A pulsed plasma thruster fault detection and isolation strategy for formation flying of satellites

Applied Soft Computing 10 (2010) 746–758

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

A pulsed plasma thruster fault detection and isolation strategy for formation flying of satellites§ A. Valdes, K. Khorasani * Department of Electrical and Computer Engineering, 1455 de Maisonneuve Blvd. W., Concordia University, Montreal, Quebec H3G 1M8, Canada

A R T I C L E I N F O

A B S T R A C T

Article history: Received 13 October 2008 Received in revised form 20 July 2009 Accepted 5 September 2009 Available online 15 September 2009

The main objective of this paper is to develop a dynamic neural network-based fault detection and isolation (FDI) scheme for pulsed plasma thrusters (PPTs) that are employed in the attitude control subsystem (ACS) of satellites tasked to perform formation flying (FF) missions. A hierarchical methodology is proposed that consists of three fault detection and isolation (FDI) approaches, namely (i) a ‘‘low-level’’ FDI scheme, (ii) a ‘‘high-level’’ FDI scheme, and (iii) an ‘‘integrated’’ FDI scheme. Based on the data from the electrical circuit of the PPTs, the proposed ‘‘low-level’’ FDI scheme can detect and isolate faults in the PPT actuators with a good level of accuracy, however the precision level is poor and below expectations with the misclassification rates as expressed by False Healthy and False Faulty parameters being too high. The proposed ‘‘high-level’’ FDI scheme utilizes data from the relative attitudes of the FF mission. This scheme has good detection capabilities, however its isolation capabilities are not adequate. Finally, the proposed ‘‘integrated’’ FDI scheme takes advantage of the strengths of each of the above two schemes while reducing their individual weaknesses. The results demonstrate a high level of accuracy (99.79%) and precision (99.94%) with a misclassification rates that are quite negligible (less than 1%). Furthermore, the proposed ‘‘integrated’’ FDI scheme provides additional and interesting information related to the effects of faults in the thrust production levels that would not have been available from simply using the low or the high level schemes alone. ß 2009 Elsevier B.V. All rights reserved.

Keywords: Fault detection and isolation Dynamic neural networks Formation flying of satellites Pulsed plasma thrusters

1. Introduction New generation of future space missions are being envisaged as groups of coordinated small-size spacecraft (micro-satellites) that can perform the same mission that large spacecraft can achieve but with much improved performance, reliability, fault tolerance, and reduced mission cost [1–6]. One type of coordinated spacecraft mission is known as the formation flying maneuvers. Examples of formation flying missions for near-Earth and deep space environments are presented in [7–11]. Malfunctions of sensors or actuators in the attitude and orbital control subsystem (AOCS) of the spacecraft can affect the performance of the entire precision formation flight. Early detection of malfunctions or faults is a mandatory requirement for safety critical systems. Fault detection and isolation (FDI) schemes for the attitude and orbital control subsystem of spacecraft have been developed in the past two

§ This research is supported in part by grants from Discovery and Strategic Projects programs of the Natural Sciences and Engineering Research Council of Canada (NSERC). * Corresponding author. Tel.: +1 514 848 2424x3086; fax: +1 514 848 2802. E-mail address: [email protected] (K. Khorasani).

1568-4946/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2009.09.005

decades. The literature on FDI for spacecraft has considered faulty components such as sensors and actuators [12–16]. The actuators that are most commonly used for large attitude maneuvers are the reaction wheels and control moment gyroscopes. Pulsed plasma thrusters (PPTs) are an alternative type of actuators that are mostly utilized for small satellites. Unfortunately, faults that produce undesirable variations in the amount of force that is generated by the PPTs are not physically measurable and observable. Therefore, the development of a FDI system for detecting and isolating faults in PPT thrusters is a challenging problem. The FDI techniques in the literature maybe divided into history-based and model-based categories. Neural networks are among the well-known history-based techniques that are capable of learning models of nonlinear systems from past input–output data. In the case of spacecraft attitude control subsystem, a number of neural network-based FDI schemes have been developed and investigated in the literature [17–21]. In this paper, a neural network-based FDI scheme that is capable of detecting and isolating faults in the PPT thrusters used in the formation flying of satellites is developed. The proposed FDI system uses a special multilayer perceptron network that is known as the dynamic neural network (DNN) [22–24].

A. Valdes, K. Khorasani / Applied Soft Computing 10 (2010) 746–758

A formation flight is defined as ‘‘two or more spacecraft that use an active control scheme to maintain the relative positions of the spacecraft’’ [25]. In this work, we consider a typical formation of three spacecraft in a near-Earth orbit for simulation experimentations. In formation flight of spacecraft at least two vehicles use an active control scheme to maintain their relative positions [26]. An alternative definition is provided in [27] where formation flying is defined as a set of more than one spacecraft in which any of the spacecraft dynamic states are coupled through a common control law. This definition is complemented with two conditions: (i) at least one spacecraft in the formation must track a desired state profile relative to another spacecraft, and (ii) the associated control law must at minimum depend upon the state of the other spacecraft. The above active control scheme or common control law can be understood as the formation flying control. There are five different formation flying control architectures that are introduced in the literature [25,27]. The leader/follower architecture is among the most common formation flying configurations. For this reason the control architecture considered and implemented in this work will be for the leader/follower scheme. For the sake of clarity, let us now explicitly state the main contributions of this work.  A novel fault detection and isolation scheme for pulsed plasma thrusters (PPTs) used in the attitude control subsystem of the formation flying of satellites is proposed by constructing four dynamic neural networks (DNNs). The proposed FDI scheme is capable of successfully detecting and isolating three classes of commonly occurring faults in PPT actuators that can adversely affect precision of the formation flying attitudes.  Notwithstanding the fact that the literature contains a large body of work on various fault diagnostic systems for common attitude control subsystem actuators (such as reaction wheels, control moment gyroscopes, and magentorquers), development of fault diagnostic systems for pulsed plasma thrusters has not been addressed. Given the fact that the forces that are generated by the PPT actuators cannot be physically measured, and given the lack of high fidelity, precise, and simple mathematical models for PPTs, development of a fault diagnostic scheme for PPTs is not a trivial task. In this work, it is demonstrated that the proposed neural network-based FDI scheme is not a complicated scheme and is indeed a reliable tool for detecting and isolating faulty PPTs.  The results obtained through extensive numerical simulation scenarios show a high level of accuracy (99.79%) and precision (99.94%) with very low and negligible misclassification rates corresponding to the False Healthy (0.03%) and the False Faulty (0.61%) metrics. The applicability of the DNN methodology for solving a fault diagnosis problem in a highly complex nonlinear system such as the formation flying systems was successfully demonstrated.  Formation flying missions are gaining more attention and interest by the space industry and academia due to a number of attractive advantages and benefits that are offered by these systems. A significant reduction in the amount of hours that could potentially be spent by the ground station personal can be achieved by implementing computationally intelligent-based methodologies that are proposed and developed in this paper. Therefore, the operational mission cost can be significantly reduced and the reliability and performance of the overall formation mission can be significantly improved. The organization of the remainder of the paper is as follows. In Section 2, the general model for the satellite’s attitude control subsystem and the dynamical model of the formation flight system

