Fault Tolerant Control Scheme Design for Formation Flight Control System of Multiple Unmanned Aerial Vehicles

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1. College of Electrical Engineering and Control Science, Nanjing Tech University, Nanjing 211816, P. R. China;2. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China

(Received 18 January 2017; revised 20 February 2017; accepted 5 March 2018)

Abstract:The command tracking problem of formation flight control system (FFCS) for multiple unmanned aerial vehicles (UAVs) with sensor faults is discussed. And the objective of the addressed control problem is to design a robust fault tolerant tracking controller such that, for the disturbances and sensor faults, the closed-loop system is asymptotically stable with a given disturbance attenuation level. A robust fault tolerant tracking control scheme, combining an observer withH∞performance, is proposed. Furthermore, it is proved that the designed controller can guarantee asymptotic stability of FFCS despite sensor faults. Finally, a simulation of two UAV formations is employed to demonstrate the effectiveness of the proposed approach.

Key words:formation control; fault tolerant control (FTC); unmanned aerial vehicle (UAV); robust control

0 Introduction

An unmanned aerial vehicle (UAV) is an aircraft without a human pilot aboard. The flying of UAV, which depends on the flight control system (FCS) including actuator, sensor and so on, is controlled either autonomously by on board computers or by the remote control of a pilot[1-5]. The UAV not only has many potential military and civil applications but also has great scientific significance in academic research. One of the interesting topics is the cooperative control of UAV in formation flight, in which a group of UAVs fly in a desired graph formation[6-10]. There are reasons to believe that the efficiency of group performance is higher than that of single UAV. There have been significant researches in the area of cooperative control of multiple vehicles system in recent years. For examples, the authors mainly focused on the motion consensus and formation control of UAV swarm in Ref.[11], and a novel formation control algorithm suitable for both leaders and followers was designed, in which leaders can be influenced by feedback from their teammates. A normal PID control law was studied for the closed formation of UAVs in Ref.[12], and the bound condition of a safe distance was given. In Ref. [13], the authors respectively proposed centralized and decentralized event-triggered control laws, which achieved the the circle formation. The fault of single UAV or agent will influence the entire formation and even safe, so it is necessary to study the fault tolerant control (FTC) problem of the formation[14]. Several approaches have been exploited to solve the problem about accomplishing a certain formation for faulty UAVs. A mathematical model using dual quaternion is employed to describe the spacecraft formation. The effectiveness of the proposed control method is further demonstrated in the presence of actuator fault, disturbance and parameter uncertainty as well. Moreover, the finite-time stability of the closed-loop system is guaranteed by the adaptive feedback FTC law[15]. The authors investigate the time varying fault tolerant control formation problem for high-order multi-agent systems with actuator failure in Ref.[16]. Using the adaptive online updating strategies, the bounds of actuator failures can be unknown. Then an algorithm is proposed to determine the control parameters of the FTC, where an approach to expand the feasible formation set and the formation feasible conditions is given. A fault tolerant formation control scheme of UAV based on Kalman filter is presented, which can deal with both GPS sensor failure and wireless communication packet losses in Ref.[17]. A centralized null space based approach is introduced to compose a complex mission for swarms. Bernulli model and Extended Gilbert model are employed to value the impact of the main packet loss in wireless networks[18]. The works mentioned above perform effectively in the design of formation controller, but the sensor faults in multi-UAV formation are not studied seriously. Once the attitude, velocity or position sensor of a UAV fails, the formation flight control system (FFCS) will be influenced, which motivates us to study further[19-21].

Motivated by the above discussion, the robust fault tolerant tracking control design is discussed in this paper for FFCS of multi-UAV in which sensor faults happen to a single UAV. The dynamics model of UAV formation including sensor faults and disturbances is established. Then an observer-based feedback control approach is employed. Finally, a simulation example is given to illustrate the validity of the proposed approach.

1 Problem Statement

In this study, the typical leader and wing-man formation patterns are adopted, and a typical triangle formation of UAV is shown in Fig.1[22].

Fig.1 Triangle formation of UAV

The aerodynamic coupling effects are ignored for the simplicity, and then a linear state space model for the FFCS of UAV is obtained[12]

(1)

where

In order to meet the requirements of actual conditions, the following important aspects should be taken into consideration simultaneously.

(1) Sensor fault. When the attitude, velocity or position sensor of a UAV occurs gain fault, the fault of FFCS will occur, and the measure values ofx,y,z,ψW,VWandζwill fully or partially be inflected. Under this condition, the output can be generally characterized as[23]

(2)

(3)

(4)

The sensor fault model parameterFcan be turned into

F=F0+Fδ=F0+diag{δ1,δ2}

(5)

(2) Input constraint. In view of the limited power of actuator, the actual control input force should be confined into a certain range, which means that

(6)

whereumaxdenotes the maximum control input.

According to the above mentioned sensor faults and disturbances, the FFCS model (Eq.(1)) of UAV can be described by

(7)

wherewis a bounded disturbance.

To design the tracking controller, a reference model is introduced for FFCS

(8)

Since UAV is a low cost aircraft and state variables are not all measured exactly, a close-loop system tracking controller based on an observer is designed as

(9)

(10)

(11)

(12)

(13)

where

(14)

The objective is now to compute the gain matricesK,Lto ensure the asymptotic stability of Eq.(13) thus guaranteeing Eq.(14). A straightforward result is summarized in the following theorem.

(15)

then the asymptotic stability of the closed-loop formation control system Eq.(13) is ensured and theH∞tracking control performance Eq.(14) is guaranteed with an attenuation levelη.

