Multiple fault isolation method for micro thrusters of drag-free systems

This paper proposes a multiple fault isolation framework for micro thrusters of drag-free systems, based on the minimum number of isolation filters. Based on the consideration of fault isolatability, the original system is reasonably divided into two completely isolated subsystems, and then two sets of fault isolation filters are designed based on the principles of duality design for state feedback decoupling, which achieves residual decoupling, and therefore, can detect and isolate multiple faults occurred on the micro thrusters. In particular, this method has low computational burden (a feature of practical importance in reducing the usage of onboard resources) and short isolation time, making it suitable for on-board fast and precise fault isolation. The simulation results demonstrate the efficacy of the proposed method when utilized for a drag-free system with 12 field emission electric propulsion (FEEP) thrusters.


Introduction
The space gravitational wave detection system is a multi-body system that achieves stable formation flight through a million-kilometer separation between three spacecrafts, and it carries six test masses (TMs) that require nano-precision motion control, which needs to be provided by drag-free control technique.Figure 1 shows the drag-free system of LISA Pathfinder (LPF).In recent decades, various countries have conducted some studies on the drag-free control problem in space gravitational wave detection, which can be found in [1][2][3].The field emission electric propulsion (FEEP) thrusters are the core of the drag-free satellite.In space gravitational wave detection task, the spacecraft is equipped with 12 FEEP thrusters to generate N   magnitude thrust to achieve ultra-high precision drag-free control.According to the task requirements, the drag-free satellite need to operate on orbit to wait for the space gravitational wave detection window, which makes it susceptible to faults especially the thrusters because of the harsh environment of the space.Thus, it is of significance to conduct fault diagnosis for the thrusters of the drag-free system.In the past decades, many studies have been conducted on fault diagnosis, which could be divided into three categories: model-based methods, signal processing methods, and artificial intelligence methods, in which model-based methods are the most suitable for the satellite on-board fault diagnosis because of its reliability and computation burden.Much work are devoted to study fault diagnosis problem about the satellite system.Tudoroiu and Khorasani [4] designed a fault diagnosis algorithm using a bank of Kalman filters for the satellite attitude control system's actuators.Henry [5] considered satellite thruster faults and designed two kinds of robust fault diagnosis algorithms based on Linear Matrix Inequality (LMI) techniques.However, most research is focused on 6 degrees-of-freedom (DOF) satellites, with limited studies on fault diagnosis for the drag-free system.Fault diagnosis for thrusters of the drag-free system has two difficulties: 1) the drag-free system has a complex dynamics model with high DOF; 2) the system carries many thrusters which are not paired arranged, thus it is difficult to isolate all thrusters faults meanwhile.The work on fault isolation mainly include three kinds of methods [6]: 1) unknown input decoupling methods; 2) the dual problem solution of decoupled controller design; 3) designing a bank of residuals generators.Among them, the first methods mainly treat the faults in the remaining dimensions and all disturbances as unknown inputs.It then decouples them from the current dimension's fault signal through unknown input observers (UIO).To achieve this, it is necessary for the number of measured system outputs to be greater than or equal to the dimensions of the unknown inputs [7].However, since the dimension of the fault signal caused by the thrusters in the drag-free system is relatively high, this method is not suitable for fault diagnosis in such systems.The third methods need to design a bank of 12 observers [8].Due to the high dimensionality of the observer, running it in parallel would result in significant computational resource consumption.
The second methods consider the fault isolation problem as the dual problem of state feedback decoupling design.By designing a set of observer gain matrix and post-filter matrix, the fault signals could achieve mutual decoupling.This method was originally proposed by Liu and Si in [9].This method only requires the design of one observer when applied, leading to small computational resource consumption.The algorithm is simple, efficient, and highly reliable.However, it has a finite maximum number of faults that can be isolated, and it needs to satisfy certain rank conditions.Li and Jaimoukha [10] consider the robust fault isolation problem against the disturbances using H  norm based on [9], and applied to a 4-order dynamic system.However, this method used LMI in the solving process, which can pose difficulties in solving problems for high-dimensional drag-free systems.
Inspired by the aforementioned research, this paper proposes a fault isolation framework which is suitable for micro thrusters of the drag-free system.This framework can detect and isolate multiple faults occurred on the 12 FEEP thrusters using the minimum number of filters, which saves computational resources.First, a 15 DOF dynamic model of the drag-free system is established considering the thrusters faults, and model-based fault isolation problem of the drag-free system is formulated.Next, a control algorithm is applied to the drag-free satellite to obtain a close-loop system.And considering the fault isolability of drag-free systems, the original system is divided into as few subsystems as possible, with each subsystem being fully fault-isolatable.Based on the design method of duality of state feedback decoupling, a bank of parallel fault isolation filters towards all the subsystems is designed to isolate all the thrusters faults.It is worth noting that the number of filters is minimum and much less than 12, which can greatly save onboard resources.
The remainder of this paper is organized as follows.Section 2 presents the dynamics of the LISA Pathfinder (LPF) drag-free satellite, which includes the thrusters configuration and fault models.In Section 3, we introduce a fault isolation framework based on parallel observer.Simulation results are shown in Section 4, followed by the conclusions and future works.