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are briefly described. In Section 3, the dynamics of the PPT’s electromechanical system that are used to simulate the satellite’s attitude maneuvers under normal and abnormal conditions are presented. In Section 4, a ‘‘low-level’’ dynamic neural networkbased fault detection and isolation scheme for the PPT of a satellite is developed. An evaluation analysis of this FDI scheme is also presented. In Section 5, a ‘‘high-level’’ dynamic neural networkbased fault detection and isolation scheme at the formation-level is developed and the simulation results are discussed and compared with the results of the ‘‘low-level’’ FDI schemes. In Section 6, the strengths and weaknesses of both FDI schemes are discussed and an ‘‘integrated’’ FDI scheme capable of taking advantage of the strengths of each scheme is proposed and evaluated through extensive simulation studies. Conclusions are presented in Section 7. 2. Formation flying and satellite attitude control subsystem Without loss of generality and for illustrative purposes, consider the formation flying of system where the leader/follower architecture is composed of three satellites. One spacecraft is designated as the ‘‘Leader’’ (s/cl) and the other two spacecraft are the ‘‘Followers’’ (s/cf1) and (s/cf2). The followers receive the attitude coordinates of the leader and according to a previously defined reference, s/cf1 and s/cf2 correct their own attitudes. From the attitude control point of view s/cl is considered as a single spacecraft. Therefore, the attitude control subsystem (ACS) of the spacecraft s/cl is operating with the absolute attitude measurements. On the other hand, the s/cf1 and s/cf2 correct their attitudes relative to the leader’s attitude and as result the ACS of s/cf1 and s/ cf2 operate with relative attitude measurements (i.e. relative to the leader’s attitude). The equations of motion describing the dynamics of a single spacecraft are given by the following nonlinear state space representation: *˙

*

* *

x ¼ f ðx Þ þ gðx Þu * y¼x

*

(1)

where the state, the control, and the output vectors are * * denoted by x ¼ ½vx ; vy ; vz ; q0 ; q1 ; q2 ; q3 T , u ¼ ½T x ; T y ; T z T and * T y ¼ ½vx ; vy ; vz ; q0 ; q1 ; q2 ; q3  , respectively. The elements of the state vector vx, vy, vz are the angular velocity of the CoM of the spacecraft with respect to the x-axis, y-axis and z-axis, respectively, and q0, q1, q2, q3 represent the quaternions [28]. The elements of the control vector Tx, Ty, Tz are the total torques that is applied about the x-, y- and z-axes of the spacecraft, respectively. Eq. (1) is used to represent the dynamics and the kinematics of each spacecraft in the formation flying system. Attitude controllers receive from ACS sensors the spacecraft’s absolute (or relative) attitude measurements and then utilize them to calculate the deviations from the desired absolute (or relative) angles. This would be translated into commands for the actuators to provide the required torques to perform absolute (or relative) attitude correction actions. Due to its simplicity and robustness, it was decided to implement a Quaternion Error Vector Command Law [29] for all the spacecraft in the formation. The only difference between the leader s/cl controller and the follower s/cf1 and s/cf2 controllers is that in the case of the followers, the measurements (angles and angular velocities) are relative to the s/cl attitude. In the formation flying mission, the three spacecraft (s/cl, s/cf1 and s/cf2) need magnetometer and gyroscope sensors. Due to the formation flying attitude control requirements, s/cf1 and s/cf2 need the s/cl attitude, therefore communication equipments such as autonomous formation flying (AFF) sensors [30] are also required. To provide the commanded torques, the three spacecraft are

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A. Valdes, K. Khorasani / Applied Soft Computing 10 (2010) 746–758 Table 1 Parameters of the parallel-plates ablative PPT electromechanic model [33].

Fig. 1. Pulsed plasma thruster (PPT) schematic diagram [31].

C f h Lc Le Lpe L0pe LT m0 ne Rc Re Rp Rpe RT Te w

Capacitance (F) Pulse frequency (Hz) Distance between electrodes (m) Internal inductance of the capacitor (H) Inductance due to wires and leads (H) Inductance due to current sheet moving down (H) Inductance per unit channel length (H m1) Total circuit inductance (H) Mass of plasma at t = 0 (kg) Electron density (m3) Capacitor resistance (V) Wire and lead resistance (V) Plasma resistance (V) Electrode resistance (V) Total circuit resistance (V) Electron temperature Width of the electrodes (m) Magnetic permeability of free space (Wb A1 m1) Characteristic pulse time (s)

m0 t

equipped with a system of pulsed plasma thrusters (PPTs). These actuators are described in detail in the next section. 3. Pulsed plasma thrusters (PPTs) actuators Attitude control for a three-axis stabilized spacecraft is typically accomplished by means of reaction wheels, magnetic torquers, and thrusters. As an alternative to this control strategy, a pulsed plasma thruster (PPT) actuator system can reach the same accuracy level but at lower cost, mass, and complexity. Furthermore, a PPT system can be used for different purposes such as station-keeping, attitude control and orbit insertion and drag make-up. Due to these reasons, in the past two decades various missions such as LES 6, LES 8/9, EO-1, TIP/NOVA and Dawgstar have been designed with PPT systems [26,31–34]. Various types of PPTs can be categorized into coaxial or parallel-plate according to the electrodes geometry and gas or ablative according to the propellant used. In this paper, we investigate a parallel-plate ablative PPT. This is the type of PPTs that are mostly used in small satellite missions and in particular is similar to the one that is utilized in the Lincoln Experimental Satellite 8/9 (LES-8/9) [31,32]. Moreover, mathematical models for the operation of these thrusters have also appeared in the literature [31,32,35]. For these reasons, we have adopted the parallel-plate ablative PPT thrusters to investigate in this work for formation flight missions. As shown in Fig. 1, the main components of a PPT are the capacitor, the electrodes, the igniter and the spring. The physics of the PPT pulse and the acceleration process are described next [31].  For a period of approximately 10 ms, the capacitor is charged to a maximum voltage of Vmax. The capacitor voltage appears across the electrodes (anode/cathode) of the thruster.  The igniter, earlier charged, receives the discharge signal. This action ablates and ionizes part of the material (e.g. Teflon) to create a conducting path which allows the discharge of the capacitor.  The discharge current ablates and ionizes the fuel bar into a plasma slug. Due to the current that flows through this circuit a self-magnetic field is produced.  Finally, the plasma is accelerated by the Lorentz force (J  B) due to the discharge current and the magnetic field. The PPT above operates electrically as an RLC circuit. By assuming that the entire fuel bar mass m = m0 is accelerated as a single unit, m is taken as a constant throughout the operational process of the PPT. A general mathematical model of the parallelplate ablative PPT thruster is described according to the following

