Fault diagnosis of AUV based on sliding-mode observer

. Designing of a sliding mode observer was finished based on the motion model of autonomous underwater vehicle (AUV). A Flutter decrease strategy for the sliding mode observer was discussed and the sliding mode observer was applied to the fault diagnosis of the AUV. Simulation experiment without faults was designed. Simulation experiment of sensor fault diagnosis and field trial of thruster fault diagnosis were undertaken. The validity and feasibility of this method were validated based on the analysis of the simulation and field trial results.


Introduction
With the development of ocean engineering and military requirements, autonomous underwater vehicle(AUV) as a part of ocean high technology, required to satisfy much higher intelligent requirements.The AUV works in very tough environment and the surrounding circumstance changes so fast that it is crucial to design an accurate and reliable control system to guarantee its safety.To realize self-fault diagnosis of control system is the core of intelligence [1].
One effective method to achieve self-fault diagnosis of control system is based on analytic redundancy.The dynamic model of fault diagnosis plant is assumed aware-ness.Numerical redundancy relationship is constructed from input and output of the system.Residual sequences can be acquired and faults can be determined by comparing the residual and a certain threshold [2].To get the accurate model of the system is very difficult because of many factors.Therefore, to detect all the faults of the control system just based on model without any mistakes is impossible.A research subject for fault detect and diagnosis is how to increase rate of accuracy while decrease rate of omission and rate of false alarm.The sliding-mode observer(SMO) was brought in fault diagnosis of control system because of its good performance when system's uncertainty was dealt with [3].
A SMO was designed according to the motion model of the AUV.Residuals were constructed by comparing the output of observer and sensors.Fault information was distilled from the residuals and fault diagnosis was realized by analysing residuals.Simulation trials of sensor fault diagnosis and thruster fault diagnosis experiments in sea were undertaken.The validity and feasibility of this method were verified by the experiment results.

Sliding-mode observers 2.1 AUV motion model
According to equation of motion of six degrees of freedom [4], AUV's equation of motion of five degrees of freedom can be acquired when it doesn't take the disturbances into account while assume gravity and buoyancy reach balance: Thereinto, M is mass matrix which contains additional mass and additional moment of inertia; X  is (angular) acceleration matrix; vis F is viscous hydrodynamic matrix; X  is velocity matrix; X is displacement matrix; t , then the system can be expressed as follows, A nonlinear SMO was designed which was expressed as Eq.( 3): Where, is the value measured by sensors, ε is the measured noise vector, and x are the estimated value of 1 x x e (4) Error output equation of SMO can be deduced from Eqs.(2) and (3) Where, z is the measured value, so aiming at Eq.( 3), the destination is that 1 ê should be equal to zero when t belongs to a finite value.As long as 0 ˆ1 ≠ e and e max in Eq.( 7) is the maximum i e 2 .Therefore, suitable 1 Λ can be found to make sure that the sliding-mode will occur on condition that 2 e is limitary.Namely, 1 ê and 1 e T V = are defined.Therefore, we get Eq.( 6) Usually, the noise plays little function.1 e and 1 e  are assumed to zero.According to the Eqs.( 4) and ( 5), Eq.( 7) can be acquired: ) ŝgn( 11 2 e Λ e = (7) therefore: ˆŝgn( ) The differentiation equation is apparently asymptotic steady, and 2 e must be limited as long as β is limited.When designing the observer, suitable gain matrix 1

Λ and 2 Λ
should be chosen to satisfy the condition that 1i k when the AUV run normally.β is the error between the real system and estimated model.Let and ∆ is limited, therefore: F are limited according to the hydrodynamic performances of the AUV, then the Δ is limited and so does β .
When the AUV seals in normal state, 2 e is relates to the error β according to the Eq.(7). 2 / λ λ increasing.The uncertainty of the model and disturbance of the noise exist though, because the errors between the real system and estimated model are always within a finite domain, the SMO is always astringency.The sliding residuals of the SMO are astringency near the zero, namely, buffeting phenomenon exists.
Once faults occurred in some component of the control system, the errors between the real system and estimated model will exceed the threshold values.The sliding-mode state will be halted and the buffeting characteristic of sliding residuals will disappeared either.Different fault modes will produce different effects to the sliding residuals and different faults can be told by analyzing these [5].

Buffeting decrease strategy
When the system is observed by the SMO as Eq.( 3), the output of the observer exists buffeting near the sliding plate because of the abrupt changes of the symbol function around the zero point.A new function was adopted to place symbol function in order to decrease abrupt change.
Firstly, normalization processing was undertaken for state observer error as: x was the maximum error determined by the residual information.* x was treated as a new error and it was used in Eq.(11): when compared with symbol function.

