Integrated Fault-Tolerant Control for Close Formation Flight

This paper investigates the position-tracking and attitude-tracking control problem of close formation flight with vortex effects under simultaneous actuator and sensor faults. On the basis of the estimated state and fault information from unknown input observers and relative output information from neighbors, an integration of decentralized fault estimation and distributed fault-tolerant control is developed to deal with bidirectional interactions and to guarantee the asymptotic stability and <inline-formula><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> performance of close formations.


I. INTRODUCTION
Formation control of unmanned aerial vehicles (UAVs) has gained considerable attention in recent years. Close formation flight (CFF) is defined as formation geometry with a lateral spacing that is less than a wingspan in between UAVs. Multiple UAVs that fly in a CFF pattern can achieve a significant reduction in power demand, thereby improving cruise performances, extending mileage and increasing payload via induced drag reduction [1], [2]. This drag reduction in CFF is due to beneficial wake-vortex encounters. A Lead UAV generates vortices that induce an up-wash on the wing and a side-wash on the vertical tail behind the Wing UAVs [3], [4]. Thus, positive effects can be obtained by specifying the close formation concept.
Various control approaches, such as adaptive control [5], sliding-mode control [6] and receding horizon control [7], have been introduced by studies that focused on CFF problems. Most of the results are based on CFF models that 0018-9251 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. topology [17]. (iii) This study considers the integration of decentralized FE and distributed FTC protocols compared with the globally decentralized FE/FTC structure [27], [28]. The bidirectional interactions between FE and FTC systems are also considered. Furthermore, unlike FTC designs based on local state information [29] or estimated fault information [17], [24], [31], the proposed FTC protocol is implemented in a fully distributed manner based on the estimated information in the FE system and on the output information of the neighbors.
The remainder of this paper is organized as follows: in Section II, the model description and system formulation are introduced. Section III is devoted to the decentralized FE design. Two types of distributed control schemes including the separated and integrated FE/FTC designs are presented in Section IV to achieve the good tracking of attitude and position commands. Simulation in Section V validates the efficiency of the proposed control algorithm.
Finally, conclusions follow in Section VI.
Notations: The symbol † denotes the pseudo inverse, ⊗ denotes the kronecker product, He (X) = X + X T , and represents the symmetric part of the specific matrix.
Graph theory: An undirected graph G is a pair (ν, ς), where ν = {ν 1 , · · · , ν N } is a nonempty finite set of nodes and ς ⊆ ν × ν is a set of edges. The edge (ν i , ν j ) is denoted as a pair of distinct nodes (i, j). A graph is said to be undirected with the property (ν i , ν j ) ∈ ς that signifies (ν j , ν i ) for any ν i , ν j ∈ ν. Node j is called a neighbor of node i if (ν i , ν j ) ∈ ς. The set of neighbors of node i is denoted as N i = {j | (ν i , ν j ) ∈ ς}. The adjacency matrix A = [a ij ] N ×N is represented as the graph topology. a ij is the weight coefficient of the edge (ν i , ν j ) and a ii = 0, a ij = 1 if (ν i , ν j ) ∈ ς, otherwise a ij = 0. The Laplacian matrix L = [l ij ] N ×N is defined as l ij = i =j a ij and l ij = −a ij , i = j.

II. MODEL DESCRIPTION AND SYSTEM FORMULATION
In this section, the CFF modeling including formation-hold autopilots, kinematics, and aerodynamic coupling vortex effects are effectively established. The simultaneous actuator and sensor fault modeling in the longitudinal, lateral and vertical directions are further introduced.

