrm{col}\left(·\right)$ represents the column vector.

3. ESO Design

Most existing studies on ESOs are only effective on integral-chain systems with matched disturbance using linear observer gains and single measurement. However, considering the UF subsystems of quadrotor UAV, the difficulties are the approximations of both the matched and unmatched disturbances. By taking the advantages of coordinate transform from the feedback linearization technique, the quadrotor UF subsystems can be reformulate into Brunovsky systems. Then, the matched and unmatched disturbances from the quadrotor system are added as the augmented states, and the whole system is estimated using the ESO technique.

Considering UF subsystems (2) and (3), augment the dynamics with lumped disturbances and rewritten in ESO from as

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{\overline{x}}}_1& ={\overline{f}}_1\left({\overline{x}}_1,{\phi }_2\right)+D_1{\overline{x}}_1^a+L_{11}\left(y_{1m}-C_1{\overline{x}}_1\right)\hfill \\ \hfill {\dot{\overline{x}}}_1^a& =L_{12}\left(y_{1m}-C_1{\overline{x}}_1\right)\hfill \end{array}\right.\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{\overline{x}}}_2& ={\overline{f}}_2\left({\overline{x}}_2,{\phi }_3\right)+D_2{\overline{x}}_2^a+L_{21}\left(y_{2m}-C_2{\overline{x}}_2\right)\hfill \\ \hfill {\dot{\overline{x}}}_2^a& =L_{22}\left(y_{2m}-C_2{\overline{x}}_2\right)\hfill \end{array}\right.\hfill \end{array}$$

(16)

with

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\overline{x}}_1& =\mathrm{col}\left(x_1\dots x_4\right),{\overline{x}}_1^a=\mathrm{col}\left(d_1,d_2,d_3\right)\hfill \\ \hfill {\overline{x}}_2& =\mathrm{col}\left(x_5,x_6\right),{\overline{x}}_2^a=\mathrm{col}\left(d_4,d_5\right)\hfill \end{array}\right.\hfill \\ \hfill & \left\{\begin{array}{cc}\hfill {\overline{f}}_1\left(·\right)& =\mathrm{col}\left(x_2,g_1{\phi }_1,g_2{x}_4,g_3{\phi }_2\right)\hfill \\ \hfill {\overline{f}}_2\left(·\right)& =\mathrm{col}\left(g_4{x}_6,g_5{\phi }_3\right)\hfill \end{array}\right.\hfill \end{array}$$

where $\left\{C_1,C_2\right\}$ are the observation matrices, $\left\{D_1,D_2\right\}$ are the distribution matrix of disturbances, $\left\{y_{1m},y_{2m}\right\}$ are the observations, $\left\{L_{11},L_{12}\right\}$ and $\left\{L_{21},L_{22}\right\}$ are the observer feedback matrices.

Remark 2.

From the state space dynamics, one can find that (2) is an affine-in-control pure-feedback system, and (3) is a strict-feedback system. Additionally, (2) and (3) contain unmatched disturbances.

In order to compute the nonlinear feedback matrices, coordinate transformation is proposed to transform the pure-feedback system (2) and strict-feedback system (3) to canonical systems. According to the feedback linearization method [49], define alternative state variables ${\overline{z}}_1=\mathrm{col}\left(z_1\dots z_4\right)$ and ${\overline{z}}_2=\mathrm{col}\left(z_5,z_6\right)$ as

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{z}}_1& ={\dot{x}}_1\hfill \\ \hfill {\dot{z}}_2& =\frac{\partial f_1}{\partial x_1}{\dot{x}}_1+\frac{\partial f_1}{\partial x_2}{\dot{x}}_2\hfill \\ \hfill {\dot{z}}_3& =\frac{\partial f_2}{\partial x_1}{\dot{x}}_1+\frac{\partial f_2}{\partial x_2}{\dot{x}}_2+\frac{\partial f_2}{\partial x_3}{\dot{x}}_3\hfill \\ \hfill {\dot{z}}_4& =\frac{\partial f_3}{\partial x_1}{\dot{x}}_1+\frac{\partial f_3}{\partial x_2}{\dot{x}}_2+\frac{\partial f_3}{\partial x_3}{\dot{x}}_3+\frac{\partial f_3}{\partial x_4}{\dot{x}}_4\hfill \end{array}\right.\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{z}}_5& ={\dot{x}}_5\hfill \\ \hfill {\dot{z}}_6& =\frac{\partial f_5}{\partial x_5}{\dot{x}}_5+\frac{\partial f_5}{\partial x_6}{\dot{x}}_6\hfill \end{array}\right.\hfill \end{array}$$

(18)

where $f_1=x_2$, $f_2=g_1{\phi }_1$, $f_3=\frac{\partial \left(g_1{\phi }_1\right)}{\partial x_3}{g}_2{x}_4$ and $f_5=g_4{x}_6$. The lumped disturbances $\left\{d_1\dots d_5\right\}$ are considered independent to state variables $\left\{x_1\dots x_6\right\}$. Therefore, according to (17) and (18), the state transform matrices for which transfer $\left\{{\overline{x}}_1,{\overline{x}}_2\right\}$ to $\left\{{\overline{z}}_1,{\overline{z}}_2\right\}$ can be obtained as

$$\begin{array}{ccc}\hfill S_1& =\left[\begin{array}{cccc}I& 0& 0& 0\\ 0& J_1& 0& 0\\ 0& 0& J_2& 0\\ 0& 0& J_3& J_4\end{array}\right],S_2\hfill & \hfill =\left[\begin{array}{cc}I& 0\\ 0& J_5\end{array}\right]\end{array}$$

(19)

where $J_1=\frac{\partial f_1}{\partial x_2}=I$, $J_2=\frac{\partial f_2}{\partial x_3}=g_1{J}_{\varphi \theta }$, $J_3=\frac{\partial f_3}{\partial x_3}=J_3\left(x_3,x_4\right)$, $J_4=\frac{\partial f_3}{\partial x_4}=g_1{J}_{\varphi \theta }{g}_2$ and $J_5=\frac{\partial f_5}{\partial x_6}=g_4$. $J_{\varphi \theta }$ is given as

$$\begin{array}{c}\hfill J_{\varphi \theta }=\left[\begin{array}{cc}{C}_{\varphi }& 0\\ -S_{\varphi }{C}_{\theta }& C_{\varphi }{C}_{\theta }\end{array}\right]\end{array}$$

(20)

and the details of variables $\left\{J_1,J_2,J_3,J_4,J_5\right\}$ are shown in Appendix A.

Introducing the observer bandwidth $w_1,w_2>0$ in the high-gain observer method [50], the bandwidth matrices are defined as

$$\begin{array}{cc}\hfill W_1& =\mathrm{diag}\left(1,1,w_1,w_1,w_1^2,w_1^2,w_1^3,w_1^3\right),\hfill \\ \hfill W_2& =\mathrm{diag}\left(1,1,w_2,w_2\right).\hfill \end{array}$$

(21)

Using the coordinate transformation matrices (19) and bandwidth matrices (21), the nonlinear ESO observer feedback matrices in (15) and (16) are obtained as

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{L}_{11}\\ L_{12}\end{array}\right]& =\left[\begin{array}{c}{w}_1{S}_1^{-1}{W}_1{\chi }_1\\ w_1^2{D}_1^T{S}_1^{-1}{W}_1{D}_1{\chi }_2\end{array}\right]C_1{W}_1^{-1}{S}_1{C}_1^T\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{L}_{21}\\ L_{22}\end{array}\right]& =\left[\begin{array}{c}{w}_2{S}_2^{-1}{W}_2{\chi }_3\\ w_2^2{D}_2^T{S}_2^{-1}{W}_2{D}_2{\chi }_4\end{array}\right]C_2{W}_2^{-1}{S}_2{C}_2^T\hfill \end{array}$$

(23)

where $\left\{{\chi }_1\dots {\chi }_4\right\}$ are gain matrices with positive entries.

To prove the convergence of the proposed ESO, the canonical form of (15) and (16) can be computed using the following relationship as

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{\overline{z}}}_1& =S_1{W}_1^{-1}{\dot{\overline{x}}}_1\hfill \\ \hfill {\dot{\overline{z}}}_1^a& =D_1^T{S}_1{\left(w_1{W}_1\right)}^{-1}{D}_1{\dot{\overline{x}}}_1^a\hfill \end{array}\right.\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill {\dot{\overline{z}}}_2& =S_2{W}_2^{-1}{\dot{\overline{x}}}_2\hfill \\ \hfill {\dot{\overline{z}}}_2^a& =D_2^T{S}_2{\left(w_2{W}_2\right)}^{-1}{D}_2{\dot{\overline{x}}}_2^a\hfill \end{array}\right.\hfill \end{array}$$

(25)

Define the observer estimation error as

$$\begin{array}{c}\hfill {\epsilon }_1=\left[\begin{array}{c}{\overline{z}}_1-{\widehat{\overline{z}}}_1\\ {\overline{z}}_1^a-{\widehat{\overline{z}}}_1^a\end{array}\right],{\epsilon }_2=\left[\begin{array}{c}{\overline{z}}_2-{\widehat{\overline{z}}}_2\\ {\overline{z}}_2^a-{\widehat{\overline{z}}}_2^a\end{array}\right]\end{array}$$