system of ordinary differential equations [35], namely: x˙ 1 ¼ x3 x˙ 2 ¼ x4 0

1 L pe ½x3 2 2 m0   1 h x2  m0 x3 x4  RT x4 þ v  C w x˙ 4 ¼ LT y1 ¼ m0 x3 f x˙ 3 ¼

(2)

y2 ¼ x4 where x1 is the position, x2 is the capacitor charge, x3 is the velocity, x4 is the current, v is the capacitor voltage, y1 is the thrust, y2 is the discharge current, RT = Rc + Re + Rpe + Rp and LT = Lc + Le + Lpe. Eqs. (3) and (4) below provide expressions for Rp and Lpe(t), respectively, and the parameters appearing in Eqs. (2)–(4) are specified in Table 1. We furthermore assume, without loss of generality, that x1(0) = x2(0) = x3(0) = x4(0) = 0. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h u 1=2 i 7 3 u h tm0 ln 1:24  10 Te =ne (3) R p ¼ 2:57 3=4 t Te w L pe ¼ m0

h x1 w

(4)

In practical situations, the electrical variables of the PPT actuator can be measured and collected. In this paper, we assume that the above PPT model can be used as an actual PPT for generating data sets to train the developed neural networks under healthy operational conditions. To obtain a full three-axis control of a satellite minimum of four thrusters are needed. However, in this paper, we consider the socalled six-independent PPT configuration as shown in Fig. 2. In this configuration, each thruster only generates a torque about a single axis of the spacecraft where independent control actuation is achieved. By using this configuration the applied torques in the +x, x, +y, y, +z and z directions are performed by the thrusters PPT1, PPT2, PPT3, PPT4, PPT5 and PPT6, respectively. 4. Low-level satellite fault diagnosis analysis Autonomous spacecraft and formation of satellites demand detection and isolation of faulty components (e.g. sensors and actuators) as early as possible to avoid fatal failures. In practical operation of a PPT actuator only electrical and temperature variables are measured and monitored. Therefore, unplanned

A. Valdes, K. Khorasani / Applied Soft Computing 10 (2010) 746–758

Fig. 2. The six-independent PPT configuration.

variations in the amount of thrust cannot be measured directly. Malfunctions in a given PPT can be initiated due to different causes such as material fatigue and waste accumulation. In this paper, the following three commonly occurring faulty cases are considered. These cases have been identified as representing critical situations that can lead to major malfunctions in the PPTs as reported in [31,32]. Furthermore, it should be noted that these fault cases are selected and chosen specifically for the reason that they can be explicitly modeled for generating numerical data to be used for training of the proposed neural network-based fault detection and isolation (FDI) architectures.  Fault Case 1 (Mass Fault Reduction): Due to reduction in the amount of propellant’s mass that is utilized during the generation of a pulse, the amount of thrust generated is also reduced. This propellant mass reduction can be due to the loss of thruster’s spring elasticity. Under this undesirable condition, the fuel bar is not properly positioned and therefore the amount of propellant mass available will be less than the amount under normal conditions. This can consequently lead to undesirable thrust generation.  Fault Case 2 (Faulty Ablation Process): The ablation process transforms the solid propellant into the exhaust plasma, but at times small portions of the propellant may not get transformed, resulting in particles which are added to the inner face of the electrodes. After several pulses, the amount of particles added to the electrodes may reach a level to be sufficient for being ablated and ionized during the generation of new pulses, resulting in an augmentation in the amount of thrust that is generated by the pulses. This again represents an undesirable behavior in the operation of the thrust generation.  Fault Case 3 (Conductivity Fault Reduction): Due to wear and tear, conductivity of the wires, capacitor and electrodes may decrease. Consequently, the amount of discharge current produced during the generation of pulses is reduced, resulting in the reduction of the amount of thrust produced. To develop the proposed ‘‘low-level’’ or ‘‘component-level’’ FDI system, a data set composed of the capacitor voltage v(t) and discharge current y2(t) is constructed. From the PPT model one can also collect the thrust variable y1(t) for evaluation purposes, although in real practice this quantity is not a measurable variable. It should be noted that a PPT pulse is considered healthy if the thrust produced (unobservable) is within an interval V that is given by (5), otherwise, the pulse is considered as being faulty, namely we have y1min ¼ ð1  0:15Þy1nominal y1max ¼ ð1 þ 0:15Þy1nominal V ¼ ½y1min ; y1max 

The reliability of a model-based FDI scheme is strongly dependent on the accuracy and fidelity of the underlying available mathematical model. Developing precise models for complex nonlinear systems can be quite difficult and availability of mathematical models for all the system components is not always possible. Neural networks have the capability to learn a component model from historical input–output data and represent and provide an implicit model. Standard multilayer perceptron networks can be used to model only static nonlinear maps. To introduce dynamic properties to the network, which can then be used for representing a dynamical process or a system, one needs to employ dynamic networks. Dynamic neural networks (DNNs) are essentially multilayer perceptron networks that are instead embedded with dynamic neurons. A dynamic neuron can be constructed by adding an infinite impulse response (IIR) filter to a static neuron. This will ensure that the activation of a neuron depends on its internal states [22–24,36]. We have had very successful experience with DNNs that were used for fault diagnosis of various actuators and sensors in the attitude control subsystem of satellites as reported in [18,20,21]. In Ref. [17] we have applied these DNNs to actual and real telemetry data for fault diagnosis of reaction wheels (an actuator in the satellite’s attitude control subsystem) in the Canadian Space Agency’s RADARSAT-1 satellite with great success. Fig. 3 shows the general structure of the so-called dynamic neuron model (DNM) [22–24,36]. The variables U(k) = [u1(k), u2(k), . . ., un(k)]T and W ¼ ½w1 ; w2 ; . . . ; wn T are the input and weight vectors, respectively. The block fjF() is the activation function of the neuron. Parameter g, is the slope of the nonlinear activation function F(.). The dynamical model of the above neuron is described by the following set of equations:

xðkÞ ¼

n X wi ui ðkÞ i¼1

˜ yðkÞ ¼

r r X X ˜  iÞ þ ai yðk bi xðk  iÞ i¼1

(6)

i¼0

˜ yðkÞ ¼ Fðg  yðkÞÞ where the signal x(k) represents the input to the filter, the coefficients ai, i = 1, 2, . . ., r and bi, i = 0, 1, . . ., r are the feedback and feed-forward filter parameters, respectively, and r is the order of the filter. Finally, ?(k) represents the output of the filter which is the input to the activation function. Dynamic neural networks (DNNs) are therefore constructed as multilayer backpropagation networks having dynamic neurons and are employed as a modeling tool for representing the dynamics of the PPT thruster (2)–(4). The goal of the proposed FDI scheme is that by using the DNN network desirable residual signals are generated that would allow one to detect the existence of a fault in the PPT actuators without measuring the amount of the thrust that is produced in each PPT. The construction of the proposed neural network-based FDI system can be divided into four consecutive steps as described next in detail.