A. Close formation modeling
It is first envisaged that the Lead UAV is equipped with the Mach, Heading and Altitude hold autopilots, respectively [8].v where v 0 , ψ 0 and h 0 are the Lead s velocity, heading angle and altitude respectively, v 0c , ψ 0c and h 0c denote the reference inputs, τ v , τ ψa , τ ψb , τ ha and τ hb are constant parameters.
According to the property of the Coriolis equation, the velocity of the Lead UAV in the Wing-frame is described This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. where v, ω and R denote the velocity, angular velocity and position, respectively. The superscript W represents the Wing-frame and the subscripts l and w refer to the Lead and Wing UAVs, respectively. The vectors (2) in the longitudinal, lateral and vertical directions are represented as where x W , y W and z W denote the relative separations between the Lead and Wing UAVs in the Wing-frame.
Furthermore, the velocity of the Lead UAV in the Wing-frame is given by v W l = C W L v L l with the rotation matrix C W L from the Lead-frame to the Wing-frame in the following form: On substituting (3) and (4) into (2), the nonlinear kinematics in the three directions are given bẏ The flight control of the Wing UAV is essential to be accommodated in close formation geometry to account for aerodynamic coupling vortices from up-washes and side-washes of the Lead UAV. Here, the stability derivatives C Dw , C Lw and C Sw in the Wing s drag, lift and side force are modeled in the following forms [8], [23]: whereȳ = y W /b andz = z W /b, C L l and C Lw denote the lift coefficients of the Lead and Wing UAVs, a w and a vt denote the lift curve slopes of the wing and vertical tail, A R is the aspect ratio of the wing, η is the aerodynamic This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. efficiency factor of the tail, S vt is the area of the vertical tail, b is the wing span, and h z is the height of the vertical tail.
The existing vortices from the Lead UAV can reduce the induced drag and increase the lift of the Wing UAV in the close formation geometry as shown in Figure 1. To achieve the minimal drag and maximum lift for fuel saving, it is determined that the derivatives of the stabilities are given by and hence it can be shown that the optimal separations between the Lead and Wing UAVs areȳ = ±π/4 and z = 0 in solving the equality constraints (7). To determine the changes in the drag, lift and side force, linearization is performed on the basis of the optimal close formation geometry with the relative separations in the lateral and vertical directions asȳ = π/4 andz = 0. where and ∂ C Dw /∂z = ∂ C Lw /∂z | (ȳ = π/4,z = 0) = 0.
On the basis of the formation-hold autopilots of the Lead UAV (1) and the optimal stability derivatives (8), the formation-hold autopilots of the i-th Wing UAV in-line (i = 1, · · · , N ) as shown in Figure 1 are represented aṡ where y i,i−1 and z i,i−1 denote the relative separations between the i-th UAV and the (i − 1)-th UAV in the lateral and vertical directions, q is the dynamic pressure, S is the surface area of the elliptical wing, and m is the total mass of each UAV.
Define the errors , where x c , y c and z c denote the optimal separations, and x i,i−1 denotes the relative separation in the On the basis of the optimal separations x c and y c , and trimming velocity v c , the kinematics of the i-th UAV in-line can be rewritten aṡ Remark 2.1: (i) The vortex effects are difficult to measure and model. The strongly nonlinear and coupling characteristics in CFF can be represented by nonlinear but linearly parameterized functions [5], [8] or be treated as unknown functions in a nonparametric form [9]. Real-time accurate knowledge of the aerodynamic effects in CFF This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. is generally unavailable; thus, linearization based on the optimal CFF geometry is used in this study. (ii) Unlike the directed networks communicating with one preceding UAV through sensors [9], black and red arrows in Figure   1 show the undirected data transmission amongst the neighboring Wing UAVs (e.g., interval position, velocity, heading angle and angular velocity information).

B. Simultaneous actuator and sensor fault modeling
Define the input vectors u iX , u iY and u iZ , state vectors x iX , x iY and x iZ , output vectors y iX , y iY and y iZ , system uncertainties d iX , d iY and d iZ , and nonlinear item g(x iY ) for the Mach, Heading and Altitude hold autopilots, i.e., (X, Y, Z) channels in the longitudinal, lateral and vertical directions.
Assume that each UAV suffers from additive actuator and sensor faults, the dynamic models of the i-th UAV in-line (i = 1, · · · , N ) in the X, Y and Z channels are described aṡ where scalars f a vi , f a ψi and f a hi denote the additive actuator faults of the i-th UAV in the X, Y and Z input channels, f a h(i−1) denotes the additive actuator fault of the (i − 1)-th UAV in the Z input channel, f s iX ∈ R q Xi , f s iY ∈ R q Y i and f s iZ ∈ R q Zi denote the sensor faults in the output channels. Matrices A X , B X , D X , A Y , B Y , D Y , A Z , B Z and D Z are appropriate gains under specific flight conditions in (9) and (10). Matrices C X , F sX , C Y , F sY , C Z and F sZ are of known and appropriate dimensions. 0018-9251 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. are not considered herein. (iii) The nonlinear term g(x iY ) in the Y channel of the i-th UAV in-line satisfies the Moreover, the initial condition g(0) = 0 is assumed for simplicity.
Definition 2.1 [33]: Let γ > 0 and > 0 be given constants, the closed-loop system can achieve a H ∞ performance index no larger than γ, i.e., G zd < γ if the following form holds: Lemma 2.1 [34]: There exists a zero eigenvalue for the Laplacian matrix L with 1 N as a corresponding right eigenvector and all nonzero eigenvalues have positive real parts in the undirected graph G. Assume that λ i denotes This study aims to stabilize the dynamics of the i-th UAV in-line (12)- (14) in the X, Y and Z channels through an FE/FTC design involving (i) the decentralized FE protocol to estimate the state and fault information, and (ii) the distributed FTC protocol based on estimated information and relative output information of neighbors. Furthermore, the proposed controllers in CFF models are developed so that the Wing UAV s velocity, heading angle to track with the relative signals of the Lead UAV and separations in the longitudinal, lateral and vertical directions are invariable while the Lead UAV is being maneuvered.