(26)

using the mean value theorem [51], the error dynamics of augmented subsystem (2) and (3) under ESO (15), (16) can be obtained from (15)–(26) as

$$\begin{array}{cc}\hfill {\dot{\epsilon }}_1& =w_1{\overline{A}}_1{\epsilon }_1+{\overline{D}}_1\frac{{\dot{d}}_1}{w_1^2}+{\overline{D}}_2\frac{{\dot{d}}_2}{w_1^3}+{\overline{D}}_3\frac{{\dot{d}}_3}{w_1^4}\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\dot{\epsilon }}_2& =w_2{\overline{A}}_2{\epsilon }_2+{\overline{D}}_4\frac{{\dot{d}}_4}{w_2}+{\overline{D}}_5\frac{{\dot{d}}_5}{w_2^2}\hfill \end{array}$$

(28)

with

$$\begin{array}{c}\hfill {\overline{A}}_1=\left[\begin{array}{cc}{A}_1-{\chi }_1{C}_1& D_1\\ -{\chi }_2{C}_1& 0^{3\times 3}\end{array}\right],{\overline{A}}_2=\left[\begin{array}{cc}{A}_2-{\chi }_3{C}_2& D_2\\ -{\chi }_4{C}_2& 0^{2\times 2}\end{array}\right]\end{array}$$

where $\left\{{\overline{D}}_1,{\overline{D}}_2,{\overline{D}}_3\right\}$ are the columns of $\left[\begin{array}{c}{0}^{8\times 6}\\ D_1^T{S}_1{D}_1\end{array}\right]$ and $\left\{{\overline{D}}_4,{\overline{D}}_5\right\}$ are the columns of $\left[\begin{array}{c}{0}^{4\times 4}\\ D_2^T{S}_2{D}_2\end{array}\right]$. $\left\{A_1,A_2\right\}$ are the transfer matrices in canonical form as

$$\begin{array}{c}\hfill A_1=\left[\begin{array}{cccc}0& I& 0& 0\\ 0& 0& I& 0\\ 0& 0& 0& I\\ 0& 0& 0& 0\end{array}\right],A_2=\left[\begin{array}{cc}0& I\\ 0& 0\end{array}\right].\end{array}$$

Remark 3.

The result of adding the observer bandwidth is a scalar amplification of the location of poles or a scalar amplification for all eigenvalues, which can be seen from (27) and (28).

Since multiple disturbances appeared in the subsystem (2) and (3), observability is essential and can be checked from

$$\begin{array}{c}\hfill A_i^{\prime }=\left[\begin{array}{cc}{A}_i& D_i\\ 0& 0\end{array}\right],C_i^{\prime }=\left[\begin{array}{cc}{C}_i& 0\end{array}\right],i=1,2\end{array}$$

where $A_i^{\prime }$ and $C_i^{\prime }$ are the transfer matrix and observation matrix of the augmented systems, separately. From subsystems (2), (3) and Assumption 1, the observability matrices can be calculated and have a rank of 7 for (2) and 4 for (3). Therefore, the corresponding states and lumped disturbances are observable. Note that the observations can be reduced as long as the observability is satisfied.

The gain matrices $\left\{{\chi }_1\dots {\chi }_4\right\}$ are selected as follows such that ${\overline{A}}_1$ and ${\overline{A}}_2$ are Hurwitz matrices.

$$\begin{array}{cc}\hfill {\chi }_1& =\left[\begin{array}{cccc}I& I& 0& 0\\ 0& 2I& I& 0\\ 0& 0& 2I& I\\ 0& 0& 0& 2I\end{array}\right],{\chi }_2=\left[\begin{array}{cccc}0& I& 0& 0\\ 0& 0& I& 0\\ 0& 0& 0& I\end{array}\right],\hfill \\ \hfill {\chi }_3& =\left[\begin{array}{cc}2I& I\\ 0& 2I\end{array}\right],{\chi }_4=\left[\begin{array}{cc}I& 0\\ 0& I\end{array}\right].\hfill \end{array}$$

(29)

Hence, the observer bandwidth becomes the only tuning parameter of the observer and is determined by the system dynamics. Now, the following is the proof of stability of proposed ESO method.

Theorem 1.

Considering the ESO error dynamics in (27) and (28), there exist two positive definite matrices $P_1$ and $P_2$ satisfying ${\overline{A}}_1^T{P}_1+P_1{\overline{A}}_1=-I$ and ${\overline{A}}_2^T{P}_2+P_2{\overline{A}}_2=-I$. Under Assumptions 1, 2 and 3, the uniformly bounded stability of proposed nonlinear ESOs for quadrotor UAV system can be guaranteed if the observer bandwidth $\left\{w_1,w_2\right\}$ is selected such that $l_1=min\left(\frac{m_1}{{\lambda }_{max}\left(P_1\right)},\frac{m_2}{{\lambda }_{max}\left(P_2\right)}\right)>0$ where $m_1=\left(w_1-\frac{{\lambda }_1}{w_1^2}-\frac{{\lambda }_2}{w_1^3}-\frac{{\lambda }_3}{w_1^4}\right)$, $m_2=\left(w_2-\frac{{\lambda }_4}{w_2}-\frac{{\lambda }_5}{w_2^2}\right)$ and ${\lambda }_i=\left\{\parallel P_1{\overline{D}}_1\parallel ,\parallel P_1{\overline{D}}_2\parallel ,\parallel P_1{\overline{D}}_3\parallel ,\parallel P_2{\overline{D}}_4\parallel ,\parallel P_2{\overline{D}}_5\parallel \right\}$.

Proof of Theorem 1. 

Choose the following Lyapunov candidate function as

$$\begin{array}{c}\hfill V={\epsilon }_1^T{P}_1{\epsilon }_1+{\epsilon }_2^T{P}_2{\epsilon }_2.\end{array}$$

(30)

Substituting (27) and (28) into the derivative of V yields

$$\begin{array}{cc}\hfill \dot{V}=& -w_1\parallel {\epsilon }_1{\parallel }^2-w_2{\parallel {\epsilon }_2\parallel }^2+2{\epsilon }_1^T{P}_1{\overline{D}}_1\frac{{\dot{d}}_1}{w_1^2}+2{\epsilon }_1^T{P}_1{\overline{D}}_2\frac{{\dot{d}}_2}{w_1^3}+2{\epsilon }_1^T{P}_1{\overline{D}}_3\frac{{\dot{d}}_3}{w_1^4}\hfill \\ \hfill & +2{\epsilon }_2^T{P}_2{\overline{D}}_4\frac{{\dot{d}}_4}{w_2}+2{\epsilon }_2^T{P}_2{\overline{D}}_5\frac{{\dot{d}}_5}{w_2^2}\hfill \\ \hfill \le & -w_1\parallel {\epsilon }_1{\parallel }^2-w_2\parallel {\epsilon }_2{\parallel }^2+2\frac{{\lambda }_1}{w_1^2}\parallel {\epsilon }_1\parallel {\overline{d}}_1+2\frac{{\lambda }_2}{w_1^3}\parallel {\epsilon }_1\parallel {\overline{d}}_2+2\frac{{\lambda }_3}{w_1^4}\parallel {\epsilon }_1\parallel {\overline{d}}_3\hfill \\ \hfill & +2\frac{{\lambda }_4}{w_2}\parallel {\epsilon }_2\parallel {\overline{d}}_4+2\frac{{\lambda }_5}{w_2^2}\parallel {\epsilon }_2\parallel {\overline{d}}_5\hfill \\ \hfill \le & -l_{1}V+l_2\hfill \end{array}$$

(31)

where $l_1=\min \left(\frac{m_1}{{\lambda }_{max}\left(P_1\right)},\frac{m_2}{{\lambda }_{max}\left(P_2\right)}\right)$ with $m_1=\left(w_1-\frac{{\lambda }_1}{w_1^2}-\frac{{\lambda }_2}{w_1^3}-\frac{{\lambda }_3}{w_1^4}\right)$, $m_2=\left(w_2-\frac{{\lambda }_4}{w_2}-\frac{{\lambda }_5}{w_2^2}\right)$ and $l_2=\frac{{\lambda }_1}{w_1^2}{\overline{d}}_1^2+\frac{{\lambda }_2}{w_1^3}{\overline{d}}_2^2+\frac{{\lambda }_3}{w_1^4}{\overline{d}}_3^2+\frac{{\lambda }_4}{w_2}{\overline{d}}_4^2+\frac{{\lambda }_5}{w_2^2}{\overline{d}}_5^2$. From (31), $\dot{V}<0$ if $V>\frac{l_2}{l_1}$, and therefore, it can be concluded that the error dynamics (27) and (28) are bounded stably as

$$\begin{array}{c}\hfill \Vert {\epsilon }_1\Vert +\Vert {\epsilon }_2\Vert \le \sqrt{\frac{2l_2/l_1}{\min \left({\lambda }_{min}\left(P_1\right),{\lambda }_{min}\left(P_2\right)\right)}}\end{array}$$

(32)

where ${\lambda }_{max}\left(P\right)$ and ${\lambda }_{min}\left(P\right)$ denote the maximum and minimum eigenvalues of the matrix P. The proof is finished. □

4. Controller Design

In this section, the prescribed performance tracking control strategy for the quadrotor UAV is proposed based on the DSC technique with attitude and input constraints. The estimated state vector $x_i,i=1\dots 6$, and lumped disturbances, $d_i,i=1\dots 5$, are obtained from the proposed ESO in previous section. For clarity, the following notations are defined in this section:

(1)

${\widehat{x}}_i,i=1,\dots ,6$ are the estimated variables,

(2)

${\tilde{x}}_i=x_i-{\widehat{x}}_i,i=1\dots 6$ are the estimated errors,

(3)

${\lambda }_m\left(A\right)$ and ${\lambda }_n\left(A\right)$ are the maximum and minimum eigenvalues of matrix A.

