Fault Tolerant Formations Control of UAVs Subject to Permanent and Intermittent Faults

The paper addresses the formation control of unmanned aerial vehicles (UAVs) in the presence of permanent and intermittent faults in each UAV. A fault tolerant control (FTC) scheme is developed to accommodate the permanent fault. It further shows that for the intermittent fault, the formation stability can be maintained under some conditions of fault appearance and disappearance without requiring to take any FTC action. Simulation results show the efficiency of the proposed method.


Introduction
Formation flight of UAVs has attracted a great deal of interest in recent years.The structure of formation can be generally classified as leaderfollower, virtual-leader and so on [1].At present, studies of UAVs formation commonly use simple first-order or second-order kinematic UAV model, which can not describe dynamic behaviors of UAVs in details.Moreover, these models cannot reflect the influences of UAV's own controller on formation flight and the type of fault as well.
On the other hand, a fault is an unpermitted deviation of at least one characteristic property or parameter of the system from the standard condition.The impact of a fault can be a small reduction in efficiency, but could also lead to overall system failure.Thus, an FTC scheme could have been designed to accommodate the fault.Faults can be classified according to their time characteristics as permanent and intermittent.They can also be classified according to their location of occurrence in the system as actuator faults, sensor faults and component faults [3].
As for the permanent faults, once they occur, they will exist in the system all the time.It is necessary to design a fault tolerant controller to stabilize the faulty system [4][5][6][7].However, unlike permanent faults, the intermittent fault we pay attention to is a kind of fault that may be active at one instant of time causing a malfunction of system or may be inactive at another instant allowing the system to operate correctly.The intermittent fault often exists in electronic equipments and may be caused by noise, wind, magnetic or any other disturbance in the environment.It is well known that FTC takes time and cost.It is often not admissible to apply FTC scheme every time, since intermittent fault may occur frequently.
In this paper we consider the FTC problem of a UAVs formation in leader-follower structure, where each UAV may have both permanent and intermittent faults.Inspired by the idea proposed in [2], we divide the UAVs formation into outerloop and inner-loop: the outer-loop controls the whole formation; the inner-loop controls UAV's own dynamics and kinematics behavior.
The main contributions of this paper are as follows: 1.As for the permanent fault, a compensation term is added to the nominal controller of the UAV to eliminate the influences caused by the fault such that formation stability is still maintained.2. The fault tolerance under intermittent faults is analyzed by the switched system approach.It shows that under some conditions of fault appearance and disappearance, the formation stability can be maintained without requiring to take any FTC action.
The rest of this paper is arranged as follows.
Section 2 provides some preliminaries.Section 3 discusses the design method of inner-loop and outer-loop.Sections 4 and 5 respectively focus on the FTC design for permanent faults and intermittent faults.Section 6 provides simulation results, followed by conclusions in Section 7.

Outer-Loop Model
Consider the flight formation consists of q(q ≥ 2) UAVs, the topology considered here is leaderfollower structure, and each UAV has only one reference UAV.UAVi's kinematic model is where x i , y i represent UAV i's position, v i is forward speed, φ i is the angle between x-axis and v i , w i are angular velocities.
In standard leader-follower formation model, actual and desired distance, as shown in Fig. 1.The information of UAV j are all known.The error model can be described as: Fig. 1 Formation geometry Deriving Eq. 2 with regard to time leads to the dynamics of outer-loop model as follows: Note that v i and w i of UAVi can be regarded as the inputs of outer-loop model, v * i , w * i in Eq. 3 are the virtual and ideal control laws.Designing v * i , w * i properly can make the system stable.This will be discussed in Section 3.

Inner-Loop Model
Different from outer-loop model, the inner-loop model describes UAVs' own dynamical and kinematical behavior, and can be written as follows where

Permanent Fault Model
The model of permanent fault discussed here is actuator fault.The type of fault under consideration is the loss of actuator effectiveness.Let δ F i (t) represent the signal from the ith actuator that has failed.Then the permanent fault can be described as follows: where 0 < ρ ij ≤ 1, j = 1, 2, 3, 4. If ρ i equals to a unit matrix, there is no fault.

