# Faults Detection Using Sliding Mode Observer and Its Application on Elevating Servo Systems

Authors: Lingfei Tian; Linfang Qian; Miao Li; Jiawei Fu
Journal: Journal of Control Science and Engineering
 (2019)

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## Abstract

The elevating servo system (ESS) of vehicle-mounted howitzer (VMH) is a typical closed-loop electrohydraulic position servo system, and the faults of its actuator and sensor seriously affect the safety and reliability of the system. In practice, model uncertainty, nonlinearities, unknown disturbance, and output noise present enormous challenges to conduct fault detection of the system. In the current paper, an online fault detection scheme using the sliding mode technology is proposed. Not only the derivation method of state equation and some common fault expressions but also a new design of sliding mode observer with the ability to eliminate the influences of the above factors on detection results is given. The observer’s parameter matrices are obtained by the linear matrix inequality. To promote the fault detection capability, a statistical-based dynamic threshold is developed to detect actuator faults and sensor faults simultaneously. Finally, experimental studies are implemented on a test rig for validating the system model, and the results of four experiments show the effectiveness of proposed methods.


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## Body

## 1. Introduction

Concerns on the safety and reliability of weapon equipment systems facilitate the development of fault detection techniques for complex industrial systems, which has been considered as a significant approach in pursuing a practical solution to guarantee the continuous and stable operation of the actual systems [1]. The ESS plays an important role in the pointing control of barrels, which is a typical closed-loop control system. To avoid the trend of serious performance degradation, the schemes proposed should be capable of preventing the dissemination of fault effects when possible failure of certain components happens.Mostly, the ESS is working in the wild field with heavy electromagnetic interference (EMI) and environment temperature range of −40°C to 50°C. Model parameters perturbation and output disturbance are inevitable as the viscosity of hydraulic fluid varies with temperature, and sensor performances change due to EMI. The model linearization method has been studied in modeling electrohydraulic servo systems [2–4]. However, linearization errors between the linearized model and the actual nonlinear one must be compensated for better performance and accuracy. Knowledge-based fault diagnosis methods, such as neural network and support vector machine, have been employed in system modeling [5–8]. However, a large number of sample data needs to be provided in advance for training the neural network, which is impractical and inconvenient for the weapon equipment system with harsh trial conditions as well as insufficient trial times.In recently decades, sliding mode variable structure technology has been widely applied in various fields for its suppression of external disturbances and insensitivity to bounded disturbances. Fault detection methods based on the sliding mode observer have been extensively studied. Li et al. [9] designed a sliding mode observer of a buck-boost converter and addressed the problem of sensor faults detection and estimation. Liu and Shi [10] proposed a sliding mode control scheme against sensor faults and disturbances simultaneously. Mao et al. [11] developed a sensor fault detection scheme for rail vehicle suspension in the presence of uncertainty, noises, and stochastic process signals. The methods mentioned above focus on sensor faults detection only, which omit actuator fault detection and have not been implemented on test rigs.Model uncertainty, nonlinearity, and output stochastic noises are the main reasons for the difficult extension of fault detection technology in practice [12–15]. By taking the sensor fault vector as a part of an extended state vector and constructed an augmented singular system, Yang et al. [12] developed a robust sliding mode observer to detect actuator faults, in which only the actuator faults and unknown inputs are considered.Faults in the system can be roughly divided into two categories: actuator faults and sensor faults. To the whole systems, the objective is not only to consider the nonlinear characteristics of the system and the working environment but also to establish a unified fault detection method for the two types of faults. The fault detection observer proposed in this work is partially motivated by the study in [12], employing a different structure with the unknown input, actuator, and sensor faults. Without loss of generality, the output noise is also considered in this paper for the enhancement of output disturbance under harsh environments. Firstly, the sensor faults and output noise are considered as a part of new states, and a new augmented system is reconstructed. Secondly, a fault detection observer is designed for this augmented system to detect sensor faults and actuator faults simultaneously. Finally, based on the experiment data, a dynamic threshold is developed to reduce the false alarm. Moreover, all the results are verified on an actual test rig.The paper is organized as follows. Section2 describes the experimental setup and system modeling and gives the derivation process of state equation and some common fault expressions. Section 3 presents a fault detection algorithm to the ESS, in which a sliding mode observer and a dynamical threshold are developed. In Section 4, the performance of the proposed method is demonstrated on the test rig. Conclusions are stated in Section 5.


## 2. Experimental Setup and System Modeling

### 2.1. Description of Experimental Test Rig

Figure1 shows the test rig used in these experiments. The main hardware components of the elevating servo system of VMH are as follows: an elevating hydraulic cylinder, a proportional valve, a pair of hydraulic lock valve, an accumulator, a hydraulic pump station with a gear pump, a relief valve, and other accessories. The flow rate and operation pressure of the hydraulic pump station are 80 l/min and 120 bar, respectively. The proportional valve determines the flow rate controlling the extension and retraction of the piston rod of the elevating cylinder that changes the elevating angles of the barrel of the VMH. By comparing with the command signals, the controller gives a feedback signal and finally controls the barrel to reach the target elevating angle. In order to bear the load of the heavy barrel, the ordinary dual-acting elevating cylinder is designed with a third independent chamber. The chamber connected with the precharged accumulator through an independent pipeline is called the elevating-counterbalance cylinder.Figure 1
Experimental test rig of ESS.Figure2 is the diagram of ESS, where O defines the trunnion between the top carriage and elevating parts, A and B are the hinge joints between the elevating cylinder and the cradle and between the elevating cylinder and the top carriage, respectively, OA¯ = b, OB¯ = a, ps and pr are the proportional valve entrance port and return port pressures, respectively, p1 and p2 are the working chambers pressures, respectively, p3 denotes the pressure of third independent chamber, A1 and A2 are effective areas of the two elevating cylinder chambers, respectively, A3 is the area of third independent chamber, θ0 is the included angle between OA¯ and OB¯ while elevating hydraulic cylinder has the minimum displacement, θ is the rotation angle (elevating angle) of elevating parts around trunnion O and Fc is the force applied on the cylinder by the elevating parts.Figure 2
Diagram of ESS.