Drag-free system dynamics
In this section, a 15-DOF dynamic model of the drag-free system is established.Then, the thruster configuration is presented with explicit consideration of four types of thruster faults.Finally, the fault isolation problem of the micro thrusters of the drag-free system is formulated as an observer design problem based on fault detection filter (FDF).

Dynamics of drag-free systems
We take the LPF drag-free system as a case study, which a three-body system consisting of a spacecraft (S/C) and two test masses (TMs), as shown in figure 2. Before proceeding, we define two coordinate systems: the heliocentric inertial (HCI) frame and the body-fixed (BF) frame of the satellite.
1) Heliocentric inertial frame .Since the LPF satellite operates in the heliocentric orbit, we chose the HCI frame as the inertial reference frame.The center of the Sun serves as the origin O , and the X axis is aligned with the ecliptic plane and points towards the J2000 equinox.The Z axis is perpendicular to the eclipticplane, while the Y axis completes the right-handed coordinate frame.2) Spacecraft body-fixed frame .The origin O of the coordinate system is fixed on the center of mass (CoM) of the satellite, aligning the triad axes with its principal axes of inertia.The X axis is parallel to the line between the test mass 1 (TM1) and the test mass 2 (TM2), and points from TM2 to TM1.The Z axis is perpendicular to the upper plane of the satellite, and the Y axis completes the right-handed coordinate frame.
Based on the assumptions of rigid body and small-angle maneuvers, the 15-DOF dynamics of the LPF satellite is modeled as [11] is the attitude of the S/C, whereas as [ , , ] The matrix E is the unit matrix of appropriate dimension, a , , i a , i α denote the accelerations of spacecraft and the th i TM.They can be further defined in a more concise form, i.e.where M and ( ) are the mass matrices of the S/C and the two TMs, f and ( ) and ( ) are the total force and torque applying on the S/C and TMs, which can be further divided into two parts: actuation and stiffness contributions (4) where 2 ,, j j DF SUS = Φ are the stiffness matrices.According to the task requirements, drag-free and suspension state vectors can be defined as : Thus, the dynamic model of the drag-free system can be expressed as follows

Thruster configuration and fault modeling
The drag-free system is outfitted with 12 FEEP thrusters, organized into three clusters of four thrusters each, to achieve ultra-high precision control.This study solely considers additive faults occuring on the 12 FEEP thrusters.The various thruster faults typically fall into four categories, as classified by [12]: 1) Stuck-open: the thruster consistently delivers maximum thrust, regardless of the required demand; 2) Blocked-closed: the thruster does not generate any thrust, regardless of the level of demand; 3) Loss: the thruster encounters a decrease in efficiency; 4) Bias: the actual thrust produced by the thruster deviates slightly from the commanded value.
An additive approach can be utilized to model the above faults ( ) ( ) ( ) ( ) where i  the loss rate of the th i thruster, max u is the maximum output provided by thrusters.According to equations ( 6)-( 8), a linear time-invariant (LTI) state space model of the drag-free system can be derived as follows A B , C and f B are system matrices of appropriate dimensions.

Fault isolation problem formulation
Based on the FDF technique and the state space model ( 9), we design a state observer to generate fault isolation residuals ˆˆ() ˆ= x Ax Bu R y Cx y Cx (10) The residual for fault isolation is defined as : xx as the state estimation error.In view of ( 9)-( 11), the system residuals are expressed as HCξ (12) According to (12), the transfer function form of Here, rf T are the transfer functions from f to r , and R and H are the matrices to be determined so that the residual i r is only affected by the th i fault and decoupled from all other faults, i.e., rf T is diagonal.

Multiple fault isolation framework
In this section, a fault isolation framework is proposed to isolate multiple micro thrusters faults of the drag-free satellite.Before doing so, a proportional-integral-derivative (PID) controller is presented to stabilize the drag-free system.Then, a bank of fault isolation filters is designed to achieve fault isolation of all the 12 FEEP thrusters.Note that the number of filters is much less than 12, which can save the onboard resources.The block diagram of the proposed fault isolation framework is shown in figure 3.