(5)

and y1nominal is the nominal thrust (considered to be 100 mN in this paper).

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Fig. 3. Structure of a DNM with n inputs.

A. Valdes, K. Khorasani / Applied Soft Computing 10 (2010) 746–758

750

Fig. 4. Series-parallel identification model during the training phase.

4.1. Step I: neural network training phase In this phase the parameters of the dynamic neural network (DNN) are adjusted until the required modeling accuracy for the PPT thruster is achieved. As discussed in [20], the network scheme used during the training phase is the series-parallel identification model that is shown in Fig. 4, where the inputs unorm(k) and y2norm(k) to the neural network are the normalized representation of the capacitor voltage and delayed discharge current, respectively. The PPT thruster output y2norm(k + 1) represents the normalized discharge current, and yest(k + 1) is the neural network output which represents the estimate of the discharge current. After a number of simulation trials and errors (not included here due to space limitations) it was concluded that the best results are obtained by selecting a 2-10-1 structure network (i.e. the neural network has two neurons in the input layer, ten neurons in the hidden layer, and one neuron in the output layer). The number of input and output neurons is fixed and is governed by the physics of the process. In this case, we have two measurements from the PPT that are used as inputs to the network. The number of output neurons is determined by the number of the actual process output measurements. In this case, one is provided with only one output variable, therefore the output layer has one neuron. The number of hidden neurons was determined after testing the network performance with a number of different selections. It was concluded that ten neurons yielded the best performance in modeling the PPT dynamics. Furthermore, it was experimentally found that a second order IIR filter with a hyperbolic tangent sigmoidal and linear activation functions for the neurons in the hidden layer and output layer, respectively yield the best results. To adjust the network parameters (input weights and filter coefficients that are collectively denoted by W), pairs of healthy input–output of the PPT thruster data set is used. By comparing the actual PPT system output with the estimated neural network output, the error signal e(k + 1) is calculated and then backpropagated through various layers to update the network parameters W. The training algorithm that is used is the extended dynamic back propagation (EDBP) [36]. The training phase is performed until a termination criterion (t.c.) is satisfied. For t.c. the mean square error (mse) criterion is used. The mse for the entire data set is calculated and when mse  t.c. the training is stopped. The termination criterion used is mse = 1  104. Fig. 5 shows for illustrative purposed a typical response of the actual PPT electromechanical model output and the estimated output that is generated by the dynamic neural network. To validate the modeling capabilities of the proposed DNN, 120 healthy data sets are constructed. To simulate these pulses, the capacitor voltage was initiated at different values (from 857 to 980 V). According to the healthy interval V as specified in Eq. (5) and the nominal thrust of 100 mN, the upper and lower bounds of V are considered as 115 and 85 mN, respectively.

Fig. 5. (a) Actual PPT discharge current (normalized) and (b) neural network estimated discharge current (normalized).

4.2. Step II: neural network validation phase The trained DNN in the previous step is now tested with new data sets that the network has not seen before to demonstrate the modeling accuracy of the network for different healthy conditions. Once the training phase is completed, the parameters of the dynamic neural network are fixed and the validation phase is initiated. A new set of data is applied to the DNN and its modeling performance is investigated and evaluated. During the validation phase, a modified architecture, as depicted in Fig. 6, is applied [20]. This is motivated by the observation that after the training phase is completed, one can assume that the difference between the actual output y2norm(k + 1) and its estimated value yest(k + 1) is sufficiently small. Therefore, one can employ yest(k + 1) instead of y2norm(k + 1) as an input to the neural network. Another motivating factor is the fact that for the purpose of fault diagnosis, as discussed in Steps III and IV below, the residual signal that is constructed should be defined in such a way that in case of a fault in the PPT thruster the difference between the PPT system output and the neural network output estimate is significant (i.e. ‘‘sufficiently’’ large). If instead one used the ‘‘faulty’’ y2norm(k) as an input to the neural network (as opposed to the proposed solution which is yest(k)), the sensitivity of the residual signal to the presence of a fault would have been significantly reduced. Note that the goal of the neural network is to produce an output that is as close as possible to the healthy PPT system being approximated. By using y2norm(k) as an input, the neural network

Fig. 6. Series-parallel neural network-based identification model during the validation phase.

A. Valdes, K. Khorasani / Applied Soft Computing 10 (2010) 746–758

would have been forced to produce an output which is as close as feasible to the ‘‘faulty’’ PPT thruster output, therefore compromising the neural network fault diagnosis capability (the neural network and the faulty PPT output would not have been distinguishable). For the above reasons, for the purpose of fault diagnosis the series-parallel architecture that is depicted in Fig. 6 is subsequently adopted.

By comparing the actual and the estimated outputs (i.e. the PPT system and the DNN’s output, respectively), the health status of the PPT system can be determined. This error signal is designated as the residual error. The parameter Threshold that is defined below determines if the data set presented to the neural network corresponds to a healthy or a faulty scenario. The Threshold is calculated by using the residual error that is collected from the complete set of the healthy data that was used previously for training the neural network. Specifically, we have Pn

i¼1

Pn   MAEðiÞ MAEðiÞ þ s maxðMAEÞni¼1  i¼1 n n

Table 2 Parameters OF PPT3 and PPT4. 100 [mN] 910 [V] 11,090 [A]

Max. thrust Capacitor voltage (t = 0 [s]) Max. discharge current

Three fault scenarios as described earlier in this section are considered. These are discussed further below.

4.3. Step III: threshold determination

Threshold ¼

751

(7)

where MAE(i) is the mean absolute error (residual error) of the healthy data consisting of n sets, that is i = 1, 2, . . ., n. The coefficient s is a constant which is used to adjust and influences the sensitivity of the FDI scheme.

