Magnesium alloy is the lightest metal structural materials in engineering applications, which is considered as the green engineering metal material of the 21st century with the most full of development and application potential. They have been widely used in the national defense, aerospace, high-speed trains, cars and electronic communications and other fields [1].
With the increasing demand to magnesium alloy sheets, there rises higher requirement for the rolling technology of magnesium sheet mill. So improving the operating precision of the whole mill system will become necessary.
Hydraulic automatic gauge control (HAGC) system is the key component of magnesium sheet mill. The performance of AGC has a direct impact on product quality and operation efficiency. HAGC system is the difficulties and focus of strip mill because of its high failure rate and the complexity of the failure causes [2, 3]. That is also the key reason of failure cut-off and decline in product quality [4].
Many difficulties are brought to condition monitoring and fault analysis because of the complex fault status and interference factors [5]. It is important to improve the running accuracy of the magnesium sheet mill by exploring suitable fault diagnosis method. So far, model-based method is still the most effective method of fault diagnosis [6]. The
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main problem of this method is that it has a strong dependence on model. To a certain extent, the accuracy of the model affects the accuracy of fault diagnosis.
In recent years, the unknown input observer (UIO) begins to be used for fault diagnosis of uncertain systems [7-9]. UIO is an estimator which can act on the system but be decoupled from the unknown input [10]. UIO has become a robust design methods commonly used in FDI.
This paper uses UIO method to study the problem of fault diagnosis of the hydraulic automatic gauge control system of magnesium plate mill. Based on the mathematical model and fault analysis of HAGC system, a fault model was established, and a group of unknown input observers were designed to realize fault diagnosis and isolation. 2. Mathematical Model of HAGC System
The control object of hydraulic AGC system is the oil cylinder which is driven by electro-hydraulic servo valve. Its task is to achieve the purpose of closed loop control for position or pressure. By controlling the position or pressure of cylinder at required value accurately, the purpose to control thickness can be realized.
A complete hydraulic AGC system is mainly composed of servo valve, oil supply pipe, hydraulic valve, rolling mill, oil return pipe, sensors, control regulator, etc. The hydraulic servo pressure system is composed of hydraulic part and control part. The hydraulic parts include hydraulic oil system and hydraulic servo cylinder system. The control part mainly adopts the position feedback control mode. Fig. 1 is the principle diagram of the hydraulic position servo control system.
Fig. 1. Principle diagram of AGC position closed-loop control.
PI controller is used in the system as the position controller, and the transfer function of the PI controller is:
c p
i
1( ) (1 )G s K
T s , (1)
where Kp is the proportional gain coefficient, Ti is the integral time constant, and s is Laplace operator.
The bandwidth of the servo amplifier is much higher than the electro-hydraulic servo valve, and the
response of the servo amplifier is fast. So the time constant can be ignored, and the servo amplifier can be approximated as a proportional amplifier [11]. The transfer function is:
vi( )G s K , (2)
where Kvi is the gain of the servo amplifier. Due to the servo valve always works near the
equilibrium point (zero point) while the mill is running in the normal rolled condition, thus the traditional small incremental linear analysis method can be used for approximate linearization in a certain range around the operating point. The linearized equation of the output flow QL is:
L sv0 i LQ Q K P , (3)
where QL is the load flow of hydraulic cylinder; Qsv0
is the no-load flow of servo valve; Ki is the pressure flow amplification coefficient of the hydraulic cylinder; PL is the load pressure variation.
According to the size of the inherent frequency of power mechanism, when the natural frequency h of
a hydraulic actuator is greater than 50 Hz, the dynamic characteristic of the servo valve can be described as the following second-order link;
sv0 sv
2sv
2sv sv
21
Q K
sI
, (4)
where Ksv is the static flow amplification factor of the servo valve, ωsv is the inherent frequency of servo valve; ξsv is the damping coefficient.
The flow from the valve to oil cylinder not only drives piston movement, but also compensates various leak in cylinder, liquid amount of compression and the expansion of the pipelines, and so on. So, the continuity equation of the hydraulic cylinder flow can be described as:
p 0 LL p i L
e
d d
d 4 d
x V PQ A K P
t t , (5)
where xp is the displacement of the hydraulic cylinder piston, Ap is the area of the hydraulic cylinder piston, Ki is internal leakage coefficient of the hydraulic cylinder, βe is the hydraulic oil elastic modulus, V0 is the initial volume of the hydraulic cylinder control chamber, PL is outlet pressure of the valve.
If the rolling mill is regarded as an approximation of a single degree of freedom model, the force balance equation can be described as:
2p p
P L e e e p d2
d d
d d
x xF A P M B K x F
t t , (6)
where F is the output force of the hydraulic cylinder, Me is the total mass of the moving parts in rolls, Be is
controllercontroller servo
amplifier servo device
hydrauliccylinder
displacementsensor
instructions
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the viscosity coefficient of moving parts, Ke is the stiffness coefficient of elastic load, Fd denotes other loads applied to the piston.
