Finite Time Output Feedback Attitude Tracking Control for Rigid Body Based on Extended State Observer

In this paper, the attitude tracking control problem of output feedback is investigated. A finite time extended state observer (FTESO) is designed through the homogeneous Lyapunov method to estimate the virtual angular velocity and total disturbances. Based on these estimated states, a finite time attitude tracking controller is developed. (e numerical simulations are given to illustrate the effectiveness of the proposed control scheme.


Introduction
Recently, the issue of attitude control of rigid body has received more and more attention due to its wide practical applications such as spacecraft, robotics, and maglev planar motor [1][2][3]. Also, the complex nonlinearity of the attitude dynamics raises potential value in theory. A great deal of control methods have been proposed to deal with the attitude control problems.
In [3], two time-varying terminal sliding mode attitude control algorithms were designed to deal with the system uncertainties and external disturbances. In [4], an adaptive regulation technique was proposed to reject the disturbances whose frequency is unknown. In [5], another adaptive attitude tracking control was implemented for the stability of a rigid spacecraft with time-varying inertia components. In [6], a backstepping method was used to design the robust attitude control for the agile satellite.
Various attitude control algorithms above need the information of full states including attitudes and angular velocities for controller design. However, in actual engineering applications, due to energy cost, weight, and other factors, an angular velocity sensor cannot be installed in the mechanical systems, or the angular velocity sensors fail to work. us, the design of output feedback attitude control in which just attitude measurements are available is required.
Many great research studies have been proposed to deal with the problem of output feedback attitude control. ese methods can be divided into two strategies: the passivitybased controller and the observer-based controller. In [7], the passivity of attitude dynamics represented by quaternion was first proved and a passivity-based attitude control was proposed to guarantee the stability of rigid body attitude.
is work was further researched in [8]; the passivity of attitude dynamics represented by modified Rodrigues parameters was proven. In [9], an auxiliary dynamical system was derived to compensate the unknown angular velocities. In [10], a passivity filter which adopted attitude Euler angles was introduced; compared to other passivity-based controllers, the proposed output feedback controller can deal with external disturbances. e examples of observer-based attitude controller can be found in [11][12][13][14]. In [11], a finite time observer was proposed through geometric homogeneity and Lyapunov methods; a virtual angular velocity is introduced in this method. In [12], a continuous finite time angular velocity observer was designed, but the settling time could not be given. In [13], an observer was designed by adding a power integrator, thus the settling time can be estimated and the disturbances could be rejected by this method. In [14], a continuous terminal sliding mode observer was designed.
However, the u-modeled system dynamics and external disturbances are also the difficulties in the attitude control design. In [15][16][17], the unknown dynamics and external disturbances are addressed by simple structure estimators for nonlinear systems. Extended state observer (ESO) is another method to compensate these problems. e main characteristic of ESO is that all disturbances and uncertainties can be regarded as total disturbances. e linear ESO (LESO) was introduced in [18] and the structure of ESO became more concise and simple, thus ESO is preferred by engineering applications whose plants are too complex [19][20][21].
Inspired by [22,23], a FTESO is introduced for the attitude control without angular velocity in this paper. e main contributions of this article are listed as follows: Compared with other observer-based controllers, the presented control scheme can estimate angular velocity and disturbances of system simultaneously.
e disturbances of the attitude system can be compensated by the outputs of proposed FTESO when the attitude controller is designed. A finite time attitude controller which combined with FTESO is designed; it is an improvement of [11]. Due to the introduction of ESO, the proposed control framework has higher control accuracy and faster response.
is article is organized as follows. In Section 2, system dynamical models, some definitions, and lemmas are shown; the proposed FTESO-based controller is shown in Section 3; simulations are given in Section 4 and Section 5 shows the conclusions and future work.

Equations of Attitude Kinematics and Dynamics.
In this section, attitude system equations of a rigid body, which include attitude kinematics and dynamics, are presented. e modified Rodrigues parameters (MRPs) are introduced to express the attitude of the rigid body; the equations of attitude kinematics are given as follows: where σ � [σ 1 , σ 2 , σ 3 ] T ∈ R 3 is the MRPs which denotes the rotation from the body frame to the inertial frame and ω � [ω 1 , ω 2 , ω 3 ] T ∈ R 3 is the angular velocity of the rigid body expressed in the body frame. e matrix T(σ) is defined as follows: where the operator σ × represents the skew symmetric matrix of vector σ and has the following form: e attitude dynamics of rigid body is modeled as: 3 denotes the external disturbances applied to the rigid body, and J ∈ R 3×3 denotes the inertial matrix of the rigid body which is symmetric.