### 2.2. System Modeling

The valve spool’s first-order dynamics is considered since the frequency response of the used proportional valve is far greater than the hydraulic actuator’s. Thus,(1)u=τkvx˙v+1kvxv,where xv is the proportional valve spool displacement, u is the control input, τ is the time constant, and kv is the gain.The flow rates through the proportional valve are as follows:(2)Q1=kqxv−dvps−p1,xv>dv,0,xv≤dv,kqxv+dvp1−pr,xv<−dv,(3)Q2=kqxv−dvp2−pr,xv>dv,0,xv≤dv,kqxv+dvps−p2,xv<−dv,where xv±dv is the orifice open size managed by the spool overlap dv. The flow rate coefficient kq=Cdω2/ρ, Cd is the discharge coefficient, ω is the proportional valve area gradient, and ρ is the fluid density.Taking leakage and fluid compressibility into consideration, the flow continuity equation of the hydraulic cylinder can be written as(4)Q1=V01+A1xpp˙1βe+A1x˙p+Cip1−p2,(5)Q2=A2x˙p+Cip1−p2−Cep2−V02−A2xpp˙2βe,where Q1 and Q2 are flow into chamber without rod and chamber with rod, respectively, V01 and V02 are volume of extension chamber and retraction chamber, respectively, xp denotes the displacement of the piston, Ci is the internal leakage coefficient, Ce is the external leakage coefficient of the cylinder, and βe is the fluid effective bulk modulus.According to Newton’s second law, the equation describing the piston motion is given by(6)p1A1−p2A2=mex¨p+Bpx˙p+Ff+Fc−F,where me is the equivalent mass of the objects moving with the piston, Bp accounts for the effective viscous coefficient, Ff is the unknown friction, F denotes the pressure of third independent chamber, and F=p3A3. Defining Fd=Ff+Fc, equation (4) can be rewritten as(7)p1A1−p2A2=mex¨p+Bpx˙p−F+Fd.Let the state variablex=x1x2x3x4T=x˙pp1p2xvT. From equations (1)–(7), the state space model of the entire system can be expressed as(8)x˙=Αx+Bu+gx+DFd,y=Cx+Edd,A=−BpmeA1me−A2me0−A1βeV01−CiβeV01CiβeV010A2βeV02CiβeV02−Ci+CeβeV020000−1τ;B=000kvτ;C=103000010−6000010−60000103;D=−1me000;gx=Fmeg1xβeV01−g2xβeV020,g1x=kqx4−dvps−x2,xv>dv,kqx4+dvx2−pr,xv<−dv,g2x=kqx4−dvx3−pr,xv>dv,kqx4+dvps−x3,xv<−dv,where x∈R4×1 is the state variable, y∈R4×1 is the output vector, Ed is the known disturbance distribution matrix, d is the output disturbance or measurement noise vector (Gauss white noise included in the brief), and its derivatives d˙ and L2 norms are bounded. There is a geometrically one-to-one correspondence between elevating angle θ measured by an angular sensor and the piston displacement. The piston velocity can be obtained by the derivative of the piston displacement xp. Spool displacement sensor is embedded in proportional valve.


### 2.3. Fault Analysis and Modeling

The high-frequency faults occurred during the operation of ESS are system pressure, cylinder leakage, proportional valve amplifier drift, and sensor drift. The relevant state variables and the model parameters included in the state equation will change when faults occur. Defining the matrix deviation of the parameter matrixΑ, B, and gx in the state equation as ΔΑ, ΔB, and Δgx, respectively:(9)ΔΑ=00000−ΔCiβeV01ΔCiβeV0100ΔCiβeV02−ΔCi+ΔCeβeV0200000,(10)ΔB=000ΔkvτT,(11)Δgx=0Δg1xβeV01−Δg2xβeV020.Three types faultsfaii=1,2,3 are considered as follows:(12)fa1=fa11,fa21,(13)fa2=fa12=−βeΔCix2−x3V01,xv>dv,fa22=βeΔCix2−x3V02−βeΔCex3V02,xv<−dv,(14)fa3=Δkvτu,where(15)fa11=kqΔx4ps−x2+kqx4−dvΔps−Δx22ps−x2βeV01,xv>dv,kqΔx4x2−pr+kqx4+dvΔx22x2−prβeV01,xv<−dv,fa21=−kqΔx4x3−pr+kqx4−dvΔx32x3−prβeV02,xv>dv,−kqΔx4ps−x3+kqx4+dvΔps−Δx32ps−x3βeV02,xv<−dv.where ΔCi, ΔCe, and Δps are the deviations of internal leakage coefficient, external leakage coefficient, and supply pressure, respectively, Δx2, Δx3, and Δx4 are the changes of system state vector component, and Δx˙p is the sensor deviation. Return pressure pr is regarded as a small positive constant because of low pressure. Thus, the above faults, equations (9)∼(14), can be expressed by(16)Fa=000100010001,fa=fa11+fa12fa21+fa22fa3T.The entire state space equation with the faults can be written as(17)x˙=Αx+Bu+gx+DFd+Fafa,y=Cx+Edd+Fsfs,where Fafa=ΔΑx+ΔBu+Δgx and Fa∈R4×3 is the actuator fault distribution matrix. Assume that the actuator fault vector fa∈R3×1 satisfies fa≤αM, where αM is a known real constant. The i-th fault is expressed as faii=1,2,3. Unknown disturbance Fd satisfies Fⅆ≤βM. Fs∈R4×1 is the sensor fault distribution matrix, and Fs=1000T. fs∈R1 is the sensor fault vector, and the i-th fault is expressed as fsii=1,2,… fs=Δx˙p. Eⅆ∈R4×1 is the output disturbance distribution matrix, and d∈R1 is the output disturbance vector.