Simulation experiment without faults
The research plant was an AUV equipped with eight thrusters.According to the thrusters' functions, we set them into four groups: horizontal plane main-thrusters, vertical plane thrusters, lateral thrusters and vertical plane thrusters.Each group of thrusters was made up of two thrusters.Horizontal plane and vertical plane thrusters adopt pipe thrusters, lateral and vertical thrusters adopted groove thrusters.Four main-thrusters fault diagnosis were studied in this paper.
The SMO was applied to the simulation experiment of faults diagnosis, parameters were chosen based on the experience: k is equals to 7 in Eq.( 12) while the initial value of the observer were all zero.
The simulation results were shown in Fig. 2. The solid line and dotted line were representing the residual curves without buffeting decrease and with buffeting decrease respectively.Compared with the symbol function of sgn( x ) , ( , 7) f x * didn't exist abrupt changes, so buffeting could be avoid effectively.
The output results of the SMO without faults were shown in Fig. 3(a) to (f).The difference between the observer and real system was very small.The residuals within the unsteady state were greater than the steady state attributed to the lagging phenomenon of the observer.

Simulation experiment of sensor fault diagnosis
Assuming that sensor i occurs a step fault at a certain time.The effect of the fault can be treated as the noise disturbance.Namely, the ratio of i ε will beyond the normal value.
Sliding state will stopped immediately, this sensor faults can be detected [6][7].
If a drifting fault exists in sensor i , the residuals of observer can also be applied to the fault diagnosis.In this case, i e 2 will increase with the augment of fault.Once the permitted range is exceeded by i e 2 , buffeting of the observer will stopped.Characteristics of drifting fault mode can be distilled from the residuals.Sensor faults can be detected from sliding residuals.Setting two sliding residual thresholds ), and mark i t 1 and 5 E3S Web of Conferences 360, 01094 (2022) https://doi.org/10.1051/e3sconf/202236001094VESEP2022 i t 2 which are the time when the residuals reach thresholds respectively, and another time threshold i h t is set up.The fault diagnosis process can be expressed as follows: (1) if ˆi e is less than 1 i h , sensor i is normal, turn to (1); otherwise, turn to (2); (2) if determine that sensor i occurs a fault, record the time i t 1 and i t 2 ; (3) if 2 1 i i t t − is less than i h t , step faults occur; otherwise, drifting faults occur.
SMO was designed for fault diagnosis.Simulation trial was undertaken with the AUV sailed at 1.5m/s in horizontal plane.The data of sensor one were yaw angle information.The sliding residuals outputted by the SMO shown in Fig. 4 represented the normal voyage of AUV.Sliding residuals were always about zero and the variation range within 0.018 rad.
SMO would depart from the sliding plate immediately when a step fault happened while SMO would depart from the sliding plate slowly, and the corresponding residuals would increase.The fault can be detected from this phenomenon shown in Fig. 5 to Fig. 7.

Field trial results of thruster fault diagnosis
Thrust matrix t F will occur abrupt changes when a thruster of AUV has faults.The error β between real system and model will appear abrupt changes according to the Eq.( 10) and the sliding residuals output by the SMO will increase known from Eqs.(8) and (9).By analyzing the sliding residuals, whether the thrusters have faults or where the faults are can be deduced.The AUV undertook its field trials in Liaoning province in China.Figure 8 were the fault diagnosis results.During this experiment, the AUV sailed from static to uniform velocity, we gave right main-thruster a zero voltage at 180 clap to simulate the right mainthruster occurring a fault.
From zero to 179 clap as shown in Fig. 8, there were no faults occurred, and residual information approaching zero value.But when main-thrusters occurred faults, bias would be appeared in longitudinal velocity residuals.
From zero to 65 clap in Fig. 8, the AUV sailed from static to uniform velocity.The longitudinal velocity residual exceeded its threshold value while yaw angle residual nearly zero.When faults occurred, the longitudinal velocity residual and yaw angle residual both would exceed their threshold value.Fault diagnosis could be undertaken according to these principles.
Though the right-main thruster occurred a fault, the AUV could sail at a certain velocity.Because the logging of the control, the velocity of the AUV decreased after the fault maintained a longtime.

Conclusions
A sliding-mode observer was designed based on the motion model of an AUV.Buffeting decrease strategies of the SMO was conducted.The SMO which was designed in this paper was applied to the AUV's fault diagnosis.Residuals were constructed by comparing the outputs between the SMO and the real sensors.Fault diagnosis of the AUV was realized by distilled the fault information from the residuals.Simulation experiments without faults were designed.Simulation experiments of sensor fault diagnosis and field trial of thruster fault diagnosis were achieved.The validity and feasibility of the method were verified based on the analysis of the experiment results.Buffeting decrease strategies in this paper only was a rough method and better approaches will be found to deal with this problem.

2 e
will limited to an open region on condition that gain matrix 1 Λ , 2 Λ are suitable selected.The radius of the open region will decrease as the ratio i i 1

Fig. 1 .Fig. 2 .
Fig. 1.Curves of two kinds of function.Fig. 2. Curves of sliding residual in simulation.Parameter of k in Eq.(11) can be adjusted and k is bigger than one.The gradient of function of ) , ( k x f * in zero point is increasing with the gain of k .Two different of curses are shown in Fig.1.There are no abrupt changes near the zero point in function ( , 7) f x *