Definitions and Lemmas.
For simplicity, some definitions about vector x � [x 1 , x 2 , . . . , x n ] T ∈ R n are given: where sign(·) represents the signum function and diag(·) represents the diagonal matrix.
Lemma 2 (see [25]). For any x i ∈ R, i � 1, 2, . . . , n, Lemma 3 (see [22]). Consider the system like equation (4); assume that a Lyapunov function V(x) is defined on a neighbourhood of U ⊂ R n of the origin, and where m ≥ 1, n ∈ (0, 1) and l 1 > 0, l 2 > 0. en, the system is locally finite time stable, and the V(x) which starts from U can arrive at V(x) ≡ 0 in finite time T: where V 0 is the starting value of V(x), and F(·) denotes the Gaussian hypergeometric function.

Finite Time Extended State Observer Design.
In this section, only the attitude MRPs σ is the available information; an ESO which ensures finite time convergence is presented to estimate the virtual angular velocity x 2 and the external disturbances d * (t).
Designate the external disturbances d * (t) as an extended state x 3 ; the system in equation (4) can be transformed to a new third-order dynamics equation which is shown in the following formulas: where w(t) � _ d * (t), and the following assumption is given.
e external disturbances d * (t) are unknown but continuously differential and bounded; its first time derivative w(t) is also unknown and bounded; the following inequalities hold: where d * and w represent the upper bound of d * (t) and w(t), respectively. Let z 1 , z 2 , and z 3 represent the estimates of x 1 , x 2 , and x 3 in equation (7), respectively, then the FTESO is designed as follows: where ϵ 1 � x 1 − z 1 represents the estimation error of attitude MRPs and θ 1 , θ 2 , and θ 3 are the parameters to deal with the external disturbances. 3 , and the error dynamical equations of the proposed FTESO is expressed as: e following compact set is constructed for any constant Δ: is compact set can be easily obtained by designing a proper desired attitude reference trajectory. If the set Ω 1 is established, there will exist a positive constant λ such that e stability properties of the extended state observer is presented in the following theorem. Theorem 1. Consider the error dynamical system given by equation (10) if the compact set Ω 1 holds, ρ, θ 1 , θ 2 , and θ 3 are sufficiently large, then the estimation errors would converge to a residual region in finite time.
Proof. Consider the error dynamics in equation (10); ignore the terms ρ|ϵ 1 (10) can be reduced to the following form: It can be acquired that equation (12) is homoge- e Lyapunov function is proposed as follows: , P denotes a positive definite symmetric matrix. Make f α represent the vector field of system equation (12); it can be obtained that V α (ζ) and L f α V α (ζ) are homogeneous of degrees 2/φ and 2/φ + α 1 − 1 with the same weights in system equation (12), respectively. According to [26,27], the inequality is obtained as follows: where Same as the previous analysis, ignore the terms . e other homogeneous system can be got from equation (10): Mathematical Problems in Engineering Similarly, the system of equation (15) is homogeneous of degree (β 1 − 1) > 0 with respect to the weight (1, α 1 , 2α 1 − 1).
Define the same Lyapunov function as equation (13) as follows: tive definite symmetric matrix. Make f β represent the vector field of system equation (15); it can be obtained that V β (ζ) and L f β V β (ζ) are homogeneous of degrees 2/φ and 2/φ + β 1 − 1 with the same weights in system equation (15), respectively.
Once again the following inequality is derived according to [26,27] L where μ 2 � − max Considering the error dynamics in equation (10), the following Lyapunov function is chosen as where tive definite symmetric matrix same as in equations (13) and (16), then take first time derivative of V: Substituting equations (14) and (17) into equation (19), the following formula can be got: where λ max (P) refers to the maximum eigenvalue of matrix P. e following inequality is derived according to Lemma 1: wherej � 1, 2, 3, and the following result is got: e following inequality is derived according to Lemma 2: e following inequalities can be derived in the same way according to Lemma 2: So, substituting equations (22)-(25) into the inequality equation (20), the following inequality can be got: where , φλ min (P) 1− (φ/2) , Two cases are discussed in further analysis.
where V (0) denotes the initial value of V. Case 2. V < 1. In this case, equation (26) can be simplified to where the parameter ρ 0 is selected to ensure that 0 < ρ 0 < 1. 6 ) ≥ 0, the Lyapunov function V will converge to the residual region Ω 2 in finite time T 1 � t 1 + t 2 : where V (t 1 ) represents the value of V at time t 1 . e estimation error can be written as In next step, the parameters θ 1 , θ 2 , and θ 3 will be analyzed to ensure (ϵ 1 , ϵ 2 , ϵ 3 ) converge to zeros within finite time.
Define the following Lyapunov function: Take time derivative of V ϵ 1 : According to equation (33), ϵ 2 will converge to the bounded region Ω 3 , and then selecting the parameter θ 1 > Ω 3 , the inequality equation (35) can be simplified to (39) So, ϵ 1 can converge to zeros within time T 2 � T 1 + t 3 : 1 � 0 is got; according to [28], discontinuous terms include that sign(ϵ 1 ) can be considered as equivalent input; the error dynamical system equation (10) can be reduced to
At this point, the proof is complete.