Control laws
In this subsection, we design a fault isolation filter for the closed-loop system to prevent the subsequent introduction of control signals from affecting the fault isolation performance.To this end, three PID controllers are designed to stabilize the drag-free system.According to the science mode of the dragfree system, the control tasks can be divided into three parts by input decoupling: 1) Drag-free loop: the drag-free control DOF are fully controlled by the FEEP; 2) Suspension loop: the suspension DOF are fully controlled by the electrostatic actuator along the suspension axes; 3) Attitude loop: the satellite attitude DOF are controlled by the electrostatic actuator along the dragfree axes, then the drag-free loop will control the S/C to follow the drag-free coordinates.According to the above description, the controllers can be expressed as [ , , , ] is the desired thrust outputs of 12 thrusters.

Fault isolation framework
Consider the state space equation (12) and suppose the initial state error is zero, thus () t ξ can be expressed as 0 ( )( ) ( ) ( )dτ where fi B is the th i column of f B .For the aforementioned state-space dynamic model of the drag-free system, we can design fault isolation filter following the line of [9].Before this, we should analyze the fault isolability of the drag-free system.According to [9], the fault detectability indexes Based on this, the fault detectability matrix is defined as Therefore, fault isolability can be determined by the rank of , then the maximum number of fault that can be isolated is m .
According to the definition of the fault isolability, we are not sure to isolate 12 FEEP thrusters faults meanwhile by one fault isolation filter.Based on this situation, we design multiple sets of parallel running filters to isolate the entire fault set, and the fault subsets isolated by each fault isolation filter do not overlap with each other.For each fault isolation filter, the thrusters which are not isolated by this filter are considered to be fault free.Thus, the fault matrix f B and fault signal f can be reorganized and divided into r subsets: where r is the total number of the fault isolation filters, and should be minimized while ensuring the isolation performance for the sake of conserving on-board resources; i E and i υ , 1,2, , ir = is composed of certain columns from f B and f .As such, the original system will also be partitioned into It is obvious that we can always find a set of subsets partitioning such that for each state space subsystem, the fault detectability matrix D is of full rank.Thus for each subsystem, the following theorem can be utilized to design fault isolation filter: Theorem 1: Given system state space model as (20), and the form of fault isolation filter is ˆ() ˆˆ()

R A e A e A e E Λ D Z I DD H WD Z I DD (22)
where W is any non-singular diagonal matrix, 1 Z and 2 Z are arbitrary matrices, 1 () Proof: This proof is similar to [9] and is omitted here.From the proof process of Theorem 1, it can be observed that if the filter parameters are chosen as (22), it is possible to make the transfer function matrix from f to r diagonal, thus the r sets of fault isolation filters can make each dimension of all filters residuals influenced by only one thruster fault signal, and the signals from the other thrusters are decoupled.Thus, the multiple faults isolation problem proposed in Section 2.3 is solved.Besides, a simple threshold logic is applied for the decision-making task, such as where 0   is a user-defined threshold.

Numerical simulation
In this section, we conduct numerical simulations to showcase the efficacy of the fault isolation framework when applied to the drag-free system.We consider the presence of bias-type faults by default and perform simulations under the remaining three fault conditions as described in equation (7).The simulation step size is set to 0.001s , and the total simulation time is 50s .It's worth noting that only blocked-closed type faults are demonstrated here.According the practical drag-free dynamics, the fault detectability matrix satisfies rank( ) 6 = D (24) Thus, we need to design two fault isolation filters to isolation 12 FEEP thrusters.
Figure 4 shows the residuals behaviors of the 12 FEEP thrusters in both fault-free and faulty situations with the 4th , 8th and 11th thrusters closing on 25s .As expected, all the residuals keep below the threshold, while the 4th , 8th and 11th residuals exceed the threshold at 26.595 s , 26.492 s and 26.942 s , respectively, indicating that the corresponding three thrusters experienced faults.This demonstrates that the proposed fault isolation framework can effectively isolate multiple micro thruster faults in the drag-free system.

Conclusion
This paper establishes a multiple faults isolation framework suitable for the micro thrusters faults of the drag-free system.Considering the unique structure and thrusters layout of the drag-free system, two sets of parallel fault isolation filters are designed to isolate all FEEP thrusters faults.The blocked-closed type faults of three thrusters are simulated to test its effectiveness, and the results shows that the framework can isolate faults occurring on the multiple thrusters, and all the faults can be isolated in 2s .Our future work will focus on the fault estimation problem based on the current framework.
relative attitude and position of the th i test mass (TM) with respect to their nominal attitude and position in their housing in frame .Here, Oi r represents the skew-symmetric matrix of Oi r in the frame , that is ,

B
are appropriate matrices associated with the S/C and TMs, which can be derived by equations (1)-(5).

Figure 3 .
Figure 3. Block diagram of the proposed fault isolation framework.