The displacement sensor is generally seen as an inertial link and modeled as:
s
s
( )1
KG s
T s
, (7)
where Ks is the feedback gain coefficient of the displacement sensor, and Ts is the time constant. According to the above-mentioned partial models and the relationship between them, the mathematical
model of the whole hydraulic cylinder displacement control system can be built as shown in Fig. 2, where
( ) nx t R is the state vector, ( ) mu t R is the input
vector; ( ) py t R is the output vector; ( ) ld t R is
the disturbance vector, which includes system noise, unknown input and model error; f(t) is the unknown inputs as faults of component and actuator. A, B, C, E and G are constant matrices with compatible dimensions.
Typically, the roll gap is made fine adjustments around the setting value of the roll gap. For the hydraulic AGC system shown in Fig. 2, Fd=0 can be accepted. So, the system input u is the given signal ui.
2
2
21
SV
sv
sv sv
k
s
p
2e e e
A
M s B s K
1S
S
K
T s
0
1
4 ie
Vs K
1
pA
LQLP
Fig. 2. Mathematical model of hydraulic cylinder displacement control system.
The state variables of the system are chosen as: X1=uf, X2=ui-uf, X3=QL, X4= LQ , X5=PL, X6=xp, X7= px .
Choose X1, X4, X5, X7 as the output variables, then the state equation of the hydraulic AGC system can be described as:
s1 1 6
s s
2 i 1
3 4
2 24 sv vi p sv i sv vi p sv 1
p 2 2sv vi sv 2 sv 3 sv sv 4
i
e e e5 3 i 5 P 7
0 0 0
6 7
e eP7 5 6 7
e e e
1
=
2
4 4 4
KX X X
T T
X u X
X X
X K K K u K K K X
KK K X X X
T
X X K X A XV V V
X X
K BAX X X X
M M M
(8)
3. Model of Fault Detection and Analysis
Consider the effect of additional disturbances and the faults on components, the state equation of a general system can be described as [13-15]:
( ) ( ) ( ) ( ) ( )
( ) ( )
x t Ax t Bu t Ed t Gf t
y t Cx t
(9)
The output equation of the system is described as:
T1 4 5 7[ ]y X X X X (10)
Fault of electro-hydraulic servo valve is reflected by the static flow amplification coefficient Ksv; fault of the hydraulic cylinder is reflected by internal leakage coefficient of the hydraulic cylinder Ki and the hydraulic oil elastic module βe. As the quality of the roll system is relatively stable, its fault is reflected by the damping coefficient of the roll system Be and the load spring stiffness Ke.
The above-mentioned fault parameters will be analyzed and described in the following, and the direction vector of the each fault parameter will be determined. The direction vector of every fault (fi, i=1,2,…,n) is represented by gi (i=1,2,…,n).
For the fault of electro-hydraulic servo valve caused by the change of Ksv
T
1 1 sv vi
p2 2 2p sv i p sv sv
i
0 0 0 1 0 0 0 ( )
( )f
g f K K
KK u K u u
T
(11)
For the fault caused by the change of Ki
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T e
2 2 i L0
40 0 0 0 1 0 0 ( )g f K P
V
(12)
For the fault caused by the change of βe
T e2 3 L i L P P
0
40 0 0 0 1 0 0 ( )g f Q K P A X
V
(13)
For the equivalent damping coefficient Be of the
rolling mill, as the state variable X7 is the differential calculus of the output roll gap which cannot be detected directly, the fault condition cannot be detected. So the fault of roll system is described by the change of the load spring stiffness Ke, namely
T e3 4 p
T
0 0 0 0 0 0 1 ( )K
g f XM
(14)
So the fault vector can be expressed as:
1
2 3
4
0 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 0 1
Tf
Gf f f
f
(15)
As a result, the available matrix A, B, C, G can be
derived respectively as
s
p2 2 2sv vi p sv sv vi sv sv
e
0
s
s
sv sv
e ei p
0 0
p e e
e e e
10 0
1 0 0
0 0 0
40 0
0 0 0
0 0 0
0 0 0
0 0 0 0
1 0 0 0
2 0 0 0 +
4 40 0
0 0 0 1
0
I
T
KK K K K KA
T
V
K
T
K AV V
A K B
M M M
(16)
2 T
sv vi p sv[0 1 0 0 0 0]B K K K (17)
1 0 0 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 0 1
C
(18)
T0 0 0 1 0 0 0
0 0 0 0 1 0 0
0 0 0 0 0 0 1
G
(19)
4. Construction of Fault Observer
For the system described by equation (8), the UIO observer can be described as
( ) ( ) ( ) ( )
ˆ( ) ( ) ( )
z t Fz t TBu t Ky t
x t z t Hy t
, (20)
where ˆ( ) nx t R is the estimated state vector and
( ) nz t R is the state of the observer. The observer
state-space matrices F, T, K and H will be designed to decouple the disturbance from the state estimation error completely. The structure of the UIO observer is shown in Fig. 3.
x
( )z t ˆ( )x t
ˆ( )y t
Fig. 3. Structure of UIO observer.