## 2.1. Description of Experimental Test Rig

Figure1 shows the test rig used in these experiments. The main hardware components of the elevating servo system of VMH are as follows: an elevating hydraulic cylinder, a proportional valve, a pair of hydraulic lock valve, an accumulator, a hydraulic pump station with a gear pump, a relief valve, and other accessories. The flow rate and operation pressure of the hydraulic pump station are 80 l/min and 120 bar, respectively. The proportional valve determines the flow rate controlling the extension and retraction of the piston rod of the elevating cylinder that changes the elevating angles of the barrel of the VMH. By comparing with the command signals, the controller gives a feedback signal and finally controls the barrel to reach the target elevating angle. In order to bear the load of the heavy barrel, the ordinary dual-acting elevating cylinder is designed with a third independent chamber. The chamber connected with the precharged accumulator through an independent pipeline is called the elevating-counterbalance cylinder.Figure 1
Experimental test rig of ESS.Figure2 is the diagram of ESS, where O defines the trunnion between the top carriage and elevating parts, A and B are the hinge joints between the elevating cylinder and the cradle and between the elevating cylinder and the top carriage, respectively, OA¯ = b, OB¯ = a, ps and pr are the proportional valve entrance port and return port pressures, respectively, p1 and p2 are the working chambers pressures, respectively, p3 denotes the pressure of third independent chamber, A1 and A2 are effective areas of the two elevating cylinder chambers, respectively, A3 is the area of third independent chamber, θ0 is the included angle between OA¯ and OB¯ while elevating hydraulic cylinder has the minimum displacement, θ is the rotation angle (elevating angle) of elevating parts around trunnion O and Fc is the force applied on the cylinder by the elevating parts.Figure 2
Diagram of ESS.


## 2.2. System Modeling

The valve spool’s first-order dynamics is considered since the frequency response of the used proportional valve is far greater than the hydraulic actuator’s. Thus,(1)u=τkvx˙v+1kvxv,where xv is the proportional valve spool displacement, u is the control input, τ is the time constant, and kv is the gain.The flow rates through the proportional valve are as follows:(2)Q1=kqxv−dvps−p1,xv>dv,0,xv≤dv,kqxv+dvp1−pr,xv<−dv,(3)Q2=kqxv−dvp2−pr,xv>dv,0,xv≤dv,kqxv+dvps−p2,xv<−dv,where xv±dv is the orifice open size managed by the spool overlap dv. The flow rate coefficient kq=Cdω2/ρ, Cd is the discharge coefficient, ω is the proportional valve area gradient, and ρ is the fluid density.Taking leakage and fluid compressibility into consideration, the flow continuity equation of the hydraulic cylinder can be written as(4)Q1=V01+A1xpp˙1βe+A1x˙p+Cip1−p2,(5)Q2=A2x˙p+Cip1−p2−Cep2−V02−A2xpp˙2βe,where Q1 and Q2 are flow into chamber without rod and chamber with rod, respectively, V01 and V02 are volume of extension chamber and retraction chamber, respectively, xp denotes the displacement of the piston, Ci is the internal leakage coefficient, Ce is the external leakage coefficient of the cylinder, and βe is the fluid effective bulk modulus.According to Newton’s second law, the equation describing the piston motion is given by(6)p1A1−p2A2=mex¨p+Bpx˙p+Ff+Fc−F,where me is the equivalent mass of the objects moving with the piston, Bp accounts for the effective viscous coefficient, Ff is the unknown friction, F denotes the pressure of third independent chamber, and F=p3A3. Defining Fd=Ff+Fc, equation (4) can be rewritten as(7)p1A1−p2A2=mex¨p+Bpx˙p−F+Fd.Let the state variablex=x1x2x3x4T=x˙pp1p2xvT. From equations (1)–(7), the state space model of the entire system can be expressed as(8)x˙=Αx+Bu+gx+DFd,y=Cx+Edd,A=−BpmeA1me−A2me0−A1βeV01−CiβeV01CiβeV010A2βeV02CiβeV02−Ci+CeβeV020000−1τ;B=000kvτ;C=103000010−6000010−60000103;D=−1me000;gx=Fmeg1xβeV01−g2xβeV020,g1x=kqx4−dvps−x2,xv>dv,kqx4+dvx2−pr,xv<−dv,g2x=kqx4−dvx3−pr,xv>dv,kqx4+dvps−x3,xv<−dv,where x∈R4×1 is the state variable, y∈R4×1 is the output vector, Ed is the known disturbance distribution matrix, d is the output disturbance or measurement noise vector (Gauss white noise included in the brief), and its derivatives d˙ and L2 norms are bounded. There is a geometrically one-to-one correspondence between elevating angle θ measured by an angular sensor and the piston displacement. The piston velocity can be obtained by the derivative of the piston displacement xp. Spool displacement sensor is embedded in proportional valve.


## 2.3. Fault Analysis and Modeling

The high-frequency faults occurred during the operation of ESS are system pressure, cylinder leakage, proportional valve amplifier drift, and sensor drift. The relevant state variables and the model parameters included in the state equation will change when faults occur. Defining the matrix deviation of the parameter matrixΑ, B, and gx in the state equation as ΔΑ, ΔB, and Δgx, respectively:(9)ΔΑ=00000−ΔCiβeV01ΔCiβeV0100ΔCiβeV02−ΔCi+ΔCeβeV0200000,(10)ΔB=000ΔkvτT,(11)Δgx=0Δg1xβeV01−Δg2xβeV020.Three types faultsfaii=1,2,3 are considered as follows:(12)fa1=fa11,fa21,(13)fa2=fa12=−βeΔCix2−x3V01,xv>dv,fa22=βeΔCix2−x3V02−βeΔCex3V02,xv<−dv,(14)fa3=Δkvτu,where(15)fa11=kqΔx4ps−x2+kqx4−dvΔps−Δx22ps−x2βeV01,xv>dv,kqΔx4x2−pr+kqx4+dvΔx22x2−prβeV01,xv<−dv,fa21=−kqΔx4x3−pr+kqx4−dvΔx32x3−prβeV02,xv>dv,−kqΔx4ps−x3+kqx4+dvΔps−Δx32ps−x3βeV02,xv<−dv.where ΔCi, ΔCe, and Δps are the deviations of internal leakage coefficient, external leakage coefficient, and supply pressure, respectively, Δx2, Δx3, and Δx4 are the changes of system state vector component, and Δx˙p is the sensor deviation. Return pressure pr is regarded as a small positive constant because of low pressure. Thus, the above faults, equations (9)∼(14), can be expressed by(16)Fa=000100010001,fa=fa11+fa12fa21+fa22fa3T.The entire state space equation with the faults can be written as(17)x˙=Αx+Bu+gx+DFd+Fafa,y=Cx+Edd+Fsfs,where Fafa=ΔΑx+ΔBu+Δgx and Fa∈R4×3 is the actuator fault distribution matrix. Assume that the actuator fault vector fa∈R3×1 satisfies fa≤αM, where αM is a known real constant. The i-th fault is expressed as faii=1,2,3. Unknown disturbance Fd satisfies Fⅆ≤βM. Fs∈R4×1 is the sensor fault distribution matrix, and Fs=1000T. fs∈R1 is the sensor fault vector, and the i-th fault is expressed as fsii=1,2,… fs=Δx˙p. Eⅆ∈R4×1 is the output disturbance distribution matrix, and d∈R1 is the output disturbance vector.