4.1. Controller Design for Underactuated System

Define $e_1=x_{1d}-x_1$ as the first surface error. To achieve guaranteed tracking performance of $\left\{x,y\right\}$, the transformed error in (10) and its derivative are

$$\begin{array}{c}\hfill {\dot{\alpha }}_1={\gamma }_1\left({\dot{x}}_{1d}-x_2-{\rho }_1^{-1}{\dot{\rho }}_1{e}_1\right)\end{array}$$

(33)

where ${\rho }_1=\mathrm{diag}\left({\rho }_x,{\rho }_y\right)$ and ${\gamma }_1=\mathrm{diag}\left({\gamma }_x,{\gamma }_y\right)$.

The virtual control variables ${\overline{x}}_2$ is designed as

$$\begin{array}{c}\hfill {\overline{x}}_2=K_1{\alpha }_1+{\dot{x}}_{1d}-{\rho }_1^{-1}{\dot{\rho }}_1{\widehat{e}}_1\end{array}$$

(34)

where $K_1\in {\mathbb{R}}^{2\times 2}$ is a positive definite matrix.

Following the principle DSC technique, introduce a variable $x_{2d}\in {\mathbb{R}}^2$ and pass the virtual control ${\overline{x}}_2$ though a first-order low-pass filter with the positive definite time constant matrix ${\tau }_1\in {\mathbb{R}}^{2\times 2}$ as

$$\begin{array}{c}\hfill {\tau }_1{\dot{x}}_{2d}+x_{2d}={\overline{x}}_2,x_{2d}\left(0\right)={\overline{x}}_2\left(0\right).\end{array}$$

(35)

Let $z_1={\overline{x}}_2-x_{2d}$ denote the filtering error, then the filtering error dynamics can be derived as

$$\begin{array}{cc}\hfill {\dot{z}}_1=& -{\tau }_1^{-1}{z}_1+{\dot{\overline{x}}}_2\hfill \\ \hfill =& -{\tau }_1^{-1}{z}_1+O_1\left({\ddot{x}}_{1d},{\dot{\alpha }}_1,e_1,{\dot{e}}_1,{\tilde{x}}_1,{\dot{\tilde{x}}}_1\right)\hfill \end{array}$$

(36)

where $O_1\left(·\right)$ is a continuous function.

Consider the Lyapunov function candidate $V_1=\frac{1}{2}{\alpha }_1^T{\gamma }_1^{-1}{\alpha }_1+\frac{1}{2}{z}_1^T{z}_1$, and its derivative with respect to time is

$${\dot{V}}_1={\alpha }_1^T\left(-K_1{\alpha }_1+z_1+e_2+{\rho }_1^{-1}{\dot{\rho }}_1{\tilde{x}}_1\right)+z_1^T\left(-{\tau }^{-1}{z}_1+O_1\right).$$

(37)

Invoking Young’s inequality, error transform (26) and bounds (14), the inequality above can be further expressed as

$$\begin{array}{cc}\hfill {\dot{V}}_1\le & -{\alpha }_1^T\left(K_1-\frac{1}{2}I\right){\alpha }_1-z_1^T\left({\tau }_1^{-1}-\frac{1}{2}I\right)z_1\hfill \\ \hfill & +\frac{1}{2}{O}_1^T{O}_1+\frac{1}{2}{e}_2^T{e}_2 + \parallel {\alpha }_1\parallel \parallel z_1\parallel + l_1^2\parallel {\alpha }_1\parallel \parallel {\epsilon }_1\parallel .\hfill \end{array}$$

(38)

Define $e_2=x_{2d}-x_2$ as the second surface error, and its derivative is

$$\begin{array}{c}\hfill {\dot{e}}_2={\dot{x}}_{2d}-g_1{\phi }_1-d_1.\end{array}$$

(39)

The virtual control variables ${\overline{\phi }}_1$ are designed as

$$\begin{array}{c}\hfill {\overline{\phi }}_1=g_1^{-1}\left(K_2{\widehat{e}}_2+{\dot{x}}_{2d}-{\widehat{d}}_1\right)-K_{{\zeta }_1}{\zeta }_1\end{array}$$

(40)

where $K_2,K_{{\zeta }_1}\in {\mathbb{R}}^{2\times 2}$ are positive definite matrices, and ${\zeta }_1\in {\mathbb{R}}^2$ is a variable from the following auxiliary system for compensating the effect of state constraints.

$${\dot{\zeta }}_1=\left\{\begin{array}{cc}-{\kappa }_1{\zeta }_1-\frac{|e_2^T{g}_1{ϵ}_1|+\frac{1}{2}{ϵ}_1^T{ϵ}_1}{\parallel {\zeta }_1{\parallel }^2}{\zeta }_1+{ϵ}_1,\hfill & \mathrm{if}\parallel {\zeta }_1\parallel \ge {\delta }_1\hfill \\ 0,\hfill & \mathrm{if}\parallel {\zeta }_1\parallel < {\delta }_1\hfill \end{array}\right.$$

(41)

where ${ϵ}_1={\overline{\phi }}_1-u\left({\phi }_1\right)$ is the error of the state constraints, ${\kappa }_1\in {\mathbb{R}}^{2\times 2}$ is a positive definite matrix, and ${\delta }_1$ is a small positive constant.

Remark 4.

As for the variable ${\zeta }_1$ in (40), which is defined in (41), its value is dependent on ${ϵ}_1$. If the derivative of ${\zeta }_1$ is not equal to zero in the auxiliary system, the result of ${\zeta }_1$ might render the virtual control ${\overline{\phi }}_1$ smaller and closed to $u\left({\phi }_1\right)$. Therefore, the saturation error can be compensated by ${\zeta }_1$. When the derivative of ${\zeta }_1$ is equal to zero, the result of ${\zeta }_1$ is a small constant value and it may affect the virtual control ${\overline{\phi }}_1$ slightly since ${\delta }_1$ is a small constant.

Remark 5.

The combination of Euler angle ϕ, θ and ψ have orders to form the rotation matrix $R_t$ in (1). Thus, the actual inclination angles along the x-axis and y-axis with respect to the earth-frame are ϕ and $si{n}^{-1}\left(C_{\varphi }{S}_{\theta }\right)$, separately. Here, the trigonometric variables are used as $S_{\varphi }$ and $C_{\varphi }{S}_{\theta }$, which is ${\phi }_1$ in (2). The state constraint exerts to $\parallel {\phi }_1\parallel$ to ensure the total inclination angle within $si{n}^{-1}\left(M_{\eta }\right)$.

Introducing a variable $x_{3d}\in {\mathbb{R}}^2$, we pass the constrained virtual control $u\left({\phi }_1\right)$ though a first-order low-pass filter with the positive definite time constant matrix ${\tau }_2\in {\mathbb{R}}^{2\times 2}$ as

$$\begin{array}{c}\hfill {\tau }_2{\dot{x}}_{3d}+x_{3d}=u\left({\phi }_1\right),x_{3d}\left(0\right)=u\left({\phi }_1\left(0\right)\right).\end{array}$$

(42)

Let $z_2=u\left({\phi }_1\right)-x_{3d}$ denote the filtering error, then the filtering error dynamics can be derived as

$$\begin{array}{cc}\hfill {\dot{z}}_2=& -{\tau }_2^{-1}{z}_2+{\dot{\phi }}_1\hfill \\ \hfill =& -{\tau }_2^{-1}{z}_2+O_2\left({\dot{e}}_2,{\ddot{x}}_{2d},{\dot{d}}_1,{\dot{\tilde{d}}}_1,{\dot{\tilde{x}}}_2,{\dot{\zeta }}_1\right)\hfill \end{array}$$

(43)

where $O_2\left(·\right)$ is a continuous function.