Intermittent Fault Model
The intermittent fault consider here occurs in the output channels of the onboard control processor, under which the inner-loop model changes into where A realistic model to represent the appearance and disappearance property of intermittent fault is continuous-parameter Markov chain [9,10] as shown in Fig. 2. Mode "0" and "1" represent healthy and faulty situations respectively.The probability for going from 0 to 1 at any time is λ, and the probability for going from 1 to 0 at any time is μ.The equations for these probabilities are [15][16][17] Fig. 2 Continuous two-state model where 0 ≤ λ < 1 represent the fault appearance rates, 0 ≤ μ < 1 represent the fault disappearance rates.t ≥ 0 is the infinitesimal transition time interval.Assume that the initial situation is healthy.

Problem Formulation
According to the analysis in Section 2.2, one can obtain that w i = ψi − βi .If ψ i − β i can track the given signal, then ψi − βi can track w * i as well.The ultimate goal is to design δ i (t) in Eq. 4 to as sure that the error system (3) is asymptotically stable in both normal and faulty situations.
Given a reference signal y ir (t), define where ) and ( 5) can be combined and the following augmented system can be obtained: Where the augmented system can be described as: is the initial state.At the occurrence of a fault, the purpose of FTC scheme is to make lim This work assumes that the appearance and disappearance of the fault can be detected rapidly by using a certain fault diagnosis scheme which is not the main focus of the paper.Interested readers are referred to, e.g., [3,8,11,12] for detailed information.

Design of Controller
First of all, we design v * i , w * i for outer-loop model (3).Then the controller δ i is designed for the inner-loop system to make sure that the state variables m i are bounded and v i , w i can track v * i , w * i .

Outer-Loop Controller Design
For the designed communication topology, UAV i has only one reference vehicle j.The desired v * i , w * i are as follows Where Apply Eqs. 10 to 3, one can get Obviously, the above system is input-to-state stable with regard to approaches 0, then the formation error converges to 0.

Inner-Loop Controller Design
In order to make the inner-loop outputs track the given ideal signal v * i w * i T , we design the following feedback controller of nominal system (10): where K i1 is the feedback gain of the controller, then the closed-loop system can be described as: Lemma 1 For a given constant γ , if there exist symmetric matrices Z ∈ R 10×10 and W ∈ R 4×10 such that the following linear matrix inequality (14) holds Then there exists control law such that η i is asymptotically stable, and m i is bounded.
Proof The proof is similar to [14], and thus is omitted.
Lemma 2 (Bellman-Gronwall) If some numbers t b ≥ t a , some constant C ≥ 0, N ≥ 0, and some non-negative, piecewise-continuous-function g : Proof The proof is similar to [25], and thus is omitted.
Theorem 1 If there exists a symmetric matrix P i such that the following inequality A T P i + P i A + (BK i1 ) T P i + P i BK i1 < 0 holds, then the error η i is exponential stable and m i is bounded.
Proof We choose a Lyapunov function for η i as The time derivative of V i along the solution of Eq. 10 with Eq. 13 is It follows that where λ i1 > 0 is the eigenvalue of Q i , which means that the error η i (t) of m i (t) is exponential stable.
As for m i (t) of m i (t), the reference signal y ir (t) is not considered at first, then Assuming that the station transition matrix of closed-loop system ( 17) is φ i (t) = e Am i t , and the following inequality holds φ i (t) = e A m i t ≤ m 0 e −αt , α > 0, ∀t ≥ 0 the solution of Eq. 17 can be described as: According to Bellman-Gronwall Lemma, we have Then take the reference y ir (t) into consideration, y ir (t) and matrix G are known, they are all bounded, so m i (t) is still bounded.According to the above analysis, one can see that the error η i of the inner-loop system is exponential stable and the states m i of inner-loop system are bounded.The proof is completed.