III. DECENTRALIZED FAULT ESTIMATION DESIGN
Define the extended states and system uncertainties as and augment the dynamics of the i-th UAV (12)-(14) intȱ 0018-9251 (c) 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. where the gain matrices are described as . Furthermore, the subscript j represents the X and Z channels, respectively.
The augmented dynamics (17) is the special case of the dynamics (18) is satisfied. Furthermore, given the accessible output information rather than the state information in the real-time applications, the dynamics (18) is taken into consideration. Thus, the statex iY of the i-th dynamic UAV model in the Y channel needs to be estimated by the i-th unknown input observer in the decentralized fashion, which means that the designed observer only requires the information from the corresponding UAV rather than its neighboring where z iY ∈ R 6+q Y i is the state of the i-th unknown input observer, in order to decouple the effects of states, system uncertainties and nonlinear items.
Then, with the following equation constraints the i-th estimation error dynamics are obtained aṡ where The designed matrices M Y , G Y and J 2Y can be obtained with the derived matrices J 1Y and H Y . Furthermore,

and it follows thaṫ
Here, a sufficient condition for the existence of a robust unknown input observer (19) is given.
Theorem 3.1: There exists a robust unknown input observer (19) if the estimation error system (25) is robustly asymptotically stable with the constraints in (21)- (24).
Proof: With the definitions in (21)-(24), the estimation error system (25) is equivalent to the original estimation error dynamics (20). Hence, if (25) is robustly asymptotically stable, then (20) is also robustly asymptotically stable, indicating that lim t→∞ e iY = 0 in the presence of system uncertainties and nonlinear items. Furthermore, the objective of obtaining the unknown input observer is to design H Y and J 1Y such that (26) is robustly asymptotically stable.  (26) is robustly asymptotically stable. However, (25) and (26) show that the FE performance is influenced by the system uncertaintyd iY and the nonlinear error ∆ḡ i . (ii) The prior information of the nonlinear error ∆ḡ i and actuator and sensor faults in the system uncertaintyd iY is not required in this study. This positive effect is evident compared with the assumptions of bounded system uncertainties and nonlinearities [17], [24], [32]. (iii) Unlike the Luenberger observer, which generates residual signals and fault estimators to detect, isolate and estimate the faults [20], unknown input observers are proposed in this study. The system uncertaintyd iY and the nonlinear error ∆ḡ i can be dealt with instead of being decoupled by the following separated and integrated FE/FTC strategies.

IV. DISTRIBUTED FAULT-TOLERANT CONTROL DESIGN
In this section, the undirected topology G in Figure 1 implies that each Wing UAV in-line can receive the relative output information rather than the state information of its neighboring Wing UAVs. On the basis of the estimated information in the unknown input observers (19) and the relative output information of neighbors, two distributed protocols are put forward, namely, the separated FE/FTC and the integrated FE/FTC designs.
Consider that the Y channel represents the general description, the distributed fault-tolerant controller for the i-th UAV in the lateral direction is designed as where K Y = [K xY 1 0 1×q Y i ] denotes the augmented gain with the state feedback gain K xY ∈ R 1×5 , a ij denotes the (i, j)-th entry of the adjacency matrix A, K gY ∈ R 1×3 denotes the distributed gain and g Y is a positive scalar.
Then, the closed-loop system (13) is rewritten aṡ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.