## 3. Observer and Threshold Design

### 3.1. Design of Sliding Mode Observers

To realize the fault detection of actuators and sensors, a robust sliding mode observer is designed to make the observer residual insensitive to the change of the disturbance but sensitive to the change of the two types of faults. The design idea of the observer is to take the sensor fault vector and output noise as a part of an augmented state vector and construct a new state space equation with the original state variables. To realize simultaneous multiple faults detection of actuators and sensors, some assumptions are made as follows.Assumption 1.
The nonlinear function vectorgx satisfies Lipschitz conditions locally, ∀x,x^∈Rn(18)gx−gx^≤γx−x^,where x^ is state estimation of x and γ is a positive real Lipschitz constant.Assumption 2.
For every complex numbers with nonnegative real part, the rank condition holds:(19)rsI−AFaC0=n+rFa.Assumption 3.
The matrixCFa is a full column rank as(20)rCFa=rFa.Remark 1.
Assumptions 2 and 3 are necessary and sufficient conditions for the design of a stable sliding mode observer when the system has matched uncertainty or unknown input [13].
Assume there exist nonsingular matricesR and S [16]. Define z=Rx=z1z2 and w=Sy=w1w2, formula (17) is transformed into(21)z˙=A1A2A3A4z+B1B2u+R1R2gR−1z+0Fa2fa+D1D2Fd,w=C100C2z+Fs10fs+Ed10d.
Equation (21) can be rewritten as(22)z˙1=A1z1+A2z2+B1u+R1gR−1z+D1Fd,w1=C1z1+Fs1fs+Ed1d,(23)z˙2=A3z1+A4z2+B2u+R2gR−1z+Fa2fa+D2Fd,w2=C2z2.
Equations (22) and (23) describe a singular system with stochastic noises.
Construct the augmented system statez¯ with the new state vector z¯1 and the transformed system state z2 as(24)z¯=z¯1z¯2T,z¯1=z1fsdT,z¯2=z2,E¯1z¯˙1=A¯1z¯1+A¯2z¯2+B¯1u+R¯1gR−1z¯+D¯1Fd+Nsfs+Ndd,w1=C¯1z¯1,(25)z¯˙2=A¯3z¯1+A4z¯2+B2u+R2gR−1z¯+Fa2fa+D2Fd,w2=C2z¯2,where(26)E¯1=I00000000,A¯1=A1000−I000−I,Ns=0I0,A¯2=A200,C¯1=C1Fs1Ed1,B¯1=B100,A¯3=A300,R¯1=R100,D¯1=D100,Nd=00I.
Equations (24) and (25) will be unified as follows:(27)E¯z¯˙=A¯z¯+B¯u+R¯gR−1z¯+F¯afa+F¯sfs+E¯dd+D¯Fd,w¯=C¯z¯,where(28)E¯=E¯10303I3,A¯=A¯1A¯2A¯3A4,B¯=B¯1B2,R¯=R¯1R2,F¯a=03Fa2,F¯s=Ns03,E¯d=Nd03,D¯=D¯1D2,C¯=C¯100C2.
Equation (27) includes the system states, actuator faults, sensor faults, and output noise.