Consider the Lyapunov function candidate $V_2=\frac{1}{2}{e}_2^T{e}_2+\frac{1}{2}{z}_2^T{z}_2+\frac{1}{2}{\zeta }_1^T{\zeta }_1$, when $\parallel {\zeta }_1\parallel \ge {\delta }_1$. Its derivative with respect to time is

$$\begin{array}{cc}\hfill {\dot{V}}_2=& e_2^T\left(-K_2{e}_2+g_1{z}_2+g_1{e}_3+g_1{K}_{{\zeta }_1}{\zeta }_1+g_1{ϵ}_1-K_2{\tilde{x}}_2-{\tilde{d}}_1\right)\hfill \\ \hfill & +z_2^T\left(-{\tau }^{-1}{z}_2+O_2\right)+{\zeta }_1^T\left(-{\kappa }_1{\zeta }_1-\frac{|e_2^T{g}_1{ϵ}_1|+\frac{1}{2}{ϵ}_1^T{ϵ}_1}{\parallel {\zeta }_1{\parallel }^2}{\zeta }_1+{ϵ}_1\right).\hfill \end{array}$$

(44)

According to the definition of matrix $g_1$, one can conclude that $\parallel g_1{\parallel }_{max}=g_{1m}=g$. Invoking Young’s inequality, the ${\dot{V}}_2$ can be further expressed as

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -e^T\left(K_2-\frac{g_{1m}^2}{2}I\right)e_2-z_2^T\left({\tau }_2^{-1}-\frac{1}{2}I\right)z_2-{\zeta }_1^T\left({\kappa }_1-\frac{1}{2}I\right){\zeta }_1\hfill \\ \hfill & +\frac{1}{2}{O}_2^T{O}_2+\frac{1}{2}{e}_3^T{e}_3+g_{1m}\parallel e_2\parallel \parallel z_2\parallel +g_{1m}{\lambda }_m\left(K_{{\zeta }_1}\right)\parallel e_2\parallel \parallel {\zeta }_1\parallel \hfill \\ \hfill & +\frac{w_1{\lambda }_m\left(K_2\right)}{{\lambda }_n\left(J_1\right)}\parallel e_2\parallel \parallel {\epsilon }_2\parallel +\frac{w_1^2}{{\lambda }_n\left(J_1\right)}\parallel e_2\parallel \parallel {\epsilon }_{d_1}\parallel .\hfill \end{array}$$

(45)

When $\parallel {\zeta }_1\parallel < {\delta }_1$, the last term is ${\zeta }_1^T{\dot{\zeta }}_1=0$, thus, its derivative with respect to time is

$$\begin{array}{cc}\hfill {\dot{V}}_2=& e_2^T\left(-K_2{e}_2+g_1{z}_2+g_1{e}_3+g_1{K}_{{\zeta }_1}{\zeta }_1+g_1{ϵ}_1-K_2{\tilde{x}}_2-{\tilde{d}}_1\right)\hfill \\ \hfill & +z_2^T\left(-{\tau }^{-1}{z}_2+O_2\right)\hfill \\ \hfill \le & -e^T\left(K_2-g_{1m}^{2}I\right)e_2-z_2^T\left({\tau }_2^{-1}-\frac{1}{2}I\right)z_2\hfill \\ \hfill & +\frac{1}{2}{ϵ}_1^T{ϵ}_1+\frac{1}{2}{O}_2^T{O}_2+\frac{1}{2}{e}_3^T{e}_3+g_{1m}\parallel e_2\parallel \parallel z_2\parallel + g_{1m}{\lambda }_m\left(K_1\right)\parallel e_2\parallel \parallel {\zeta }_1\parallel \hfill \\ \hfill & +\frac{w_1{\lambda }_m\left(K_2\right)}{{\lambda }_n\left(J_1\right)}\parallel e_2\parallel \parallel {\epsilon }_2\parallel +\frac{w_1^2}{{\lambda }_n\left(J_1\right)}\parallel e_2\parallel \parallel {\epsilon }_{d_1}\parallel .\hfill \end{array}$$

(46)

Synthesizing (45) and (46), the inequality becomes

$$\begin{array}{cc}\hfill {\dot{V}}_2\le & -e_2^T\left(K_2-g_{1m}^{2}I\right)e_2-z_2^T\left({\tau }_2^{-1}-\frac{1}{2}I\right)z_2-{\zeta }_1^T\left({\kappa }_1-\frac{1}{2}I\right){\zeta }_1\hfill \\ \hfill & +\frac{1}{2}{ϵ}_1^T{ϵ}_1+\frac{1}{2}{O}_2^T{O}_2+\frac{1}{2}{e}_3^T{e}_3+g_{1m}\parallel e_2\parallel \parallel z_2\parallel + g_{1m}{\lambda }_m\left(K_{{\zeta }_1}\right)\parallel e_2\parallel \parallel {\zeta }_1\parallel \hfill \\ \hfill & +\frac{w_1{\lambda }_m\left(K_2\right)}{{\lambda }_n\left(J_1\right)}\parallel e_2\parallel \parallel {\epsilon }_2\parallel +\frac{w_1^2}{{\lambda }_n\left(J_1\right)}\parallel e_2\parallel \parallel {\epsilon }_{d_1}\parallel .\hfill \end{array}$$

(47)

Define $e_3=x_{3d}-{\phi }_1$ as the third surface error, and its derivative is

$$\begin{array}{c}\hfill {\dot{e}}_3={\dot{x}}_{3d}-J_{\varphi \theta }{g}_2{x}_4-J_{\varphi \theta }{d}_2\end{array}$$

(48)

where $J_{\varphi \theta }$ is the Jacobian matrix of ${\phi }_1$ in (20) with determinant equals to $C_{\varphi }^2{C}_{\theta }$. Therefore, $J_{\varphi }$ is non-singular under Assumption 3.

The virtual control variable ${\overline{x}}_4$ is designed as

$$\begin{array}{c}\hfill {\overline{x}}_4=g_2^{-1}\left(J_{\varphi \theta }^{-1}\left(K_3{\widehat{e}}_3+{\dot{x}}_{3d}\right)-{\widehat{d}}_2\right)\end{array}$$

(49)

where $K_3\in {\mathbb{R}}^{2\times 2}$ is a positive definite matrix.

Introducing a variable $x_{4d}\in {\mathbb{R}}^2$, we pass the constrained virtual control $g\left({\overline{x}}_4\right)$ though a first-order low-pass filter with the positive definite time constant matrix ${\tau }_3\in {\mathbb{R}}^{2\times 2}$ as

$$\begin{array}{c}\hfill {\tau }_3{\dot{x}}_{4d}+x_{4d}=g\left({\overline{x}}_4\right),x_{4d}\left(0\right)=g\left({\overline{x}}_4\right)\left(0\right).\end{array}$$

(50)

Let $z_3={\overline{x}}_4-x_{4d}$ denote the filtering error, then the filtering error dynamics can be derived as

$$\begin{array}{cc}\hfill {\dot{z}}_3=& -{\tau }_3^{-1}{z}_3+{\dot{\overline{x}}}_4\hfill \\ \hfill =& -{\tau }_3^{-1}{z}_3+O_3\left(e_3,{\ddot{x}}_{3d},{\dot{d}}_2,{\dot{\tilde{e}}}_3,{\dot{\tilde{d}}}_2\right)\hfill \end{array}$$

(51)

where $O_3\left(·\right)$ is a continuous function.

Consider the Lyapunov function candidate $V_3=\frac{1}{2}{e}_3^T{e}_3+\frac{1}{2}{z}_3^T{z}_3$. Its derivative is

$$\begin{array}{cc}\hfill {\dot{V}}_3=& e_3^T\left(-K_3{e}_3+J_{\varphi \theta }{g}_2{z}_3+J_{\varphi \theta }{g}_2{e}_4-K_3{\tilde{x}}_3-J_{\varphi \theta }{\tilde{d}}_2\right)\hfill \\ \hfill & +z_3^T\left(-{\tau }^{-1}{z}_3+O_3\right).\hfill \end{array}$$

(52)

From (2) and (20), after the matrix operation, one can conclude that ${\left(J_{\varphi \theta }\right)}_{ij}<1$ and ${\left(J_{\varphi \theta }{g}_2\right)}_{ij}<1$ are satisfied. Invoking Young’s inequality, ${\dot{V}}_3$ can be further expressed as

$$\begin{array}{cc}\hfill {\dot{V}}_3\le & -e_3^T\left(K_3-\frac{1}{2}I\right)e_3-z_3^T\left({\tau }_3^{-1}-\frac{1}{2}I\right)z_3+\frac{1}{2}{O}_3^T{O}_3+\frac{1}{2}{e}_4^T{e}_4\hfill \\ \hfill & +\parallel e_3\parallel \parallel z_3\parallel +\frac{w_1^2{\lambda }_m\left(K_3\right)}{{\lambda }_n\left(J_2\right)}\parallel e_3\parallel \parallel {\epsilon }_3\parallel +\frac{w_1^3}{{\lambda }_n\left(J_2\right)}\parallel e_3\parallel \parallel {\epsilon }_{d_2}\parallel .\hfill \end{array}$$

(53)

Define $e_4=x_{4d}-x_4$ as the fourth surface error, and its derivative is

$$\begin{array}{c}\hfill {\dot{e}}_4={\dot{x}}_{4d}-g_3{\phi }_2-d_3.\end{array}$$

(54)

The virtual control variables ${\overline{\phi }}_2$ is designed as

$$\begin{array}{c}\hfill {\overline{\phi }}_2=g_3^{-1}\left(K_4{\widehat{e}}_4+{\dot{x}}_{4d}-{\widehat{d}}_3\right)\end{array}$$

(55)

where $K_4\in {\mathbb{R}}^{2\times 2}$ are positive definite matrices. The control input ${\overline{\phi }}_2$ represents the input torque along x-axis and y-axis with respect to body frame of quadrotor UAV which are compound control variables of four thrust force generated by rotors shown in (5). Thus, the saturation constraints cannot be implemented directly. The design of the input saturation is introduced in the following Section C.

4.2. Controller Design for Fully Actuated System

Define $e_5=x_{5d}-x_5$ as the fifth surface error. To achieve guaranteed tracking performance of $\left\{\psi ,z\right\}$, the transformed error in (10) and its derivative are

$$\begin{array}{c}\hfill {\dot{\alpha }}_5={\gamma }_2\left({\dot{x}}_{5d}-g_4{x}_6-d_4-{\rho }_2^{-1}{\dot{\rho }}_2{e}_5\right)\end{array}$$

(56)

where ${\rho }_2=\mathrm{diag}\left({\rho }_z,{\rho }_{\psi }\right)$ and ${\gamma }_2=\mathrm{diag}\left({\gamma }_z,{\gamma }_{\psi }\right)$.