 Fault Scenario #1: A decrease in the amount of thrust due to the mass reduction fault of PPT4 is simulated. The mass m0 = 2.58e8 kg is linearly decreased until it reaches m0 = 2.00e10 kg. This progressive reduction in the amount of mass is modeled according to the expression m0 ¼ 2:58  108  ið0:04  108 Þ;

i ¼ 1; . . . ; 41

 Fault Scenario #2: A variation in the amount of thrust, due to the particles being added to the inner surface of the PPT4 electrodes is simulated. The width of the electrode is initially at w = 0.0254 m and is linearly decreased according to the expression w ¼ 0:0254  ið0:0004Þ;

i ¼ 1; :::; 41

 Fault Scenario #3: Changes in the conductivity of the electrical components of the thruster affect the generation of the desired thrust. This faulty scenario is simulated by injecting a fault in the thruster PPT3. The resistance of the electrode Rpe = 0.00 V is gradually increased according to R pe ¼ 0:0  ið0:00015Þ;

i ¼ 1; . . . ; 64

4.4. Step IV: FDI scheme implementation The implementation of the entire FDI system constitutes the final step. The proposed DNN FDI scheme is shown in Fig. 7. The voltage u(k) = v(k) and the estimated current yest(k) = y2est(k) are normalized before being presented to the DNN. Using the actual output y2(k + 1) and the output of the network yest(k + 1) a decision making process calculates the MAE for the entire data set (corresponding to a single pulse of the thruster). The MAE is compared with the Threshold to detect if the pulse is healthy (MAE < Threshold) or faulty (MAE > Threshold). The classification signature is called the Health Status. Generally, a six-independent configuration PPT system is composed of three clusters of two thrusters sharing the same capacitor. The proposed FDI scheme can be implemented by applying and assigning one DNN to each cluster. In this way, not only a fault can be detected, but also the faulty thruster can be identified (isolated). For sake of illustration, let us present the results corresponding to thrusters PPT3 and PPT4 which generate the rotation around the pitch axis. The parameters of the thrusters are shown in Table 2.

Table 3 shows a summary of the results that are obtained under the above three scenarios. Fig. 8 shows the FDI results that are obtained by utilizing the proposed neural network scheme as given in Fig. 7. Fig. 8(a) depicts the results corresponding to the first fault scenario. According to this figure, the behavior of the PPT4 is considered healthy for the first 28 pulses and faulty for the last 13 pulses. The FDI scheme does not detect faults in any of the pulses that are generated by the PPT3. Fig. 8(b) shows the results corresponding to the second fault scenario. According to this figure, the behavior of PPT3 is considered healthy for all the 64 pulses that are generated. In the case of PPT4, the FDI scheme classifies the first 28 pulses as healthy and the last 13 pulses as faulty. Fig. 8(c) depicts the classification of the pulses that are generated by the PPT3 and PPT4. In case of PPT3, the first 51 pulses are classified as healthy while the remaining 13 pulses are classified as faulty. The 41 pulses generated by PPT4 are all classified as healthy. To evaluate the performance of the proposed FDI scheme, the confusion matrix approach is used in this work. The confusion matrix criterion is a common tool that is used in classification and Table 3 Actual and detected status of the generated pulses.

Fig. 7. ‘‘Low-level’’ or the ‘‘component-level’’ FDI scheme for the PPT system.

Fault scenario

PPT number

Actual healthy pulses

Actual faulty pulses

Detected healthy pulses

Detected faulty pulses

#1

PPT3 PPT4

64 22

00 19

64 28

00 13

#2

PPT3 PPT4

64 8

00 33

64 28

00 13

#3

PPT3 PPT4

46 41

18 00

51 41

13 00

A. Valdes, K. Khorasani / Applied Soft Computing 10 (2010) 746–758

752

Fig. 8. FDI results for the PPT3 and the PPT4 under (a) fault scenario #1, (b) fault scenario #2, and (c) fault scenario #3. (a) Scenario #1, (b) scenario #2 and (c) scenario #3.

pattern recognition research. The specific evaluation terms that are defined corresponding to this matrix are: Accuracy, True Healthy, False Healthy, True Faulty, False Faulty, and Precision. These notions are defined explicitly below. Specifically, we have Accuracy ¼

AþD ; AþBþCþD

True Healthy ¼

D ; CþD

A ; True Faulty ¼ AþB

False Healthy ¼

B AþB

C False Faulty ¼ ; CþD

D Precision ¼ BþD

and the actual confusion matrix is defined as shown below

Table 4 Confusion matrix results. Accuracy True healthy False healthy True Faulty False Faulty Precision

82.08% 55.71% 11.23% 88.77% 44.29% 55.71%

The above results indicate that the proposed FDI scheme is capable of correctly classifying only 55.71% of the healthy pulses and 88.77% of the faulty pulses. The accuracy level of 82.08% and the percentage of misclassification of faulty pulses (11.23%) are not ideally acceptable. The main disadvantages include a poor precision that could only reach a level of 55.71%, and a large percentage of healthy pulses that are misclassified as faulty (44.29%). We can therefore conclude that the proposed FDI scheme is not fully capable of correctly classifying all the generated faults. It turns out that the utilization of a fixed Threshold is the main reason in resulting an inadequate capability in detecting faults under the above three different scenarios. 5. Formation-level fault diagnosis analysis

Based on the total number of pulses that are generated by the pair of thrusters as given in Table 3, the results associated with the confusion matrix are shown in Table 4.

In this section, a ‘‘high-level’’ or ‘‘formation-level’’ FDI approach is introduced and developed. Dynamic neural networks are employed to model and represent the relative attitude of a

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Table 5 Data set used for the formation level FDI scheme. follower; j q1 : leader

angular rotation about the x-axis (s/cf,j w.r.t. s/cl)

follower; j q2 : leader

angular rotation about the y-axis (s/cf,j w.r.t. s/cl)

follower; j q3 : leader

angular rotation about the z-axis (s/cf,j w.r.t. s/cl)

follower; j leader

Dvx :

angular velocity about the x-axis (s/cf,j w.r.t. s/cl)

follower; j leader

Dvy :

angular velocity about the y-axis (s/cf,j w.r.t. s/cl)

follower; j leader

Dvz :

angular velocity about the z-axis (s/cf,j w.r.t. s/cl)

follower spacecraft with respect to the leader spacecraft in a formation flying mission. Using this neural network model, residual signals are generated for detecting the existence of faults in the actuators of the followers by assuming that the leader satellite is fault free. Note that this is a reasonable assumption in a leader/follower configuration, given the well-known fact that a fault in the leader will compromise the entire mission due to a single point of failure situation. An important advantage of the proposed high-level FDI scheme is that only data from the follower’s attitude and rates are used to detect anomalies in the follower PPT actuators. Table 5 specifies the set of ACS signals that are used in the approach that is proposed in this section. The details on the structure of the dynamic neural model and the training and validation procedures that are provided in Section 4 for the low-level FDI scheme still remain valid for the high-level scheme here. The proposed high-level FDI is composed of three dynamic neural networks (DNNs). Figs. 9–12 show the schematic of the DNNs that are used for the roll angle axis during the training phase. The same schematics can also be used to describe the proposed DNNs for the other two angles, namely the pitch and the yaw axes. Fig. 10 shows the details of the Leader ACS level and Fig. 11 shows the details of the Follower 1 ACS level as presented in Fig. 9.

Fig. 9. The high-level identification model proposed during the training phase.

Fig. 11. The Follower 1 ACS model proposed during the training phase.

Fig. 12. The high-level neural network identification model during the training phase (FDI scheme for the roll angle of follower 1).