### 3.2. Observer Design

To facilitate observer design, the augmented system statez¯ can be transformed into z˜ by z˜=Πz¯, where(29)∏=∏000I=E¯1+QC¯100I,z˜=z˜1z˜2=∏0z¯1z¯2.For the new state variablez˜, equations (24)∼(25) are transformed into(30)z˜˙1=A¯1∏0−1z˜1+A¯2z˜2+B¯1u+R¯1gR−1E0∏−1z˜+Qw˙1+D¯1Fd+Nsfs+Ndd,w1=C¯1∏0−1z˜1,(31)z˜˙2=A¯3∏0−1z˜1+A4z˜2+B2u+R2gR−1E0∏−1z˜+Fa2fa+D2Fd,w2=C2z˜2,where E0=I0. Two sliding mode observers are presented to the system of equations (30) and (31):(32)z˜^˙1=Ksz˜^1+A¯2z˜^2+B¯1u+R¯1gR−1E0∏−1z˜^+Qw˙1+K1w1−KsQw1+GL1w2−C2z˜^2,(33)z˜^˙2=A¯3∏0−1z˜^1+A4z˜^2+B2u+R2gR−1E0∏−1z˜^+GL2w2−C2z˜^2+υa,where z˜^1 and z˜^2 represent the estimation of z˜1 and z˜2 respectively, and K1 and Ks are unknown matrix. The discontinuous output error injection term υa is defined as(34)υa=ρaP2e˜2P2e˜2,e˜2≠0,0,others,where ρa is the observer gain and P2 is the symmetric positive definite matrix.If the state estimation error is defined ase˜=e˜1e˜2T, the dynamics can be obtained as(35)e˜˙1=z˜˙1−z˜^˙1=Kse˜1+A¯2−GL1C2e˜2+D¯1Fd−Ks−A¯1∏0−1−KsQC¯1∏0−1+K1C¯1∏0−1z˜1+Nsfs+R¯1gR−1E0∏−1z˜−gR−1E0∏−1z˜^+Ndd,(36)e˜˙2=z˜˙2−z˜^˙2=A¯3∏0−1e˜1+A4−GL2C2e˜2+R2gR−1E0∏−1z˜−gR−1E0∏−1z˜^+Fa2fa+D2Fd−υa.Equations (32) and (33) can be used as fault detection observers when the following necessary conditions are satisfied:(37)KsE¯1+K1C¯1=A¯1,(38)GL1=A¯2C2−1,(39)GL2=A4−As2C2−1.Equation (37) holds, and the rank condition of the matrix is satisfied as follows:(40)rE¯1C¯1A¯1T=rE¯1C¯1T.After assigning the parameter values,rE¯1C¯1A¯1T=3 and rE¯1C¯1T=3 are obtained. Thus, the above equation is verified.The derivative of the error can be simplified as(41)e˜˙1=Kse˜1+D¯1Fd+Nsfs+Ndd+R¯1gR−1E0∏−1z˜−gR−1E0∏−1z˜^,(42)e˜˙2=As2e˜2+A¯3∏0−1e˜1+Fa2fa+D2Fd−υa+R2gR−1E0∏−1z˜−gR−1E0∏−1z˜^.Theorem 1.
Under necessary conditions equations (37)∼(39), the error dynamics in equations (41) and (42) approach stable asymptotically, if the inequality(43)∑=Γ¯1Γ2P1D¯1P1NdP1NsΓ3Γ¯4P2D200D¯1TP1D2TP2−σ2I00NdTP100−σ2I0NsTP000−σ2I<0,holds for a symmetric positive definite matrix P and the scalar σ > 0, where P=diagP1,P2,(44)Γ1=KsTP1+P1Ks+κγ2R−1E0∏−12+κ−1P1R¯1P1R¯1T,Γ2=P2A¯3∏0−1+κ−1P1R¯1P2R2T,Γ3=P2A¯3∏0−1T+κ−1P2R2P1R¯1T,Γ4=As2TP2+P2As2+κγ2R−1E0∏−12+κ−1P2R2P2R2T,Γ¯1=Γ1+C¯1∏0−1C¯1∏0−1T,Γ¯4=Γ4+C2C2T.The proof of this theorem is given in Appendix.Equation (43) is a linear matrix inequality. According to the Schur complement lemma, the parameter matrices P and σ can be solved using the MATLAB LMI toolbox.Theorem 2.
[3]. Under assumptions 1–3 and the observer equations (32) and (33), the trajectories of the error dynamics equations (35) and (36) can be driven to the sliding surface S=e2e2=0 in finite time if the gain holds as(45)ρa≥αM+A¯2∏0−1+γR2R−1E0∏−1εM+D1βM,where εM is the system error upper bound.Proof.
Based on theorem 1, the time derivative of the Lyapunov functionV2=e˜2TP2e˜2 is(46)V˙2=e˜2TAs2TP2+P2As2e˜2+2e˜2TP2A¯3∏0−1e˜1+2e˜2TP2D2Fd+2e˜2TP2R2gR−1E0∏−1z˜−gR−1E0∏−1z˜^−2e˜2TP2υa+2e˜2TP2Fa2fa.
With Cauchy–Schwartz inequalities,e˜=e˜1+e˜2≤εM, and As2TP2+P2As2<0, the time derivative is as follows.
ForFa2=1,(47)V˙2≤2e˜2TP2A¯3∏0−1e˜1+γR2R−1E0∏−1e˜+D2Fd−α0M=2e˜2TP2A¯3∏0−1e˜1+γR2R−1E0∏−1e˜+D2Fd−α0M≤−2e˜2TP2α0M−D2βM−A¯3∏0−1+γR2R−1E0∏−1εM<0,when(48)α0M−A¯3∏0−1+γR2R−1E0∏−1εM+D2βM≥0.
The inequality (45) satisfies(49)ρa≥αM+A¯2∏0−1+γR2R−1E0∏−1εM+D1βM.
This completes the proof.


### 3.3. Dynamic Threshold Design

For the ideal state, the observer residual signal should be zero while no fault occurs. However, in actual application, the residual will be approximately to zero, because the state of complete decoupling almost does not exist. The traditional fixed thresholds are less sensitive to faults due to its wide range, especially when incipient faults occur. The dynamic threshold can solve this problem and has the characteristics of a narrow threshold and high sensitivity. Therefore, it is necessary to design a threshold functionJth to make the residual satisfy(50)e≤Jth,fa=0,e>Jth,fa≠0.Velocity residuals are a nonstationary stochastic process, which corresponds to Gauss distribution [4]. The well-known algorithm, exponentially weighted moving average (EWMA) chart [17], is used to improve the signal-to-noise ratio of velocity residuals:(51)μj=λ0rj+1−λ0μj−1,j=1,…,N,where rj is the velocity residuals (k is the sampling number of velocity residuals), and the smoothing factor λ0 takes as 0.25.According to stochastic theories, the mean and variance of velocity residuals are as follows:(52)μr=1k∑j=1kμj,(53)σr2=1k−1∑j=1kμj−μr2.The confidence limit of the mean of velocity residuals is(54)Pμr−zσr<μr<μr+zσr=1−α,where α is the confidence level and z is the coefficient. In general engineering practice, α=0.025. According to the z test table, the coefficient z = 2.24.The dynamic threshold is defined as(55)Jth=μr±zσr,x¨p>x¨pc,v0,x¨p≤x¨pc,where v0 is an empirically stable stage threshold and x¨pc is the critical acceleration of the hydraulic cylinder. According to equations (50)∼(55), the fault detection can be realized.


## 3.1. Design of Sliding Mode Observers

To realize the fault detection of actuators and sensors, a robust sliding mode observer is designed to make the observer residual insensitive to the change of the disturbance but sensitive to the change of the two types of faults. The design idea of the observer is to take the sensor fault vector and output noise as a part of an augmented state vector and construct a new state space equation with the original state variables. To realize simultaneous multiple faults detection of actuators and sensors, some assumptions are made as follows.Assumption 1.
The nonlinear function vectorgx satisfies Lipschitz conditions locally, ∀x,x^∈Rn(18)gx−gx^≤γx−x^,where x^ is state estimation of x and γ is a positive real Lipschitz constant.Assumption 2.
For every complex numbers with nonnegative real part, the rank condition holds:(19)rsI−AFaC0=n+rFa.Assumption 3.
The matrixCFa is a full column rank as(20)rCFa=rFa.Remark 1.
Assumptions 2 and 3 are necessary and sufficient conditions for the design of a stable sliding mode observer when the system has matched uncertainty or unknown input [13].
Assume there exist nonsingular matricesR and S [16]. Define z=Rx=z1z2 and w=Sy=w1w2, formula (17) is transformed into(21)z˙=A1A2A3A4z+B1B2u+R1R2gR−1z+0Fa2fa+D1D2Fd,w=C100C2z+Fs10fs+Ed10d.
Equation (21) can be rewritten as(22)z˙1=A1z1+A2z2+B1u+R1gR−1z+D1Fd,w1=C1z1+Fs1fs+Ed1d,(23)z˙2=A3z1+A4z2+B2u+R2gR−1z+Fa2fa+D2Fd,w2=C2z2.
Equations (22) and (23) describe a singular system with stochastic noises.
Construct the augmented system statez¯ with the new state vector z¯1 and the transformed system state z2 as(24)z¯=z¯1z¯2T,z¯1=z1fsdT,z¯2=z2,E¯1z¯˙1=A¯1z¯1+A¯2z¯2+B¯1u+R¯1gR−1z¯+D¯1Fd+Nsfs+Ndd,w1=C¯1z¯1,(25)z¯˙2=A¯3z¯1+A4z¯2+B2u+R2gR−1z¯+Fa2fa+D2Fd,w2=C2z¯2,where(26)E¯1=I00000000,A¯1=A1000−I000−I,Ns=0I0,A¯2=A200,C¯1=C1Fs1Ed1,B¯1=B100,A¯3=A300,R¯1=R100,D¯1=D100,Nd=00I.
Equations (24) and (25) will be unified as follows:(27)E¯z¯˙=A¯z¯+B¯u+R¯gR−1z¯+F¯afa+F¯sfs+E¯dd+D¯Fd,w¯=C¯z¯,where(28)E¯=E¯10303I3,A¯=A¯1A¯2A¯3A4,B¯=B¯1B2,R¯=R¯1R2,F¯a=03Fa2,F¯s=Ns03,E¯d=Nd03,D¯=D¯1D2,C¯=C¯100C2.
Equation (27) includes the system states, actuator faults, sensor faults, and output noise.