A. Separated FE and FTC design
Note that the estimation errors in the decentralized FE system are not considered in the following separated FTC system, thus the corresponding FTC system (28) with the distributed controller (27) is derived as is the accessible output in order to verify the separated FTC performance with the matrix C xY ∈ R r Y 1 N ×5N , L is the Laplacian matrix corresponding to the undirected graph G. Hence, the objective of the separated FTC design is to devise the state feedback gain K xY and the distributed gain K gY to guarantee the robust stability of the separated FTC system (29). with Then, the state feedback gain is given by K xY =Q −1 Y 0 X 1 , and the distributed gain is given by On the basis of the condition g(0) = 0, then g(x Y ) ≤ L g x Y . According to Definition 2.1, the sufficient the Schur Lemma is used to obtain the LMI (30). This completes the proof.
Furthermore, it is shown that the nonlinear error ∆ḡ in the FTC system is not considered in the following separated FE system, thus the corresponding FE system (26) becomes where z Y 2 ∈ R r Y 2 N is the measured output with C eY ∈ R r Y 2 N ×(6+q Y i )N . Hence, the objective of the proposed FE design is to devise the gains H Y and J 1Y to guarantee the robust stability of the separated FE system (32). This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Theorem 4.2: Given a positive scalar γ 2 , matrices C eY x ∈ R r Y 2 ×5 , C eY a ∈ R r Y 2 ×1 and C eY s ∈ R r Y 2 ×q Y i , the Then, the unknown input observer gains are given by According to Definition 2.1, the sufficient condition of achieving the Thus, the Schur Lemma is applied and the proof of Theorem 4.2 is straightforward and is omitted here. Remark 4.1: (i) Graph theory is adopted to describe undirected transformation networks from an arbitrary connected topology [10] to the CFF networks in this study. (ii) Unlike the integration of fault detection and FTC mechanisms, which uses residuals between sensor measurements and desired values from monitors for detecting fault occurrence [14], [25], the proposed FE/FTC control scheme does not utilize any fault detection and isolation information to detect, identify and isolate faults. As a result, online computation is minimized and the responsiveness of distributed controllers is expedited. (iii) Unlike the design based on local state information [29], FE information [17], [24], [31] or only output estimation errors [19], [21], the proposed FTC scheme (27) is constructed in a fully distributed fashion based on estimated information in FE and on the output information of neighbors.

B. Integrated FE and FTC design
Note that the bidirectional interactions exist in both the FE and FTC systems. The estimation error e Y from the FE process influences the FTC performance and the nonlinear error ∆ḡ from the FTC process influences the FE This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
the objective of the proposed integrated FE/FTC design is to devise the state feedback gain K xY , the distributed gain K gY , and the unknown input observer gains H Y and J 1Y to guarantee the robust stability of the integrated structure system (35).

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS, MAY 2019 13
Then, the designed gains for the integrated system are given by T Yd Y +V eY +V xY < 0. Thus, the Schur Lemma is applied and the proof of Theorem 4.3 is straightforward and is omitted here.

Remark 4.2:
Note that the undirected graph plays a role in the description of the LMI formulations, i.e., where λ 2 is the smallest nonzero eigenvalue of L. In order to avoid the requirement of the global information of undirected graph, the following derivation is obtained.  (17) in the X channel is considered with available output information.
where S 3 = [0 q Xi ×4 0 q Xi ×1 I q Xi ] and S 4 = [I 5 0 5×1 I 5×q Xi ]. Note that only the estimation error e X from the FE process in the X channel influences the FTC performance and the FTC process does not influence the FE performance in turn.

Remark 4.4:
Note that the additive actuator fault f a hi occurs in the i-th UAV and the actuator fault f a h(i−1) occurs in the (i − 1)-th UAV in the Z channel. The existing fault f a h(i−1) makes the distributed FTC controller (27) not appropriate in the FTC process. Thus, the distributed fault-tolerant controller for the i-th UAV in the vertical direction is designed as Z . Furthermore, the integrated FE/FTC model in the Z channel is considered with the distributed fault-tolerant controller (41).
Note that only the estimation error e Z from the FE process in the Z channel influences the FTC performance and the FTC process does not influence the FE performance in turn.

V. SIMULATION RESULTS
In this section, an application of integrated FE and FTC scheme for CFF models with simultaneous actuator and Then, the unknown input observer gains and the FE/FTC gains are derived as      This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.     This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.   X, Y and Z channels. Compared with our previous study [23], although there exists a sharp peak in the proposed integrated FE/FTC scheme due to its rapid convergence, the integrated algorithm shows faster convergence and smaller amplitudes of the oscillations in estimated faults to an extent. i.e., (x c = 18.9m) in the X channel in Figure 11 and (y c = 7.42m) in the Y channel in Figure 12. Due to the  existence of the coupling item qS m p Dwy and the parameter selection of C xX and C eX in the separated design, more oscillation responses are shown in the separated FE/FTC in Figure 11. Compared with the separated FE/FTC in Theorems 4.1 and 4.2, the integrated FE/FTC in Theorem 4.3 shows a smaller convergence amplitude of the heading angles and separations in the X and Y channels at each fault occurring time instant because the integrated FTC system contains more information from the FE process. Figure 13 shows the position-space trajectories of each Wing UAVs for two different Lead UAV maneuvers (with actuator/sensor faults from Case 2). All the trajectories show some separation errors when the actuator or sensor faults occur, but quickly return to the rated values. Hence, the combined maneuvering cases with simultaneous actuator/sensor faults demonstrate the effectiveness of the proposed integrated FE/FTC control scheme for CFF systems, and the control objective is achieved.