The virtual control variables ${\overline{x}}_6$ is designed as

$${\overline{x}}_6=g_4^{-1}\left(K_5{\alpha }_5+{\dot{x}}_{5d}-{\widehat{d}}_4-{\rho }_2^{-1}{\dot{\rho }}_2{\widehat{e}}_5\right)$$

(57)

where $K_5\in {\mathbb{R}}^{2\times 2}$ is a positive definite matrix.

Introduce a variable $x_{6d}\in {\mathbb{R}}^2$, and pass the virtual control ${\overline{x}}_6$ though a first-order low-pass filter with the positive definite time constant matrix ${\tau }_4\in {\mathbb{R}}^{2\times 2}$ as

$$\begin{array}{c}\hfill {\tau }_4{\dot{x}}_{6d}+x_{6d}={\overline{x}}_6,x_{6d}\left(0\right)={\overline{x}}_6\left(0\right).\end{array}$$

(58)

Let $z_4={\overline{x}}_6-x_{6d}$ denote the filtering error, and following the same procedure in (36), the filtering error dynamics can be derived with (12) and (57) as

$$\begin{array}{cc}\hfill {\dot{z}}_4=& -{\tau }_4^{-1}{z}_4+{\dot{\overline{x}}}_6\hfill \\ \hfill =& -{\tau }_4^{-1}{z}_4+O_4\left({\ddot{x}}_{5d},{\dot{\alpha }}_5,e_5,{\dot{e}}_5,{\tilde{x}}_5,{\dot{\tilde{x}}}_5,{\dot{d}}_4,{\dot{\tilde{d}}}_4\right)\hfill \end{array}$$

(59)

where $O_4\left(·\right)$ is a continuous function.

Consider the Lyapunov function candidate $V_5=\frac{1}{2}{\alpha }_5^T{\gamma }_2^{-1}{\alpha }_5+\frac{1}{2}{z}_4^T{z}_4$. Its derivative with respect to time is

$${\dot{V}}_5={\alpha }_5^T\left(-K_5{\alpha }_5+g_4{z}_4+g_4{e}_6+{\rho }_2^{-1}{\dot{\rho }}_2{\tilde{x}}_5-{\tilde{d}}_4\right)+z_4^T\left(-{\tau }^{-1}{z}_4+O_4\right).$$

(60)

Define $\parallel g_4{\parallel }_{max}=g_{4m}$ and $g_{4m}=max\left\{|C_{\varphi }S{e}_{\theta }{|}_{max},1\right\}$ since the entry $C_{\varphi }S{e}_{\theta }$ in matrix $g_1$ is bounded under the state constraints. Invoking Young’s inequality, the ${\dot{V}}_5$ can be further expressed as

$$\begin{array}{cc}\hfill {\dot{V}}_5\le & -{\alpha }_5^T\left(K_5-\frac{g_{4m}^2}{2}I\right){\alpha }_5-z_4^T\left({\tau }_4^{-1}-\frac{1}{2}I\right)z_4+\frac{1}{2}{O}_4^T{O}_4+\frac{1}{2}{e}_6^T{e}_6\hfill \\ \hfill & + g_{4m}\parallel {\alpha }_5\parallel \parallel z_4\parallel + l_2^2\parallel {\alpha }_5\parallel \parallel {\epsilon }_5\parallel + w_2\parallel {\alpha }_5\parallel \parallel {\epsilon }_{d_4}\parallel .\hfill \end{array}$$

(61)

Define $e_6=x_{6d}-x_6$ as the sixth surface error, and its derivative is

$$\begin{array}{c}\hfill {\dot{e}}_6={\dot{x}}_{6d}-g_5{\phi }_3-d_5.\end{array}$$

(62)

The virtual control variables ${\overline{\phi }}_3$ is designed as

$$\begin{array}{c}\hfill {\overline{\phi }}_3=g_5^{-1}\left(K_6{\widehat{e}}_6+{\dot{x}}_{6d}-{\widehat{d}}_5\right)\end{array}$$

(63)

where $K_6\in {\mathbb{R}}^{2\times 2}$ are positive definite matrices. The control input ${\overline{\phi }}_3$ represents the input torque and input force along the z-axis with respect to the body frame of the quadrotor UAV in which the saturation constraints cannot be implemented directly. The design of the input saturation is introduced in the next step.

4.3. Constraint Design for Actuator System

The input saturation is related to the thrust force generated by each rotor of the quadrotor UAV. The virtual control variables ${\overline{\phi }}_2$ and ${\overline{\phi }}_3$ represent the input torque along three axes and the input force along the z-axis, which cannot implement saturation constraints directly. Thus, a switch matrix and an auxiliary system are proposed to fulfill the actuator saturation constraints.

Define the final real control variable ${\phi }_4$ as

$$\begin{array}{c}\hfill {\phi }_4={\mathsf{\Phi }}^{-1}\left(\left[\begin{array}{c}{\overline{\phi }}_2\\ {\overline{\phi }}_3\end{array}\right]-K_{{\zeta }_2}{\zeta }_2\right)\end{array}$$

(64)

where $\mathsf{\Phi }$ is the matrix in (5) and $K_{{\zeta }_2}\in {\mathbb{R}}^4$ is a variable from following auxiliary system for compensating the effect of input saturation.

$${\dot{\zeta }}_2=\left\{\begin{array}{cc}-{\kappa }_2{\zeta }_2-\frac{|G^{\ast }\mathsf{\Phi }{ϵ}_2|+\frac{1}{2}{ϵ}_2^T{\mathsf{\Phi }}^T\mathsf{\Phi }{ϵ}_2}{\parallel {\zeta }_2{\parallel }^2}{\zeta }_2+\mathsf{\Phi }{ϵ}_2,\hfill & \mathrm{if}\parallel {\zeta }_2\parallel \ge {\delta }_2\hfill \\ 0,\hfill & \mathrm{if}\parallel {\zeta }_2\parallel < {\delta }_2\hfill \end{array}\right.$$

(65)

where $G^{\ast }=\left[e_4^T{g}_3{e}_6^T{g}_5\right]$, ${ϵ}_2={\phi }_4-u\left({\phi }_4\right)$ is the error of the input saturation, ${\kappa }_2\in {\mathbb{R}}^{4\times 4}$ is a positive definite matrix and ${\delta }_2$ is a small positive constant.

Remark 6.

As for the variable ${\zeta }_2$ in (64), which is defined in (65), its value is dependent on ${ϵ}_2$. If the derivative of ${\zeta }_2$ is not equal to zero in the auxiliary system, the result of ${\zeta }_2$ might render the real control ${\phi }_4$ smaller and closed to $u\left({\phi }_4\right)$. Therefore, the saturation error can be compensated by ${\zeta }_2$. When the derivative of ${\zeta }_2$ is equal to zero, the result of ${\zeta }_2$ is a small constant value, and it may affect the virtual control ${\phi }_4$ slightly since ${\delta }_2$ is a small constant.

Consider the Lyapunov function candidate $V_{46}=V_4+V_6=\frac{1}{2}{e}_4^T{e}_4+\frac{1}{2}{e}_6^T{e}_6+\frac{1}{2}{\zeta }_2^T{\zeta }_2$, when $\parallel {\zeta }_2\parallel \ge {\delta }_2$. Its derivative with respect to time is

$$\begin{array}{cc}\hfill {\dot{V}}_{46}=& e_4^T\left(-K_4{e}_4+K_4{\tilde{x}}_4-{\tilde{d}}_{\varphi \theta }\right)+e_6^T\left(-K_6{e}_6+K_6{\tilde{x}}_6-{\tilde{d}}_{\psi z}\right)\hfill \\ \hfill & +G^{\ast }{K}_{{\zeta }_2}{\zeta }_2+G^{\ast }\mathsf{\Phi }{ϵ}_2+{\zeta }_2^T\left(-{\kappa }_2{\zeta }_2-\frac{|G^{\ast }\mathsf{\Phi }{ϵ}_2|+\frac{1}{2}{ϵ}_2^T{\mathsf{\Phi }}^T\mathsf{\Phi }{ϵ}_2}{\parallel {\zeta }_2{\parallel }^2}{\zeta }_2+\mathsf{\Phi }{ϵ}_2\right)\hfill \end{array}$$

(66)

Define $g_{3m}=max\left\{\frac{1}{J_x},\frac{1}{J_y}\right\}$, $g_{5m}=max\left\{\frac{1}{J_z},\frac{C_{\varphi }{C}_{\theta }}{m}\right\}$. Invoking Young’s inequality, the ${\dot{V}}_{46}$ can be further expressed as

$$\begin{array}{cc}\hfill {\dot{V}}_{46}\le & -e_4^T{K}_4{e}_4-e_6^T{K}_6{e}_6-{\zeta }_2^T\left({\kappa }_2-\frac{1}{2}I\right){\zeta }_2+{\lambda }_m\left(K_{{\zeta }_2}\right)\left(g_{3m}\parallel e_4\parallel +g_{5m}\parallel e_6\parallel \right)\parallel {\zeta }_2\parallel \hfill \\ \hfill & +\frac{w_1^3{\lambda }_m\left(K_4{J}_3\right)}{{\lambda }_n\left(J_2{J}_4\right)}\parallel e_4\parallel \parallel {\epsilon }_3\parallel +\frac{w_1^3{\lambda }_m\left(K_4\right)}{{\lambda }_n\left(J_4\right)}\parallel e_4\parallel \parallel {\epsilon }_4\parallel +\frac{w_1^4{\lambda }_m\left(J_3\right)}{{\lambda }_n\left(J_2{J}_4\right)}\parallel e_4\parallel \parallel {\epsilon }_{d_2}\parallel \hfill \\ \hfill & +\frac{w_1^4}{{\lambda }_n\left(J_4\right)}\parallel e_4\parallel \parallel {\epsilon }_{d_3}\parallel +\frac{w_2{\lambda }_m\left(K_6\right)}{{\lambda }_n\left(J_5\right)}\parallel e_6\parallel \parallel {\epsilon }_6\parallel +\frac{w_2^2}{{\lambda }_n\left(J_5\right)}\parallel e_6\parallel \parallel {\epsilon }_{d_5}\parallel .\hfill \end{array}$$