ProofConsider the following candidate Lyapunov function

(16)

The stability of the closed-loop model (Eq.(13)) is satisfied under theH∞performance (Eq.(14)) with the attenuation levelηif

(17)

Eq.(17) leads to

(18)

or equivalently

(19)

RemarkMost of former results just consider the wireless communication or GPS fault of the multiple aircrafts formation[14,17], but few attempts have been made towards solving the controller design problem with sensor faults. Thus, the novelty of model (Eq.(13)) with respect to existing results is that the sensor fault of FFCS is taken into consideration.

2 Fault Tolerant Control Design

In this section, the fault tolerant formation controller design problem of multiple UAVs is investigated. The design requirements mentioned above are analyzed separately, and the obtained results are utilized for the tracking controller design.

Lemma1[25]For real matricesX,YandS=ST>0 with appropriate dimensions and a positive constantγ, the following inequalities hold

XTY+YTX≤γXTX+γ-1YTY

(20)

XTY+YTX≤XTS-1X+YTSY

(21)

Lemma2[26]For real matricesA,B,W,Y,Zand a regular matrixQwith appropriate dimensions, one has

⟹

(22)

Lemma3[27]Let a matrixΞ<0, a matrixXwith appropriate dimension such thatXTΞX≤0, and a scalar α, the following inequality holds

XTΞX≤-α(XT+X)-α2Ξ-1

Lemmas 1—3 formulate the conditions under which the inequality meets the negative definite. Based on these propositions, the following theorem presents a controller design method via convex optimization.

(23)

(24)

(25)

where

Π11=P1(A-LC)+(A-LC)TP1，Π12=P1L(I-F)C+KTBTP2，Π22=P2(A+BK)+(A+BK)TP2+Q

For a convenient design, in-Eq.(19) is simplified as

Γ1+Γ2<0

where

Γ1=

Γ2=

Then in-Eq.(25) holds, ifΓ1<0 andΓ2<0.

Using the well-known property in Lemma 1, the following inequality can be deduced

Based on the above,Γ1<0 is equivalent to

(26)

(27)

Similarly,Γ2<0 can be turned into

(28)

Using Eqs.(3)—(5), in-Eq.(27) can be rewritten as

(29)

For Lemma 1 andS=ST>0, in-Eq.(29) is equivalent to

Applying Schur complement theorem on the above inequalities, the in-Eq.(23) is deduced, then Theorem 3 is proved. When sensor fault of FFCS is unknown and satisfies conditions (Eqs.(3)—(5)), the asymptotic stability of the close-loop FFCS (Eq.(13)) is ensured.

3 Simulation Results

Simulations are performed for FFCS consisting of two UAVs. The expected formation geometries are specified byxc=20 m,yc=20 m andzc=0 m, respectively. The initial state of leader UAV are set as:VL0=30 m/s,ψL0=10°，hL0=1 000 m. The initial state of follower UAV are set as:VW0=20 m/s,ψW0=20°,hW0=1 010 m. The initial formation geometries are defined as:x0=35 m,y0=35 m andz0=-10 m. The dynamic matricesAandBare valuated for the data listed as[8]: The velocity time constantτVW=5 s, the heading time constantτψW=0.75 s, the altitude time constantτha=0.3 s,τhb=3.85 s. Outside disturbance is assumed as:w2=0.02·[2sint,cos0.5t,3sint-cost]T. Furthermore, the parameter matrix of the reference model is given by

In the following, different simulation results are given using a robust controller without FTC and an observer-based FTC scheme developed in this study, respectively. First of all, a robust tracking controller without FTC is adopted. As shown in Figs.2,3, the robust controller without FTC cannot guarantee the state and control input of FFCS having a satisfactory dynamic performance. For example, the evolution ofxcannot asymptotically track commandxc.

Correspondingly, according to the linear matrix inequality in Theorem 3, one can deduce the fault tolerant tracking controller matrixKand observer gain matrixLas

Fig.2 State evolution of FFCS using controller without FTC

Fig.3 Control input evolution of FFCS using controller without FTC

By utilizing the robust fault tolerant controller obtained in Theorem 3, it can be seen from Fig.4 that the unknown sensor fault can be well regulated. Bothxandyasymptotically track the specified commandsxcandycwithin 6 s. Moreover, as shown in Fig.5, the robust fault tolerant tracking controller developed in this paper, makes that control input responses of the close-loop FFCS have a satisfactory dynamic performance in spite of the sensor fault, which demonstrates the efficiency of the proposed approach.

Fig.4 State evolution of FFCS using robust fault tolerant controller

Fig.5 Control input evolution of FFCS using robust fault tolerant controller

4 Conclusions

In this study, a novel fault tolerant tracking control design approach is proposed for a class of multiple UAVs formation control systems with sensor faults using both LMI technique and Lyapunov stability approach, which guarantees that the faulty UAVs formation has the good fault tolerant capability. Firstly, a formation control model of multiple UAVs considering the disturbances and sensor faults is introduced and a sensor gain fault model is also given. Then a robust fault tolerant tracking controller is designed in unknown sensor faulty case by means of a generalized observer. Finally, the simulation results are shown to exhibit the effectiveness of designed FTC approach.

Acknowledgements

This work was supported in part by the Post Doctoral Research Foundation of Jiangsu Province (No.1701140B), the National Natural Science Foundation of China (No.61403195), and the GF Research and Development Project of the Nanjing Tech Universities (No.201709).