## 3.2. Observer Design

To facilitate observer design, the augmented system statez¯ can be transformed into z˜ by z˜=Πz¯, where(29)∏=∏000I=E¯1+QC¯100I,z˜=z˜1z˜2=∏0z¯1z¯2.For the new state variablez˜, equations (24)∼(25) are transformed into(30)z˜˙1=A¯1∏0−1z˜1+A¯2z˜2+B¯1u+R¯1gR−1E0∏−1z˜+Qw˙1+D¯1Fd+Nsfs+Ndd,w1=C¯1∏0−1z˜1,(31)z˜˙2=A¯3∏0−1z˜1+A4z˜2+B2u+R2gR−1E0∏−1z˜+Fa2fa+D2Fd,w2=C2z˜2,where E0=I0. Two sliding mode observers are presented to the system of equations (30) and (31):(32)z˜^˙1=Ksz˜^1+A¯2z˜^2+B¯1u+R¯1gR−1E0∏−1z˜^+Qw˙1+K1w1−KsQw1+GL1w2−C2z˜^2,(33)z˜^˙2=A¯3∏0−1z˜^1+A4z˜^2+B2u+R2gR−1E0∏−1z˜^+GL2w2−C2z˜^2+υa,where z˜^1 and z˜^2 represent the estimation of z˜1 and z˜2 respectively, and K1 and Ks are unknown matrix. The discontinuous output error injection term υa is defined as(34)υa=ρaP2e˜2P2e˜2,e˜2≠0,0,others,where ρa is the observer gain and P2 is the symmetric positive definite matrix.If the state estimation error is defined ase˜=e˜1e˜2T, the dynamics can be obtained as(35)e˜˙1=z˜˙1−z˜^˙1=Kse˜1+A¯2−GL1C2e˜2+D¯1Fd−Ks−A¯1∏0−1−KsQC¯1∏0−1+K1C¯1∏0−1z˜1+Nsfs+R¯1gR−1E0∏−1z˜−gR−1E0∏−1z˜^+Ndd,(36)e˜˙2=z˜˙2−z˜^˙2=A¯3∏0−1e˜1+A4−GL2C2e˜2+R2gR−1E0∏−1z˜−gR−1E0∏−1z˜^+Fa2fa+D2Fd−υa.Equations (32) and (33) can be used as fault detection observers when the following necessary conditions are satisfied:(37)KsE¯1+K1C¯1=A¯1,(38)GL1=A¯2C2−1,(39)GL2=A4−As2C2−1.Equation (37) holds, and the rank condition of the matrix is satisfied as follows:(40)rE¯1C¯1A¯1T=rE¯1C¯1T.After assigning the parameter values,rE¯1C¯1A¯1T=3 and rE¯1C¯1T=3 are obtained. Thus, the above equation is verified.The derivative of the error can be simplified as(41)e˜˙1=Kse˜1+D¯1Fd+Nsfs+Ndd+R¯1gR−1E0∏−1z˜−gR−1E0∏−1z˜^,(42)e˜˙2=As2e˜2+A¯3∏0−1e˜1+Fa2fa+D2Fd−υa+R2gR−1E0∏−1z˜−gR−1E0∏−1z˜^.Theorem 1.
Under necessary conditions equations (37)∼(39), the error dynamics in equations (41) and (42) approach stable asymptotically, if the inequality(43)∑=Γ¯1Γ2P1D¯1P1NdP1NsΓ3Γ¯4P2D200D¯1TP1D2TP2−σ2I00NdTP100−σ2I0NsTP000−σ2I<0,holds for a symmetric positive definite matrix P and the scalar σ > 0, where P=diagP1,P2,(44)Γ1=KsTP1+P1Ks+κγ2R−1E0∏−12+κ−1P1R¯1P1R¯1T,Γ2=P2A¯3∏0−1+κ−1P1R¯1P2R2T,Γ3=P2A¯3∏0−1T+κ−1P2R2P1R¯1T,Γ4=As2TP2+P2As2+κγ2R−1E0∏−12+κ−1P2R2P2R2T,Γ¯1=Γ1+C¯1∏0−1C¯1∏0−1T,Γ¯4=Γ4+C2C2T.The proof of this theorem is given in Appendix.Equation (43) is a linear matrix inequality. According to the Schur complement lemma, the parameter matrices P and σ can be solved using the MATLAB LMI toolbox.Theorem 2.
[3]. Under assumptions 1–3 and the observer equations (32) and (33), the trajectories of the error dynamics equations (35) and (36) can be driven to the sliding surface S=e2e2=0 in finite time if the gain holds as(45)ρa≥αM+A¯2∏0−1+γR2R−1E0∏−1εM+D1βM,where εM is the system error upper bound.Proof.
Based on theorem 1, the time derivative of the Lyapunov functionV2=e˜2TP2e˜2 is(46)V˙2=e˜2TAs2TP2+P2As2e˜2+2e˜2TP2A¯3∏0−1e˜1+2e˜2TP2D2Fd+2e˜2TP2R2gR−1E0∏−1z˜−gR−1E0∏−1z˜^−2e˜2TP2υa+2e˜2TP2Fa2fa.
With Cauchy–Schwartz inequalities,e˜=e˜1+e˜2≤εM, and As2TP2+P2As2<0, the time derivative is as follows.
ForFa2=1,(47)V˙2≤2e˜2TP2A¯3∏0−1e˜1+γR2R−1E0∏−1e˜+D2Fd−α0M=2e˜2TP2A¯3∏0−1e˜1+γR2R−1E0∏−1e˜+D2Fd−α0M≤−2e˜2TP2α0M−D2βM−A¯3∏0−1+γR2R−1E0∏−1εM<0,when(48)α0M−A¯3∏0−1+γR2R−1E0∏−1εM+D2βM≥0.
The inequality (45) satisfies(49)ρa≥αM+A¯2∏0−1+γR2R−1E0∏−1εM+D1βM.
This completes the proof.