(67)

When $\parallel {\zeta }_2\parallel < {\delta }_2$, the last term is ${\zeta }_2^T{\dot{\zeta }}_2=0$, thus, its derivative with respect to time is

$$\begin{array}{cc}\hfill {\dot{V}}_{46}=& e_4^T\left(-K_4{e}_4+K_4{\tilde{x}}_4-{\tilde{d}}_{\varphi \theta }\right)+e_6^T\left(-K_6{e}_6+K_6{\tilde{x}}_6-{\tilde{d}}_{\psi z}\right)+G^{\ast }{K}_{{\zeta }_2}{\zeta }_2+G^{\ast }\mathsf{\Phi }{ϵ}_2\hfill \\ \hfill \le & -e_4^T\left(K_4-g_{3m}^{2}I\right)e_4-e_6^T\left(K_6-g_{5m}^{2}I\right)e_6+{\lambda }_m\left(K_{{\zeta }_2}\right)\left(g_{3m}\parallel e_4\parallel +g_{5m}\parallel e_6\parallel \right)\parallel {\zeta }_2\parallel \hfill \\ \hfill & +\frac{w_1^3{\lambda }_m\left(K_4{J}_3\right)}{{\lambda }_n\left(J_2{J}_4\right)}\parallel e_4\parallel \parallel {\epsilon }_3\parallel +\frac{w_1^3{\lambda }_m\left(K_4\right)}{{\lambda }_n\left(J_4\right)}\parallel e_4\parallel \parallel {\epsilon }_4\parallel +\frac{w_1^4{\lambda }_m\left(J_3\right)}{{\lambda }_n\left(J_2{J}_4\right)}\parallel e_4\parallel \parallel {\epsilon }_{d_2}\parallel \hfill \\ \hfill & +\frac{w_1^4}{{\lambda }_n\left(J_4\right)}\parallel e_4\parallel \parallel {\epsilon }_{d_3}\parallel +\frac{w_2{\lambda }_m\left(K_6\right)}{{\lambda }_n\left(J_5\right)}\parallel e_6\parallel \parallel {\epsilon }_6\parallel +\frac{w_2^2}{{\lambda }_n\left(J_5\right)}\parallel e_6\parallel \parallel {\epsilon }_{d_5}\parallel +\frac{1}{2}{ϵ}_2^T{\mathsf{\Phi }}^T\mathsf{\Phi }{ϵ}_2.\hfill \end{array}$$

(68)

Synthesizing (67) and (68), the inequality becomes

$$\begin{array}{cc}\hfill {\dot{V}}_{46}\le & -e_4^T\left(K_4-g_{3m}^{2}I\right)e_4-e_6^T\left(K_6-g_{5m}^{2}I\right)e_6-{\zeta }_2^T\left({\kappa }_2-\frac{1}{2}I\right){\zeta }_2+\frac{1}{2}{ϵ}_2^T{\mathsf{\Phi }}^T\mathsf{\Phi }{ϵ}_2\hfill \\ \hfill & +{\lambda }_m\left(K_{{\zeta }_2}\right)\left(g_{3m}\parallel e_4\parallel +g_{5m}\parallel e_6\parallel \right)\parallel {\zeta }_2\parallel +\frac{w_1^3{\lambda }_m\left(K_4{J}_3\right)}{{\lambda }_n\left(J_2{J}_4\right)}\parallel e_4\parallel \parallel {\epsilon }_3\parallel \hfill \\ \hfill & +\frac{w_1^3{\lambda }_m\left(K_4\right)}{{\lambda }_n\left(J_4\right)}\parallel e_4\parallel \parallel {\epsilon }_4\parallel +\frac{w_1^4{\lambda }_m\left(J_3\right)}{{\lambda }_n\left(J_2{J}_4\right)}\parallel e_4\parallel \parallel {\epsilon }_{d_2}\parallel +\frac{w_1^4}{{\lambda }_n\left(J_4\right)}\parallel e_4\parallel \parallel {\epsilon }_{d_3}\parallel \hfill \\ \hfill & +\frac{w_2{\lambda }_m\left(K_6\right)}{{\lambda }_n\left(J_5\right)}\parallel e_6\parallel \parallel {\epsilon }_6\parallel +\frac{w_2^2}{{\lambda }_n\left(J_5\right)}\parallel e_6\parallel \parallel {\epsilon }_{d_5}\parallel .\hfill \end{array}$$

(69)

According the above procedures, the structure of the proposed controller is demonstrated in Figure 3.

4.4. Stability Analysis

For clarity, define vectors $s={\left[{\alpha }_1^T,e_2^T,e_3^T,e_4^T,{\alpha }_5^T,e_6^T\right]}^T$, $z={\left[z_1^T,z_2^T,z_3^T,z_4^T\right]}^T$, $\zeta ={\left[{\zeta }_1^T,{\zeta }_2^T\right]}^T$ and $\epsilon ={\left[{\epsilon }_1^T,{\epsilon }_2^T\right]}^T$.

The whole Lyapunov function candidate is considered as

$$\begin{array}{cc}\hfill V=& \frac{1}{2}\sum _{i=1}^6{V}_i+\frac{1}{2}{\epsilon }^T{P}^{\ast }\epsilon \hfill \end{array}$$

(70)

where $P^{\ast }$ is a positive definite matrix satisfying ${\overline{A}}^{\ast T}{P}^{\ast }+P^{\ast }{\overline{A}}^{\ast }=-I$ and $A^{\ast }=\mathrm{diag}\left({\overline{A}}_1,{\overline{A}}_2\right)$.

For the low-pass filter error dynamics $z_i,i=1\dots 4$, according to the bound properties in Assumptions 1, 2, 3 and 4, there exist positive constants $O_{im},i=1\dots 4$ such that $\parallel O_i\parallel \le O_{im},i=1\dots 4$ [14,24].

According to (17), (18) and (30) that $\parallel J_1\parallel =1$, the maximum value of $\parallel J_3\parallel$ is 1, and the minimum values of $\parallel J_2\parallel$, $\parallel J_4\parallel$ and $\parallel J_5\parallel$ are related to the inclination angles of the quadrotor UAV, which are predefined within valid ranges as Assumption 3. Therefore, define $J_{in}$ as the minimum value of $\parallel J_i\parallel$ where $i=2,4,5$.

Define a set of known constants as

$$\begin{array}{cc}\hfill & {\sigma }_1=I,{\sigma }_2=g_{1m}I,{\sigma }_3=I,{\sigma }_4=g_{4m}I,{\sigma }_5={\lambda }_m\left(K_{{\zeta }_1}\right)g_{1m}I,{\sigma }_6={\lambda }_m\left(K_{{\zeta }_2}\right)g_{3m}I,\hfill \\ \hfill & {\sigma }_7={\lambda }_m\left(K_{{\zeta }_2}\right)g_{5m}I,{\sigma }_8=l_1^{2}I,{\sigma }_9=w_1{\lambda }_m\left(K_2\right)I,{\sigma }_{10}=\frac{w_1^2{\lambda }_m\left(K_3\right)}{J_{2n}}I,\hfill \\ \hfill & {\sigma }_{11}=\frac{w_1^3{\lambda }_m\left(K_4\right)}{J_{2n}{J}_{4n}}I,{\sigma }_{12}=\frac{w_1^3{\lambda }_m\left(K_4\right)}{J_{4n}}I,{\sigma }_{13}=l_2^{2}I,{\sigma }_{14}=\frac{w_2{\lambda }_m\left(K_6\right)}{J_{5n}}I,\hfill \\ \hfill & {\sigma }_{15}=w_1^{2}I,{\sigma }_{16}=\frac{w_1^3}{J_{2n}}I,{\sigma }_{17}=\frac{w_1^4}{J_{2n}{J}_{4n}}I,{\sigma }_{18}=\frac{w_1^4}{J_{4n}}I,{\sigma }_{19}=w_{2}I,{\sigma }_{20}=\frac{w_2^2}{J_{5n}}I,\hfill \\ \hfill & {\sigma }_{21}=w_1-\frac{1}{2w_1^4}\left({\lambda }_1{w}_1^2+{\lambda }_2{w}_1+{\lambda }_3\right),{\sigma }_{22}=w_2-\frac{1}{2w_2^2}\left(w_2{\lambda }_4+{\lambda }_5\right),\hfill \\ \hfill & \varsigma =\frac{1}{2}\sum _{i=1}^4{O}_{im}^2+\frac{1}{2}{ϵ}_1^T{ϵ}_1+\frac{1}{2}{ϵ}_2^T{\mathsf{\Phi }}^T\mathsf{\Phi }{ϵ}_2+\frac{{\lambda }_1}{w_1^2}{\overline{d}}_1^2+\frac{{\lambda }_2}{w_1^3}{\overline{d}}_2^2+\frac{{\lambda }_3}{w_1^4}{\overline{d}}_3^2+\frac{{\lambda }_4}{w_2}{\overline{d}}_4^2+\frac{{\lambda }_5}{w_2^2}{\overline{d}}_5^2.\hfill \end{array}$$

(71)

Theorem 2.