Finite Time Attitude Controller Design Based on FTESO.
In this section, inspired by Zou [11], a finite time controller is introduced to stabilize the rigid body to the desired attitude. Because only the attitude measurement is available, the estimation of virtual angular velocity z 2 and external disturbances z 3 are utilized in the design procedure. 1 /] 2 < 1, and ] 1 and] 2 are positive odd integers. e proposed finite time attitude controller is shown as follow: (9) is employed, the desired attitude can be tracked within finite time.

Theorem 2. Consider the system expressed in equation (4); if finite time control torque in equation (40) combined with FTESO in equation
Proof. Define the Lyapunov function: where V i is defined as Taking time derivative of Lyapunov function V f , the result can be got as follows: In this section, just the terms _ V 0 and 3 i�1 _ V i need to be derived. After tedious derivation, it could be obtained that where τ i � zη i /zt. Substitute the control torque equation (40) into equation (47). e _ V i can be derived as follows: Mathematical Problems in Engineering 7 Combining equation (48) with equations (42) and (46), Choose k 3 > 0 and let where k 4 is the positive constant. e following inequality is obtained: where Hence, the following inequality is obtained: where where (45) and (54), According to eorem 1, ϵ(t) would converge to zeros in finite time, so equation (55) becomes Because 0 < (1 + ])/2 < 1, the tracking errors e converge to zeros in finite time [11]. e proof is completed.
□ Remark 1. In practical situations, due to the absence of certain parameters, the system function f(x 1 , x 2 ) can not be calculated accurately. To deal with this problem, the system function f(x 1 , x 2 ) and external disturbances d * (t) can be regarded as total disturbances d total (t). e control algorithm proposed in this article can be written as

Mathematical Problems in Engineering
Remark 2. In order to eliminate the chattering issue caused by discontinuous terms in the observer and controller, the signum function can be replaced by the following function: where c o and δ are constants which can be designed.

Simulation Results
In this section, several numerical simulations are implemented to show the effectiveness of the FTESO-based control scheme proposed in this paper; meanwhile, the control scheme with general LESO and the control scheme without ESO in reference [11] are introduced as comparisons.
e initial states are designed as e desired attitude trajectory is designed as e upper bound of control torque is selected as |τ i | ≤ 50 Nm, where i � 1, 2, 3.
First, the proposed scheme includes that equations (9) and (40) are simulated. In this simulation, the system function f(x 1 , x 2 ) is assumed to be known; the proposed  Figures 1 and 2. According to Figure 1, the proposed control scheme can converge the tracking errors to zeros within about 4 seconds, and the extended state z 3 can also estimate the total disturbances d * (t) in finite time.
en, consider the situation that the system function f(x 1 , x 2 ) is unknown, which is regarded as total disturbances d total (t) together with external disturbances d * (t).
e simulation results are shown in Figures 3 and 4. Compared with above results, the total disturbances d total (t) is much larger than d * (t) in Figure 2, but it can be fast estimated by FTESO. Because the estimated total disturbances can be compensated in controller, even the system function f(x 1 , x 2 ) is unknown; the proposed control scheme can also converge the tracking errors to zeros within finite time. e FTESO guarantees that the two simulations in Figures 1 and 3 have the similar results.
For comparison, the introduced linear LESO is designed as follows: e corresponding controller is similar to the one designed in this article: e parameters are designed as ρ 1 � 28, ] � 7/9, k 1 � 1.1, k 2 � 1.5, and k 3 � 0.01. e other control scheme in [11] is expressed as  _ z 1 � z 2 + θc 1 ε 1 α 1 sign ε 1 , where the parameters are selected to be same as in [11]: θ � 3, c 1 � 1, c 2 � 0.2, ] � 7/9, k 1 � 1.1, k 2 � 1.5, and k 3 � 0.01. e comparison results between these three attitude control schemes are shown in Figure 5. As shown in Figure 5, the proposed control scheme in this article and the control algorithm based on LESO in equations (59) and (60) can attain faster convergence than the controller in equation (61), and the proposed control scheme gets the best disturbances rejection.

Conclusion
e finite time attitude tracking control for rigid body is studied in this article; a FTESO is designed through the homogeneous Lyapunov method. e angular velocity and total disturbances can be estimated by the proposed observer. Based on this, a finite time attitude controller is introduced. Numerical simulations show that the proposed control scheme can achieve fast convergence and good disturbances rejection. Further work will be focused on reducing power cost and dealing with the input saturation problem.

Data Availability
All data generated or analyzed during this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Mathematical Problems in Engineering 11