## 3.3. Dynamic Threshold Design

For the ideal state, the observer residual signal should be zero while no fault occurs. However, in actual application, the residual will be approximately to zero, because the state of complete decoupling almost does not exist. The traditional fixed thresholds are less sensitive to faults due to its wide range, especially when incipient faults occur. The dynamic threshold can solve this problem and has the characteristics of a narrow threshold and high sensitivity. Therefore, it is necessary to design a threshold functionJth to make the residual satisfy(50)e≤Jth,fa=0,e>Jth,fa≠0.Velocity residuals are a nonstationary stochastic process, which corresponds to Gauss distribution [4]. The well-known algorithm, exponentially weighted moving average (EWMA) chart [17], is used to improve the signal-to-noise ratio of velocity residuals:(51)μj=λ0rj+1−λ0μj−1,j=1,…,N,where rj is the velocity residuals (k is the sampling number of velocity residuals), and the smoothing factor λ0 takes as 0.25.According to stochastic theories, the mean and variance of velocity residuals are as follows:(52)μr=1k∑j=1kμj,(53)σr2=1k−1∑j=1kμj−μr2.The confidence limit of the mean of velocity residuals is(54)Pμr−zσr<μr<μr+zσr=1−α,where α is the confidence level and z is the coefficient. In general engineering practice, α=0.025. According to the z test table, the coefficient z = 2.24.The dynamic threshold is defined as(55)Jth=μr±zσr,x¨p>x¨pc,v0,x¨p≤x¨pc,where v0 is an empirically stable stage threshold and x¨pc is the critical acceleration of the hydraulic cylinder. According to equations (50)∼(55), the fault detection can be realized.


## 4. Experimental Results

To validate the derived model and evaluate the effectiveness of the proposed scheme, the experiment was conducted on a laboratory ESS test rig. The high-performance controller X20CP3585 of B&R Industrial Automation Company was employed, and the sampling period was 0.4 ms. The data acquisition module and fault diagnosis system are operated at a rate of 1 kHz. Two types of experiments were carried out: (1) the normal state; (2) the fault state.


### 4.1. Normal State

The slaving process of the elevating servo system of VMH is a typical point-to-point control. To facilitate this study and ensure the representativeness of the experiment, this article considered the upward and downward gun slaving process as a working cycle. Table1 shows the system parameters.Table 1
Parameters of system.
Parameters
Value
A
1 (m2)
0.0084
A
2 (m2)
0.0084
A
3 (m2)
0.0151
p
s (MPa)
12
p
r (MPa)
0.15
V
01 (m3)
0.0277
V
02 (m3)
0.0088
m
e (kg)
8224.4
θ
0 (°)
21.5
a (m)
1.075
b (m)
2.115
τ (s)
0.012
C
i (m3·s−1·Pa−1)
5 × 10−15
C
e (m3·s−1·Pa−1)
1 × 10−15
β
e (MPa)
1200
d
v (m)
5 × 10−5
k
v (m·V−1)
1 × 10−3The sensor output data were considered using white noise signal with mean value 0 and variance value 0.1. Figure3 shows the elevating angle of command value, measurement value, and observed value in normal state.Figure 3
Elevating angle.Generally, the hydraulic cylinder undergoes the motion of acceleration-uniform velocity-deceleration process; thereby, the working process can be separated into a stable stage (uniform velocity) and a transitional stage (acceleration or deceleration). Due to the large difference in the acceleration characteristics of the two stages, the instability of the speed residuals signal is shown in Figure4(a), which was used to distinguish between stable stage and transitional stage. According to several experiments, the critical acceleration of the cylinder was chosen with the value of ±0.002 m/s2. As shown in Figure 4(a), under the normal working condition, the dotted line shows the upper threshold and lower threshold and solid lines shows the residuals signal. As a comparison, a method based on Wald’s sequential test proposed in [18] is applied to the ESS. The result is shown in Figure 4(b). The residuals of cylinder velocity in normal state change in a wide range.Figure 4
(a) Residuals of cylinder velocity and acceleration in normal state. (b) Residuals of cylinder velocity using method in [18].
(a)
(b)


### 4.2. Fault State

The causes of parameter change in the system were complex and diverse. In order to verify the observer's ability to detect faults, the common faults were set artificially in the laboratory environment based on Table2.Table 2
Common faults implementation scheme.
Serial no.
Faults type
Faults implementation
Parameter setting
1
System pressure fault
Regulating pressure by a proportional relief valve
−1.0 MPa
2
Internal leakage
Adding an adjustable throttle valve between cylinder ports
Throttle diameter 3.5 mm
3
Proportional fault
Reducing proportional valve amplifier gain
95%
4
Angular sensor fault
Reducing sensor gain
95%The failure decision criteria were designed as follows: if the system residual signal exceeds the dynamic threshold, the system could be regarded as faulty, and an alarm signal will be sent out. Figure5 shows the system state with pressure deviation implemented by substituting the pilot relief valve with a proportional pilot relief valve. Both types of valves have the same pressure-flow rate performance with different control modes only. As seen, the alarm signal appeared in the whole working cycle. The fault detection system generated some alarm signals when the residuals went beyond the dynamic threshold obviously.Figure 5
Residuals of system pressure fault and alarm signal.An adjustable throttle valve was added between the extension chamber and retraction chamber with throttle orifice diameter of 3.5 mm to simulate a state of increasing internal leakage caused by seal wear. Figure6 shows the fluctuations in residuals exceeding the dynamic threshold with alarm signal.Figure 6
Residuals of internal leakage and alarm signal.To conduct the fault state of the flow reduction through the servo valve due to blockage, the maximum displacement of the proportional valve spool was set as 95% of the normal state. When the spool of the proportional valve is blocked or stuck, the fault detection system could detect this fault and send out an alarm signal. The experimental results are shown in Figure7.Figure 7
Residuals of proportional fault and alarm signal.The angular displacement sensor gain was adjusted to simulate the fault state that the sensing component in the sensor drifts. When the sensor output value deviated, the fault detection system was capable of discerning the fault and producing an alarm signal. The experimental results are shown in Figure8.Figure 8
Residuals of angular sensor fault and alarm signal.By comparing the methods in reference [18], the system pressure fault is taken as an example to detect the system faults. Because the observer has poor robustness to disturbance, the residual fluctuation is large in the normal state and the threshold must be set wider. As a result, the observer is insensitive to system pressure fault, and the system does not alarm. Compared with Figure 4(b) and Figures 5 and 9, the sliding mode-based observer method shows a better robustness and is more sensitive to faults.Figure 9
Residuals of system pressure fault and alarm signal.