Considering the quadrotor UAV system in the UF frame as (2), (3), under the Assumptions 1, 2 and 3, the ESO designed with proposed feedback matrices in (22), (23), the controller designed in (34), (40), (49),(55), (57), (63), (64), with low-pass filter designed in (35), (42), (50), (58), the auxiliary systems designed in (41), (65), and properly selecting gains $K_i\left(i=1\dots 6,{\zeta }_1,{\zeta }_2\right)$, ${\tau }_i\left(i=1\dots 4\right)$, ${\kappa }_i\left(i=1,2\right)$ and bandwidth $w_1$, $w_2$ such that the matrix $\mathrm{\Lambda }$ in the following is positive definite, error dynamics $\left\{s,z,\zeta ,\epsilon \right\}$ are uniformly ultimately bounded, and the prescribed performance of state $x_1$ and $x_5$ are satisfied.

$$\begin{array}{c}\hfill \mathrm{\Lambda }=\left[\begin{array}{cccc}{\mathrm{\Lambda }}_1& -{\mathrm{\Lambda }}_5& -{\mathrm{\Lambda }}_6& -{\mathrm{\Lambda }}_7\\ -{\mathrm{\Lambda }}_5^T& {\mathrm{\Lambda }}_2& 0& 0\\ -{\mathrm{\Lambda }}_6^T& 0& {\mathrm{\Lambda }}_3& 0\\ -{\mathrm{\Lambda }}_7^T& 0& 0& {\mathrm{\Lambda }}_4\end{array}\right]\end{array}$$

(72)

where

$$\begin{array}{cc}\hfill {\mathrm{\Lambda }}_1& =\mathrm{diag}\left(K_1-\frac{1}{2}I,K_2-\frac{2g_{1m}^2+1}{2}I,K_3-I,K_4-\frac{2g_{3m}^2+1}{2}I,K_5-\frac{g_{4m}^2}{2}I,K_6-\frac{2g_{5m}^2+1}{2}I\right),\hfill \\ \hfill {\mathrm{\Lambda }}_2& =\mathrm{diag}\left({\tau }_1^{-1}-\frac{1}{2}I,{\tau }_2^{-1}-\frac{1}{2}I,{\tau }_3^{-1}-\frac{1}{2}I,{\tau }_4^{-1}-\frac{1}{2}I\right),\hfill \\ \hfill {\mathrm{\Lambda }}_3& =\mathrm{diag}\left({\kappa }_1-\frac{1}{2}I,{\kappa }_2-\frac{1}{2}I\right),\hfill \\ \hfill {\mathrm{\Lambda }}_4& =\mathrm{diag}\left({\sigma }_{21}{I}^{12\times 12},{\sigma }_{22}{I}^{10\times 10}\right),\hfill \\ \hfill {\mathrm{\Lambda }}_5& =\left[\begin{array}{cccc}\frac{{\sigma }_1}{2}& 0& 0& 0\\ 0& \frac{{\sigma }_2}{2}& 0& 0\\ 0& 0& \frac{{\sigma }_3}{2}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{\sigma }_4}{2}\\ 0& 0& 0& 0\end{array}\right],{\mathrm{\Lambda }}_6=\left[\begin{array}{cc}0& 0\\ \frac{{\sigma }_5}{2}& 0\\ 0& 0\\ 0& \frac{{\sigma }_6}{2}\\ 0& 0\\ 0& \frac{{\sigma }_7}{2}\end{array}\right],\hfill \\ \hfill {\mathrm{\Lambda }}_7& =\left[\begin{array}{ccccccccccc}\frac{{\sigma }_8}{2}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& \frac{{\sigma }_9}{2}& 0& 0& 0& 0& \frac{{\sigma }_{15}}{2}& 0& 0& 0& 0\\ 0& 0& \frac{{\sigma }_{10}}{2}& 0& 0& 0& 0& \frac{{\sigma }_{16}}{2}& 0& 0& 0\\ 0& 0& \frac{{\sigma }_{11}}{2}& \frac{{\sigma }_{12}}{2}& 0& 0& 0& \frac{{\sigma }_{17}}{2}& \frac{{\sigma }_{18}}{2}& 0& 0\\ 0& 0& 0& 0& \frac{{\sigma }_{13}}{2}& 0& 0& 0& 0& \frac{{\sigma }_{19}}{2}& 0\\ 0& 0& 0& 0& 0& \frac{{\sigma }_{14}}{2}& 0& 0& 0& 0& \frac{{\sigma }_{20}}{2}\end{array}\right].\hfill \end{array}$$

(73)

Proof of Theorem 2. 

Substituting corresponding ${\dot{V}}_i$ and observer error dynamics (27), (28) into the derivative of (70) results in

$$\begin{array}{cc}\hfill \dot{V}\le & -{\varpi }^T\Lambda \varpi +\varsigma \hfill \end{array}$$

(74)

where $\varpi ={\left[s^T,z^T,{\zeta }_1^T,{\epsilon }_1^T\right]}^T$. Since the matrix $\mathrm{\Lambda }$ defined in (72) is positive definite, thus

$$\begin{array}{cc}\hfill \dot{V}\le & -{\lambda }_n\left(\mathrm{\Lambda }\right)\left(\parallel s^{\prime }{\parallel }^2+{\parallel z\parallel }^2+{\parallel \zeta \parallel }^2+\frac{{\alpha }_1^T{\gamma }_1^{-1}{\alpha }_1}{1/{\gamma }_{1n}}+\frac{{\alpha }_2^T{\gamma }_2^{-1}{\alpha }_2}{1/{\gamma }_{2n}}+\frac{{\epsilon }^T{P}^{\ast }\epsilon }{{\lambda }_m\left(P^{\ast }\right)}\right)+\varsigma \hfill \\ \hfill \le & -\vartheta V+\varsigma \hfill \end{array}$$

(75)

where $s^{\prime }={\left[e_2^T,e_3^T,e_4^T,e_6^T\right]}^T$ and $\vartheta =2{\lambda }_n\left(\mathrm{\Lambda }\right)\min \left(1,{\gamma }_{1n},{\gamma }_{2n},\frac{1}{{\lambda }_m\left(P^{\ast }\right)}\right)$. Thus, the right-hand side of (70) is upper bounded as

$$\begin{array}{c}\hfill V\left(t\right)\le V\left(0\right)\exp \left(-\vartheta t\right)+\frac{\varsigma }{\vartheta }\left(1-\exp \left(-\vartheta t\right)\right).\end{array}$$

(76)

Therefore, (70) and (76) indicate that s, z, $\zeta$ and $\epsilon$ are uniformly ultimately bounded. According to Lemma 1, the tracking error $e_1$ and $e_5$ remains within the prescribed bounds defined in (8). By appropriately choosing the constants ${\underline{\delta }}_i$, ${\overline{\delta }}_i$ and performance function ${\rho }_i$, the states $\left\{x\left(t\right),y\left(t\right),z\left(t\right),\psi \left(t\right)\right\}$, which is $x_1$ and $x_5$, can track the desired trajectory $\left\{x_d\left(t\right),y_d\left(t\right),z_d\left(t\right),{\psi }_d\left(t\right)\right\}$ with guaranteed errors. The proof is finished. □

5. Simulation Results

In this section, simulations are carried out to demonstrate the effectiveness of the prescribed performance with the attitude and input saturation controller (PPAISC) for the quadrotor UAV. Additionally, the performance of the proposed nonlinear ESO based DSC is also illustrated.

The dynamics of a typical quadrotor UAV is governed by Equation (1) with the physical parameters shown in

Table 1. Physical parameters of the quadrotor UAV.

Description	Values	Unit
Body mass	0.7735	Kg
Body inertia	$J_x=J_y=5.196\times {10}^{-3}$, $J_z=10.272\times {10}^{-3}$	Kg$·{\mathrm{m}}^2$
Rotor arm	$l=0.165$	m
Rotor thrust drag coefficient	$c=0.0046$	NA
Rotor thrust range	$T_{max}=6.1049$, $T_{min}=0.06$	N
Air drag coefficient	$K_x=K_y=K_z=0.01$, $K_p=K_q=K_r=0.01$	${\mathrm{Nms}}^2$

from a small-size quadrotor UAV platform [52]. All the physical parameters were obtained through experiment parameter identification, specifications and theoretical analysis.