Fig. 12 shows the details of the high-level FDI scheme for the roll angle of the Follower 1 as presented in Fig. 9. After a number of simulation trials and errors (not included here due to space limitations) it was concluded that the best results are obtained by selecting a 4-10-1 structure network (i.e. the neural network has four neurons in the input layer, ten neurons in the hidden layer, and one neuron in the output layer). The number of input and output neurons is fixed and is governed by the specific characteristics and physics of the process. In this case, we have four measurements from the ACS attitudes that are used as inputs to the network. The number of output neurons is determined by the number of the actual process output measurements. In this case, one is provided with only one output variable, therefore the output layer has one neuron. The number of hidden neurons was determined after testing the network performance with a number of different selections. It was concluded that ten neurons yielded the best performance in modeling the ACS dynamics. Furthermore, it was experimentally found that a second order IIR filter with a hyperbolic tangent sigmoidal and linear activation functions for the neurons in the hidden layer and output layer, respectively yield the best results. Table 6 shows the details regarding the three DNN networks.

Table 6 Input–output data set for the three DNN networks. DNNroll (Roll)

Fig. 10. The Leader ACS model proposed during the training phase.

DNNpitch (Pitch)

DNNyaw (Yaw)

u1 :

follower; j TPPT1=PPT2 leader

u1:

follower; j TPPT3=PPT4 leader

u1:

follower; j TPPT5=PPT6 leader

u2 :

follower; j q1 leader

u2:

follower; j q1 leader

u2:

follower; j q1 leader

u3 :

follower; j q2 leader

u3:

follower; j q2 leader

u3:

follower; j q2 leader

u4 :

follower; j q3 leader

u4:

follower; j q3 leader

u4:

follower; j q3 leader

yest:

follower; j leader

Dvx;est

yest:

follower; j leader

Dvy;est

yest:

follower; j leader

Dvz;est

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 Fault Scenario #1: A mass fault reduction of the thruster PPT2 of s/ cf1 is injected at time t = 0 second during a rotation of the spacecraft from the attitude [08, 08, 08] to [258, 308, 408].  Fault Scenario #2: A fault due to the accumulation of particles in the inner surface of the PPT3 thruster electrodes of s/cf1 is injected at time t = 0 second during a rotation of the spacecraft from the attitude [08, 08, 08] to [208, 358, 458].  Fault Scenario #3: A loss of conductivity in the electrical components of the PPT6 thruster of s/cf1 generates a fault which is injected at time t = 0 second during a rotation of the spacecraft from the attitude [08, 08, 08] to [258, 358, 458].

Fig. 13. The high-level FDI model proposed during the validation phase.

From Table 6 it can be observed that each DNN estimates the angular velocity of one of the three axes of the spacecraft, specifically DNNroll, DNNpitch and DNNyaw estimate the angular velocity in the x-axis, y-axis and z-axis, respectively. Once the training phase is completed, the parameters of the dynamic neural networks are fixed and the validation phase is initiated. A new set of data is applied to the DNNs and their modeling performance is investigated and evaluated. Fig. 13 shows the architecture of the DNN that is used for the roll angle during the validation phase. Similar architecture is used for the other two angles, namely the pitch and the yaw angles. Similar to the low-level FDI approach in the previous section, we calculate a Threshold that is used to determine the health condition of the spacecraft during its maneuvers in the mission. Towards this end, we need three different Thresholds, one for each DNN. For calculating these thresholds the Sum of Absolute Error (SAE) from the expected settling time (ts) to the final steady state time (tfinal), is used. Specifically, we have SAEroll :¼

tfinal   X follower;1  follower;1 leader Dvx ðkÞ  leader Dvx;est ðkÞ

(8)

k¼ts

The Thresholdroll for the DNNroll is now determined by: Pl Thresholdroll ¼

i¼1

SAEroll ðiÞ l

þ s roll maxðSAEroll ðiÞli¼1 

Pl

i¼1

SAEroll ðiÞ l

! (9)

where l denotes the number of formation flying maneuvers during a given mission. The coefficient sroll is a constant which is used to adjust the performance of the healthy/faulty classification rates. The configuration of the proposed high-level FDI scheme for the roll axis is shown in Fig. 14. Similar configurations also hold for the yaw and the pitch axes. To investigate the performance of the proposed high-level FDI scheme, we simulate the formation flying mission under the following three different fault scenarios:

Fig. 14. The high-level fault detection scheme proposed for PPT actuators of a follower spacecraft in the formation flying.

Fig. 15(a)–(c) depicts the actual sequence of pulses that are generated by the PPT1/PPT2 under the Fault Scenario #1, the actual sequence of pulses generated by the PPT3/PPT4 under the Fault Scenario #2, and the actual sequence of pulses generated by the PPT5/ PPT6 under the Fault Scenario #3, respectively. Table 7 summarizes the results obtained by using the proposed high-level FDI scheme. It is important to note that interesting information can be extracted from the above results. Specifically, according to the number of pulses that are generated by the faulty pair of thrusters in the three simulated cases above one can infer the following cause–effect relationships:  In case of a faulty pair of thrusters, one of the thrusters generates more pulses than the other and that difference is ‘‘considerable’’.  If the faulty thruster is the one which generates more pulses, then one can infer that the fault produces a reduction in the amount of thrust that is generated by each pulse.  If the faulty thruster is the one which generates fewer pulses, then one can infer that the fault produces an augmentation in the amount of thrust that is produced by each pulse. According to the above conducted simulation results and others not reported due to space limitations, the proposed high-level FDI scheme can only detect the pair of thrusters which is faulty. Furthermore, based on the number of pulses that are generated by the faulty pair, a secondary analysis can be performed. Using this analysis and depending on the faulty thruster, one can identify if the fault produces an increase or a decrease in the amount of thrust that is produced. However, so far as the isolation requirement is concerned, the proposed high-level approach cannot be used to achieve this goal. In the next section, the capabilities of the low-level and the high-level schemes are combined to propose a highly effective and reliable scheme for both detection and isolation of faults. 6. An integrated fault detection and isolation scheme In Section 3, a ‘‘low-level’’ or a ‘‘component-level’’ FDI scheme based on dynamic neural networks was developed for health monitoring the PPT actuators of a spacecraft. The performance results obtained revealed that using a fixed threshold to determine the health status of a thruster affects the precision capability of the FDI approach. In Section 4, dynamic neural networks are used again to develop a ‘‘high-level’’ or ‘‘formation-level’’ FDI scheme for detecting behavioral deviations in the relative attitudes of the formation flying satellites. The latter approach can effectively only detect faults and cannot be used to determine which of the actuator pairs is the faulty one and when the fault is injected. In this section, we take advantage of the strengths of the two schemes and propose a new integrated FDI scheme. The integrated FDI scheme uses first the high-level approach for detecting which pair of thrusters is faulty and which ones are healthy. Once the faulty thruster pair is identified, the integrated FDI scheme uses the low-level approach to investigate which thruster is the faulty one and, more specifically, which one of the generated pulses is faulty.