## 4.1. Normal State

The slaving process of the elevating servo system of VMH is a typical point-to-point control. To facilitate this study and ensure the representativeness of the experiment, this article considered the upward and downward gun slaving process as a working cycle. Table1 shows the system parameters.Table 1
Parameters of system.
Parameters
Value
A
1 (m2)
0.0084
A
2 (m2)
0.0084
A
3 (m2)
0.0151
p
s (MPa)
12
p
r (MPa)
0.15
V
01 (m3)
0.0277
V
02 (m3)
0.0088
m
e (kg)
8224.4
θ
0 (°)
21.5
a (m)
1.075
b (m)
2.115
τ (s)
0.012
C
i (m3·s−1·Pa−1)
5 × 10−15
C
e (m3·s−1·Pa−1)
1 × 10−15
β
e (MPa)
1200
d
v (m)
5 × 10−5
k
v (m·V−1)
1 × 10−3The sensor output data were considered using white noise signal with mean value 0 and variance value 0.1. Figure3 shows the elevating angle of command value, measurement value, and observed value in normal state.Figure 3
Elevating angle.Generally, the hydraulic cylinder undergoes the motion of acceleration-uniform velocity-deceleration process; thereby, the working process can be separated into a stable stage (uniform velocity) and a transitional stage (acceleration or deceleration). Due to the large difference in the acceleration characteristics of the two stages, the instability of the speed residuals signal is shown in Figure4(a), which was used to distinguish between stable stage and transitional stage. According to several experiments, the critical acceleration of the cylinder was chosen with the value of ±0.002 m/s2. As shown in Figure 4(a), under the normal working condition, the dotted line shows the upper threshold and lower threshold and solid lines shows the residuals signal. As a comparison, a method based on Wald’s sequential test proposed in [18] is applied to the ESS. The result is shown in Figure 4(b). The residuals of cylinder velocity in normal state change in a wide range.Figure 4
(a) Residuals of cylinder velocity and acceleration in normal state. (b) Residuals of cylinder velocity using method in [18].
(a)
(b)


## 4.2. Fault State

The causes of parameter change in the system were complex and diverse. In order to verify the observer's ability to detect faults, the common faults were set artificially in the laboratory environment based on Table2.Table 2
Common faults implementation scheme.
Serial no.
Faults type
Faults implementation
Parameter setting
1
System pressure fault
Regulating pressure by a proportional relief valve
−1.0 MPa
2
Internal leakage
Adding an adjustable throttle valve between cylinder ports
Throttle diameter 3.5 mm
3
Proportional fault
Reducing proportional valve amplifier gain
95%
4
Angular sensor fault
Reducing sensor gain
95%The failure decision criteria were designed as follows: if the system residual signal exceeds the dynamic threshold, the system could be regarded as faulty, and an alarm signal will be sent out. Figure5 shows the system state with pressure deviation implemented by substituting the pilot relief valve with a proportional pilot relief valve. Both types of valves have the same pressure-flow rate performance with different control modes only. As seen, the alarm signal appeared in the whole working cycle. The fault detection system generated some alarm signals when the residuals went beyond the dynamic threshold obviously.Figure 5
Residuals of system pressure fault and alarm signal.An adjustable throttle valve was added between the extension chamber and retraction chamber with throttle orifice diameter of 3.5 mm to simulate a state of increasing internal leakage caused by seal wear. Figure6 shows the fluctuations in residuals exceeding the dynamic threshold with alarm signal.Figure 6
Residuals of internal leakage and alarm signal.To conduct the fault state of the flow reduction through the servo valve due to blockage, the maximum displacement of the proportional valve spool was set as 95% of the normal state. When the spool of the proportional valve is blocked or stuck, the fault detection system could detect this fault and send out an alarm signal. The experimental results are shown in Figure7.Figure 7
Residuals of proportional fault and alarm signal.The angular displacement sensor gain was adjusted to simulate the fault state that the sensing component in the sensor drifts. When the sensor output value deviated, the fault detection system was capable of discerning the fault and producing an alarm signal. The experimental results are shown in Figure8.Figure 8
Residuals of angular sensor fault and alarm signal.By comparing the methods in reference [18], the system pressure fault is taken as an example to detect the system faults. Because the observer has poor robustness to disturbance, the residual fluctuation is large in the normal state and the threshold must be set wider. As a result, the observer is insensitive to system pressure fault, and the system does not alarm. Compared with Figure 4(b) and Figures 5 and 9, the sliding mode-based observer method shows a better robustness and is more sensitive to faults.Figure 9
Residuals of system pressure fault and alarm signal.


## 5. Conclusions

In the paper, an online fault detection scheme is presented for the ESS with model uncertainty, nonlinearities, unknown disturbance, and output noise, which is capable of improving the safety and reliability of the VMH system. A robust sliding mode-based observer is designed to generate the fault detection residual to detect actuator faults and sensor faults simultaneously. Four experiments on faults are conducted. The correctness of the derived model and some common fault expressions are also validated. A comparative experiment was performed, and the proposed method shows good performance.Despite the online fault detection scheme was demonstrated in the ESS system, it is expected to be applied to similar industrial systems [19, 20].



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*Source: 4173871-2019-10-07.xml*