The Control Strategy
Once a permanent fault occurs, a fault tolerant controller is needed to re-stabilize the system.Actuator fault is one of common permanent faults.
The augmented system (10) with permanent fault can be described as: If ψ i , β i track the signal y ir (t), ψi , βi track the desired signal as well, then w i = ψi − βi is tracked.

Fault Tolerant Controller Design
In order to eliminate the influence of the faults, a compensation controller is designed.
The whole control law δ i (t) for the system is where c i is a function that satisfies c i = 0 i f all actuators are f ault-free Once the permanent fault occurs, the system stabilized by controller δ i (t) becomes The closed-loop system can be described as where , where X 11 is a block-matrix, then the following three conditions are equivalent: Proof The proof is similar to [14], and thus is omitted.
Theorem 2 Given a constant γ f , if there exists matrix X = X T > 0, Y, such that the following inequality holds Then compensate state feedback controller (19) exists, such that the error η i (t) is asymptotically stable.The gain of controller (19) is Proof According to [13], if holds, the system satisfies H∞ performance indicators, left-multiply and right-multiply matrix diag P −1 , I , Then the proof is completed.
It can be seen that the proposed fault tolerant control method is decentralized since each UAV just needs to know the information of its neighbor [18], and the FTC scheme is needed only for the faulty UAV rather than the whole formation, this control strategy greatly reduces the amount of computation and the efficiency of achievement is extraordinary.

FTC of Intermittent Faults
The following lemma analyzes the UAV's behavior when there is an intermittent fault.

Lemma 4
If there exists an intermittent fault, even if asymmetric matrix P i and a positive def ine matrix Q i exist, the error η i of the inner-loop system may not be exponential stable.
Proof At the occurrence of intermittent faults, Eq. 15 changes into Then Equation 16 changes into where λ i2 > 0 is the eigenvalue of Q i .The proof is completed.
Let P ij (t) denote the probability for going from state i(i = 0, 1) to state j( j = 0, 1).The equations for these probabilities are Then during time period [0, t), the time period the system keeps stable is P 0,0 (t)t,and the time period the system becomes unstable is P 0,1 (t)t.

Theorem 3 Consider system (10) with intermittent.
There exists an initial condition of m i (0) and δ i1 , such that the origin is asymptotically in probability, if Proof According to Eqs. 15 and 16, while −λ i1 t 1 + λ i2 t 2 can be described as , which means that the control law ( 12) is always available in probability.Hence, we can have lim t→∞ E (V i (t)) = 0.The proof is completed.Condition (27) provides an explicit relation among healthy and faulty situations for the maintenance of the stability, which implies that the healthy situation can compensate for the negative effect of faulty situations provided that λ i1 and μ are large enough compared with λ i2 and λ.Note that any active FTC design is not needed.Such a result can be combined with other FTC design method to improve the reliability of the flight control system with respect to intermittent faults, and to make the FTC scheme more flexible.

Simulation Results
In the simulation, the formation is composed of 5 UAVs as is shown in Fig. 3. Details of such formation model can be found in [1].The system matrices take the form:  As for the normal system, the velocity error and angular velocity error are showed in Figs. 4 and 5.

Permanent Faults
The incipient permanent fault happens at 5 s.Figures 6 and 7 show velocity and angular velocity error under incipient fault respectively.Figures 8 and 9 show the changes of control surfaces δ 2e , δ 2T and δ 2a , δ 2r under incipient fault respectively.
The severe permanent fault happens at 5 s.

Intermittent Faults
At the occurrence of intermittent fault, the fault's appearance and disappearance are showed in By changing the fault model, one can see from Figs. 17, 18 and 19 that the fault model does not satisfy the probability, so the system becomes unstable.The first time the fault happens at 2 s and disappears at 10 s, the second time the fault happens at 12 s and disappears at 16 s.

Conclusion
This paper considers the FTC problem of UAVs formation in the presence of permanent and intermittent faults.FTC is achieved in each individual UAV, the future work still focus on cooperative FTC design under which the FTC goal can be achieved by cooperation among UAVs.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Fig. 19
Fig. 19 Angular velocity error of intermittent fault