The control parameters are selected, satisfying Theorem 2, as the following values:

$$\begin{array}{cc}\hfill & K_1=\mathrm{diag}\left(5,5\right),K_2=\mathrm{diag}\left(2,2\right),K_3=\mathrm{diag}\left(20,20\right),\hfill \\ \hfill & K_4=\mathrm{diag}\left(5,5\right),K_5=\mathrm{diag}\left(20,20\right),K_6=\mathrm{diag}\left(5,5\right),\hfill \\ \hfill & K_{{\zeta }_1}=\mathrm{diag}\left(0.5,0.5\right),K_{{\zeta }_2}=\mathrm{diag}\left(0.5,0.5\right),\hfill \\ \hfill & {\kappa }_1=\mathrm{diag}\left(5,5\right),{\kappa }_1=\mathrm{diag}\left(20,20\right),\hfill \\ \hfill & {\tau }_1={\tau }_2={\tau }_3={\tau }_4=\mathrm{diag}\left(0.01,0.01\right),\hfill \\ \hfill & w_1=w_2=5.\hfill \end{array}$$

(77)

In order to show the effectiveness of the proposed control scheme under attitude and input saturation, the prescribed performance control using the traditional DSC approach (PPDSC) without taking account of saturation constraints, the active disturbance rejection control (ADRC) strategies in the work of Zhang et al. [15], and, the most widely used in open source and real application, cascade-PID (CPID) control strategies [2] are also simulated as a comparison. To this end, the CPID controller is designed as follows:

$$\left\{\begin{array}{cc}\hfill e_i& =K_{F,i}\left(x_{d,i}-x_i\right)-{\dot{x}}_i\hfill \\ \hfill U_j& =K_{P,i}{e}_i+K_{I,i}\int e_{i}dt-K_{D,i}{\ddot{x}}_i\hfill \end{array}\right.$$

(78)

where $x_i,U_i,i=x,y,z,\varphi ,\theta ,\psi$ are the states and control inputs of quadrotor UAV. Additionally, the CPID controller needs a conversion algorithm between position control output $\left\{U_x,U_y,U_z\right\}$ and attitude control input $\left\{U_1,x_{d,\varphi },x_{d,\theta }\right\}$ as

$$\left\{\begin{array}{cc}\hfill U_1& =\sqrt{U_x^2+U_y^2+U_z^2}\hfill \\ \hfill x_{d,\varphi }& =ta{n}^{-1}\left(C_{\theta }\frac{U_x{S}_{\psi }-U_y{C}_{\psi }}{U_z}\right)\hfill \\ \hfill x_{d,\theta }& =ta{n}^{-1}\left(\frac{U_x{C}_{\psi }+U_y{S}_{\psi }}{U_z}\right)\hfill \end{array}\right.$$

(79)

For fair comparison, the control parameters of the CPID and ADRC are arrived at by trial and error such that these four controllers have nearly identical convergence rates, and the parameters of the PPDSC are chosen to be the same as that of the PPAISC. The original is at zero, and the target position is given at ${\left[1,-1,1\right]}^T$ with the heading angle remained at zero.

The disturbances can be caused by the external wind, air drag, blade flapping and installation error of four rotors, etc., and will cause both extra force and torque on the quadrotor UAV. Therefore, two kinds of disturbances are considered here: (1) external forces $d_{\xi }$ on the translational dynamics, which are unmatched disturbances for UF subsystem (2), and (2) external torques $d_{\eta }$ on rotational dynamics, which are matched disturbances for the UF subsystem (2). These two variables are defined as follows:

$$\begin{array}{cc}\hfill d_{\xi }& ={\left[0.6,0.5,0.2\right]}^T\hfill \\ \hfill d_{\eta }& ={\left[1.0,1.2,0.5\right]}^T\hfill \end{array}$$

(80)

The comparative results are shown from Figure 4, Figure 5, Figure 6 and Figure 7. The time response of positions is depicted in Figure 4a. With the prescribed performance approach, PPDSC and PPAISC are within the required bounds during the transit convergence and the steady state error. The response errors for all controllers are converging to a small neighborhood of the target position; here, we take the x-axis as representative as shown in Figure 4b. The steady-state errors for PPDSC and PPAISC are largely reduced at about 4.5 s compared to CPID and ADRC as shown in the zoomed section in Figure 4b. Considering PPDSC and PPAISC in the zoomed section of Figure 4b, one can conclude that the steady-state errors are slightly affected by the augmented two auxiliary systems for attitude and the input saturation constraints cause bounded variables ${ϵ}_1$ and ${ϵ}_2$.

Although the backstepping-typed control scheme shows a strong capability in stabilization nonlinear systems, the obvious drawback is the steep variation of control variables at start. For controlling a quadrotor UAV, this phenomenon reflects as a large inclination angle of body as shown in Figure 5. Under the same convergence rates, our proposed attitude saturation method effectively constrains the inclination angle, ${\sin }^{-1}\left(\parallel {\phi }_1\parallel \right)$ in (4), within the predefined requirement (15 degrees).

Figure 6 shows the simulation results of disturbance estimation for ADRC, PPDSC and PPAISC. All three controllers integrated with ESO are successfully estimates $\mathrm{\Delta }{d}_{\xi }$ and $\mathrm{\Delta }{d}_{\eta }$ at a steady state. The main difference is that, in ADRC, the six ESOs are independent to each other inside each control channel, and only single measurement are considered for the observation feedback. For the proposed nonlinear ESO in PPAISC, the translational and rotational dynamics are considered together in one high order ESO. Thus, using multiple observation feedback, a uniform convergence speed of estimation can be reached. The simulation results exhibited a small variation during response and fast tracking of disturbance estimation compared to the ADRC method. In Figure 6b, the simulation of disturbance estimation for PPDSC during 0 to 0.3 second appears as large deviations, which is caused by the unmodeled limitation of input variables as shown in Figure 7. Compared to the proposed PPAISC with input saturation, the out-ranged control variables are dominated by the output of the auxiliary systems result as constrained control variables for the estimation of disturbances in ESO. Furthermore, the system stability is also enhanced.

To quantitatively compare the response performance of various controllers, six performance indices are used as following:

(1)

Integral squared errors (ISE) of position are defined as [31]

$$ISE={\int }_0^t\left(x_e^2\left(t\right)+y_e^2\left(t\right)+z_e^2\left(t\right)\right)dt$$

where $x_e\left(t\right)$, $y_e\left(t\right)$ and $z_e\left(t\right)$ are position errors, thus the controller with lower ISE index reflects a fast convergence speed.

(2)

Integral time-multiplied absolute errors (ITAE) of position are defined as [31]

$$ITAE={\int }_0^{t}t\left(|x_e|\left(t\right)+|y_e|\left(t\right)+\left|z_e\right|\left(t\right)\right)dt.$$

Different from ISE, ITAE considers the steady-state error rather than the initial response, thus the controller with lower ITAE index reflects a smaller steady-state errors.

(3)

Maximum inclination angle (MIA) is ${\sin }^{-1}\left(\parallel {\varphi }_1\parallel \right)$.

(4)

Variance of thrust force (VTF). The controller with a lower VTF index reflects a smooth output and less aggressive maneuvers.

(5)

Root mean square error of estimated force disturbances (RMSEEFD). The observer with lower RMSEEFD index means a faster convergence speed and fewer oscillations during estimation of the force disturbances on the translational dynamics.

(6)

Root mean square error of estimated torque disturbances (RMSEETD). The observer with lower RMSEETD index means a faster convergence speed and less oscillations during estimation of the force disturbances on the rotational dynamics.

The quantitative results of performance indices are collected in

Table 2. The performance comparisons of different control scheme.

Controller	ISE	ITAE	MIA	VTF	RMSEEFD	RMSEETD
CPID	$2.3334$	$2.4060$	$19.5259$	$3.3635$	-	-
ADRC	$1.8498$	$1.9539$	$23.8298$	$3.0931$	$1.8140$	$4.0141$
PPDSC	$\mathbf{1.6102}$	$\mathbf{1.2179}$	$22.3933$	$2.9834$	$1.5764$	$3.2063$
PPAISC	$1.69476$	$1.2330$	$\mathbf{15.0335}$	$\mathbf{2.2569}$	$\mathbf{0.6319}$	$\mathbf{2.1411}$

. Since there is no observer in the CPID method, RMSEEFD and RMSEETD items are neglected. The control method of PPDSC is the same as PPAISC except attitude and input saturation constraints. Therefore, the lowest values in the term ISE and ITAE reflect faster convergence character and less steady-state errors of PPAISC and PPDSC compared with ADRC and CPID. The effectiveness of our proposed attitude and input saturation constraints are shown in the values of MIA and VTF. The values in RMSEEFD and RMSEETD show that the proposed ESO design for quadrotor UAV system has faster convergence and fewer oscillations, compared to conventional ESOs.

6. Conclusions

Prescribed performance control for quadrotor UAV under limited attitude and input saturation constraints are addressed in this paper. Based on the underactuated subsystem of the quadrotor UAV, a nonlinear extended state observer is proposed to tackle with the pure-feedback system, and multiple observations are designed to estimate the matched and unmatched disturbances on translational and rotational dynamics. In addition to the important input saturation, the safe attitude range for the quadrotor UAV is also considered as a saturation constraint in the control scheme with a compensation auxiliary system. In order to implement attitude saturation constraints, dynamic surface control is adopted with prescribed performance to guarantee the convergence. As verified by employing the Lyapunov technique, all the errors are uniformly and ultimately bounded. Finally, comparison simulations are demonstrated to verify the effectiveness of the proposed control strategy.

In future research, aiming for flying safety for the quadrotor UAV in a limited space, the fault tolerant control will be taken into account with more complex input constraints, such as loss of effectiveness, input bias and input delay [31]. Moreover, the finite-time convergence technique for ESO [53] will be considered to further improve the estimate performance. For practical implementations, the measurement noise is also a challenge issue for designing an ESO for the quadrotor UAV system [54].