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Table 7 Health analysis results obtained by our proposed high-level FDI approach (the bold figures correspond to the highest SAEs for each respective axis). Fault scenario

DNN

SAE

Threshold

Health Status

#1

DNNRoll DNNPitch DNNYaw

382.12 41.53 45.16

135.00 50.00 76.00

Faulty Healthy Healthy

#2

DNNRoll DNNPitch DNNYaw

100.21 57.88 39.75

135.00 50.00 76.00

Healthy Faulty Healthy

#3

DNNRoll DNNPitch DNNYaw

132.84 38.40 279.98

135.00 50.00 76.00

Healthy Healthy Faulty

Fig. 16. Proposed ‘‘integrated’’ FDI scheme.

 Upon the detection of a faulty pair of thrusters, the sequence of follower; j pulses that is represented by leader TPPTðþÞ=PPTðÞ is applied to the Threshold Logic Selection block.  The Threshold Logic Selection block then determines the number of pulses that are generated by each thruster (e.g. #PPT(+) and #PPT()) and identifies which thruster has generated the largest number of pulses.  If the number of pulses that are generated by the thruster PPT(+) is larger than the number of pulses that are generated by the thruster PPT(), the low-level scheme will then employ the Upper Threshold and the Lower Threshold for the health analysis of the PPT(+) and the PPT(), respectively. Otherwise, the reverse thresholds will be used. With the Threshold Logic Selection block specified, the components of the integrated FDI scheme is complete. Figs. 17

Fig. 15. The actual sequence of pulses generated by (a) PPT1/PPT2 (scenario #1), (b) PPT3/PPT4 (scenario #2), and (c) PPT5/PPT6 (scenario #3).

Based on the number of pulses that are generated by the pair of thrusters that is declared as a faulty pair, an important cause–effect relationship was established in Section 5. The analysis and use of these relationships allow the construction of a Threshold Logic Selection block which is described next and is shown in Fig. 16.

Fig. 17. High-level stage of the proposed ‘‘integrated’’ FDI scheme.

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Fig. 18. Threshold logic selection block and the low-level stage of the proposed ‘‘integrated’’ FDI scheme.

and 18 depict the proposed integrated FDI scheme. Using the three DNNs in the ‘‘high-level’’ scheme, one determines if an anomaly has occurred in the roll, pitch or yaw axes, and then identifies the pair of thrusters that is responsible for this abnormality. Based on the number of pulses that are generated by the pair of thrusters as identified in the high-level scheme, the Threshold Logic Selection block selects a proper Threshold value for each thruster. A single DNN similar to the low-level scheme is sufficient. Using this DNN to model the pair of thrusters, the residuals (MAE) are generated and using the corresponding Threshold values the health status of each pulse is then determined. To demonstrate and illustrate the capabilities of the proposed integrated FDI scheme, eight new fault scenarios are simulated with their specifics as shown in Table 8. The rotational attitude maneuver column represents the desired set point angles that are requested by the ground station command during the mission. In other words, from a given initial condition one is tasked to slew the spacecraft to the desired attitude [208, 308, 408] during which a fault is injected to the PPT1 thruster in the scenario #1. Similarly, in the scenario #2 one is tasked to slew the spacecraft from a given initial condition to the desired attitude [258, 308, 408] during which a fault does occurs in the PPT4 thruster. The same description also holds for the other fault scenarios and rotational attitude maneuvers. A typical fault scenario #6 is described in more detail in the following paragraphs. Similar conclusions can be drawn for the

Fig. 19. Actual sequence of pulses that is generated by PPT5/PPT6 for the fault scenario #6.

Table 9 Health analysis results obtained by the proposed FDI approach for the fault scenario 6. Fault scenario

DNN

SAE

Threshold

Health status

#6

DNNRoll DNNPitch DNNYaw

127.19 42.63 160.73

135.00 50.00 76.00

Healthy Healthy Faulty

other seven fault scenarios that are shown in Table 8. These results are not shown here due to space limitations. In order to describe the sequence that one should follow for the proposed integrated FDI scheme, a step-by-step task corresponding to the fault scenario #6 is now presented. Fig. 19 shows the real effect of the injected fault on the thrusters PPT5 and PPT6. In the first stage of the proposed integrated FDI scheme the three components of the angular velocity are estimated and then compared with the actual model values to generate the corresponding residual signals (SAE) as shown in Fig. 17. The results are shown in Table 9. From the above table it follows that the fault is declared in the pair of thrusters PPT5/PPT6. Based on the number of pulses that are generated by the pair, the Threshold Logic Selection block determines the Threshold values for both thrusters (refer also to Eq. (6)). Table 10 shows the resulting Threshold values. Once the Thresholds are assigned, the proposed integrated FDI scheme activates the second stage of the detection and isolation process as shown in Fig. 18. Fig. 20 shows the health status signals

Table 8 Health analysis conditions considered by the proposed ‘‘integrated’’ FDI approach. F. Sc.

Rotational attitude maneuver

Faulty actuator

Type of fault

#1 #2 #3 #4 #5 #6 #7 #8

[208, [258, [208, [258, [258, [308, [208, [308,

PPT1 PPT4 PPT5 PPT2 PPT3 PPT6 PPT3 PPT4

Faulty Faulty Faulty Faulty Faulty Faulty Faulty Faulty

F. Sc.: fault scenario. Incr.: incremental.

308, 308, 358, 358, 408, 408, 458, 458,

408] 408] 458] 458] 558] 508] 608] 558]

Case Case Case Case Case Case Case Case

1 1 2 2 3 3 2 2

Fault injection time (s) (15%) (15%) (15%) (15%) (incr.) (incr.) (incr.) (incr.)

t=0 t=0 t=0 t=0 t = 400 t = 400 t = 300 t = 300

A. Valdes, K. Khorasani / Applied Soft Computing 10 (2010) 746–758 Table 10 Logic threshold selection for the fault scenario 6.

PPT5 PPT6

757

Table 13 ‘‘Integrated’’ FDI scheme results for s/cf1 (fault scenarios 1–8).

Number of pulses

Threshold value

Fault scenario

PPT number

Additional information

256 861

0.0300 (Lower) 0.0370 (Upper)

#1

PPT1 PPT2

Fault Case 1 or 3 No fault detected

#2

PPT3 PPT4

No fault detected Fault Case 1 or 3

#3

PPT5 PPT6

Fault Case 2 No fault detected

#4

PPT1 PPT2

No fault detected Fault Case 2

#5

PPT3 PPT4

No fault detected Fault Case 1 or 3

#7

PPT3 PPT4

Fault Case 2 No fault detected

#8

PPT3 PPT4

No fault detected Fault Case 2

Table 14 Performance results for our proposed ‘‘Low-Level’’ and ‘‘Integrated’’ FDI schemes.

Accuracy True healthy False healthy True Faulty False Faulty Precision

Fig. 20. Health status signal for PPT5/PPT6 (fault scenario #6).

Table 11 ‘‘Integrated’’ FDI scheme results for s/cf1 for the fault scenario 6. Fault scenario

PPT number

Actual healthy pulses

Actual faulty pulses

Detected healthy pulses

Detected faulty pulses

#6

PPT5 PPT6

256 42

0 819

256 40

0 821

for the PPT5 and the PPT6 during the simulated period. Table 11 shows the final health monitoring results that are obtained by using the proposed integrated FDI scheme. It should be noted that one can clearly declare that the faulty actuator is the PPT6 and the type of fault yields a reduction in the amount of produced thrust (fault case 1 or case 3 as defined in Table 12 ‘‘Integrated’’ FDI scheme results for s/cf1 (fault scenarios 1–8). Actual healthy pulses

Actual faulty pulses

Detected healthy pulses

Detected faulty pulses

Fault scenario

PPT number

#1

PPT1 PPT2

0 0

2770 0

0 0

2770 0

#2

PPT3 PPT4

5 0

0 2883

5 0

0 2883

#3

PPT5 PPT6

0 541

115 0

7 541

108 0

#4

PPT1 PPT2

338 0

0 44

338 0

0 44

#5

PPT3 PPT4

112 153

433 0

106 153

439 0

#7

PPT3 PPT4

114 849

121 0

112 849

123 0

#8

PPT3 PPT4

769 76

0 115

769 71

0 120

‘‘Low-Level’’ FDI scheme performance

‘‘Integrated’’ FDI scheme performance

82.08% 55.71% 11.23% 88.77% 44.29% 55.71%

99.79% 99.39% 0.03% 99.97% 0.61% 99.94%

Section 4). The health monitoring results for the remaining seven scenarios are shown in Table 12. Table 13 provides the additional information that the ‘‘integrated’’ FDI scheme can extract from the health monitoring analysis of the considered fault scenarios. Finally, the performance of the proposed integrated FDI scheme is studied by invoking the confusion matrix criterion. Table 14 provides a comparison between the results that are obtained in this section and those that were obtained previously in Section 4. To summarize, the proposed integrated FDI scheme contains four dynamic neural networks, three in the first stage for the purpose of fault detection and one in the second stage for the purpose of fault isolation that is implemented in each follower spacecraft. This approach does not require one to use any additional sensor, since it uses the signals that are already incorporated to control other subsystems (e.g. electrical variables and relative attitude variables). The proposed FDI scheme is shown to be a reliable health monitoring tool. The results that are obtained clearly show high levels of accuracy (99.79%) and precision (99.94%) with the misclassification rates for the False Healthy (0.03%) and False Faulty (0.61%) parameters being quite negligible. The ‘‘integrated’’ FDI scheme is also capable of determining whether or not the fault produces an increase or a decrease in the amount of thrust that is generated by the faulty actuator. 7. Conclusion Although a number of results are available for fault diagnosis of conventional attitude control subsystem actuators (e.g. reaction wheels and Gyroscopes), development of a fault diagnosis system for pulsed plasma thrusters (PPTs) has not yet been investigated in the literature. This can be mainly due to the fact that the forces that are generated by these actuators cannot be directly measured, and

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furthermore precise mathematical models of these systems are not readily and always available. In this paper, we have proposed a novel fault detection and isolation (FDI) scheme for the PPTs that are utilized in the attitude control subsystem (ACS) of the formation flying of multiple satellites. An extensive set of simulation studies is conducted to demonstrate and illustrate the capabilities and advantages of the proposed methodologies. By utilizing four dynamic neural networks (DNN), the proposed FDI scheme is capable of detecting and isolating faults in the PPT actuators that affect the strict performance requirements that are envisaged for the highly precise formation flying missions. The results obtained demonstrate a high level of accuracy and precision with the misclassification rates for the False Healthy and False Faulty parameters being quite negligible. Formation flying missions are starting to gain popularity due to their many offered benefits and advantages. A significant reduction in the amount of resources utilized by the ground station personnel can be achieved by implementing the proposed DNN-based FDI schemes. This will ensure that the overall cost of the missions can be reduced significantly. References [1] M. Guelman, Spacecraft formation flying, in: Proceedings of the Satellite Formation Flying and Applications Workshop, 2005. [2] M. Mitchell, L. Breger, J.P. How, Effects of navigation filter properties on formation flying control, in: AIAA Guidance, Navigation, and Control Conference and Exhibit, August, 2004. [3] Z. Wang, Y. Zhang, Design and verification of a robust formation keeping controller, Journal of the International Academy of Astronautics, Acta Astronautica 61 (7–8) (2007) 565–574. [4] H. Wong, H. Pan, M.S. de Queiroz, V. Kapila, Adaptive learning control for spacecraft formation flying, in: Proceedings of the 40th IEEE Conference on Decision and Control, 2001. [5] H. Wong, V. Kapila, A.G. Sparks, Adaptive output feedback tracking control of spacecraft formation, International Journal of Robust and Nonlinear Control (2002) 117–139. [6] W. Ren, R.W. Beard, A decentralized scheme for spacecraft formation flying via the virtual structure approach, in: Proceedings of the 2003 American Control Conference, 2003. [7] M. Aung, A. Ahmed, M. Wette, D. Scharf, J. Tien, G. Purcell, M. Regehr, An overview of formation flying technology development for the terrestrial planet finder mission, in: Proceedings of the 2004 IEEE Aerospace Conference, 2004. [8] M. Campbell, R.R. Fullmer, C.D. Hall, The ION-F formation flying experiments, in: AAS/AIAA Space Flight Mechanics Meeting, 2000. [9] S. Persson, P. Bodin, E. Gill, J. Harr, J. Jorgensen, PRISMA—an autonomous formation flying mission, in: Proceedings of the Small Satellite Systems and Services— The 4S Symposium, 2006. [10] M.E. Campbell, V. Knagenhjelm, J. Yingling, Flight software development for the ION-F formation flying mission, in: IEEE Proceedings on Aerospace Conference, 2001. [11] R. Sanchez, P. Renard, Design of a micro-satellite for precise formation flying demonstration, in: Proceedings of the Fifth IAA International Conference on LowCost Planetary Missions, 2003. [12] T. Yairi, Y. Kawahara, R. Fujimaki, Y. Sato, K. Machida, Telemetry-mining: a machine learning approach to anomaly detection and fault diagnosis for space systems, in: Proceedings of the 2nd IEEE International Conference on Space Mission Challenges for Information Technology, 2006. [13] B. Rubin, M. Guelman, A. Kapulkin, Principles of hall thruster onboard malfunction diagnostics based on magnetic field measurements of plasma currents, in: Proceedings of the 4th International Spacecraft Propulsion Conference, 2004.

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