The sufficient condition of UIO observer described in equation (20) becoming a fault observer is: when f(t)=0, the state estimation error and output estimation error approach zero asymptotically, that is:
ˆlim ( ) lim( ( ) ( )) 0t t
e t x t x t
(21)
ˆ ˆlim( ( ) ( )) lim( ( ) ( )) 0t t
y t y t y t Cx t
(22)
The state estimation error can be made of the following equation:
1
1
2 1
( ) ( ) ( )
[ ( )] ( )
[ ( ) ] ( )
[ ( )] ( ) ( ) ( ),
e t A HCA K C e t
F A HCA K C z t
K A HCA K C H y t
T I HC Bu t HC I Ed t
(23)
where K=K1+K2. To synthesize UIO, the following relationships must hold for the observer matrices F, T, K and H:
1 2
( ) 0,
,
HC I E T I HC
F A HCA K C K FH
(24)
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In the case of meeting the above-mentioned conditions, the estimation error of the system state is:
e Fe (25)
Choose a stable matrix F, the error e can tend to zero gradually. So, on the promise of confirming the presence of UIO, an unknown input observer is designed by first selecting a stable F and then solving equations (24)-(25).
For the system described by equation (8), the sufficient and necessary condition of that the UIO observer exists is [16]
rank(CE)=rank(E) (26)
Theoretically, the residuals ˆr y Cx should be
approximately zero when there is no fault; once a fault occurs, the residual values will deviate from zero. Due to the influence of modeling precision and other factors, the residual in fault free condition is difficult to maintain zero strictly. So the method of threshold is adopted to analyze and judge. Let the threshold value be . If ir , the detection signal
is set as s=0, corresponding to a fault-free condition; if ir , the detection signal is set as s = 1,
corresponding to a fault condition. Since hydraulic automatic gauge control system
of rolling mill is a complex system integrating mechanical, electrical and hydraulic parts, its potential fault points are numerous. If the fault location is want to be found, the faults need to be positioned and isolated. In this paper, three UIO are designed for the hydraulic control system to detect and separate the faults of servo valve, the leaking fault of hydraulic system and the fault of load spring stiffness. When designing UIO, disturbances including system noise, unknown inputs and modeling deviation are omitted. Instead, the fault signal f is regard as the unknown input disturbance of the system, and its direction vector is G. The input signals of the i-th unknown input observer UIOi includes all control input and output signals except the i-th measured signals. If the i-th fault occurs, the output residual of UIOi is the least of UIOs’ because of the structural conditions of the UIOi. Using this feature, the faulty components can be found.
The fault direction of the first UIO is g1=[0 0 0 1 0 0 0]T; the fault direction of the second UIO is g2=[0 0 0 0 1 0 0]T; the fault direction of the third UIO is g3=[0 0 0 0 0 0 1]T. Upon examination, rank (Cgi)=rank(gi) =1 (i=1,2,3), the necessary and sufficient conditions of UIO existence is met. According to the above-mentioned solving method of UIO fault observer, the UIO observer parameters can be obtained. The main parameters used in the design are shown in Table 1.
In order to test the performance of the designed UIO fault observers, three groups of faults are simulated with MATLAB/Simulink.
Table 1. Main parameters of the rolling mill system.
Variables Unit Value Me kg 1.874×105
Be (Ns)/mm 3.04×104 Ke N/mm 3.18×105 V0 mm3 0.7239×108
Ki mm5/Ns 5000 Ksv mm/sA 6.48×109 Ks V/mm 0.01 Ts s 0.002 βe N/mm2 1.379×103
Ti s 0.04 Kp V/mm 0.13 F N 1.47×107
The first group simulates the fault of servo gain. Ksv is decreased by 30 % on the base of normal value. The residuals generated by the three UIOs are shown in Fig. 4. It can be seen that the residual of fault g1 which is observed by UIO1 fluctuates around zero, but the residuals of UIO2 and UIO3 are far away from zero, which means UIO1 detected and located the fault. By comparing the residuals of the three UIO, it can be determined that the servo gain is in fault.
0 0.5 1 1.5 2 2.5 3 3.5 4-8-6-4-2024
x 10-12
Th
e R
esi
du
al
of
UIO
1 g
1
0 0.5 1 1.5 2 2.5 3 3.5 4
-10
-5
0
x 1011
Th
e R
esi
du
al
of
UIO
2 g
1
0 0.5 1 1.5 2 2.5 3 3.5 4
-10
-5
0x 1011
Time/s
Th
e R
esi
du
al
of
UIO
3 g
1
Fig. 4. UIO output residuals with servo gain failure.
The second group simulates the leakage fault of hydraulic system. Ki is decreased by 30 % on the base of normal value. The corresponding residuals generated by the three UIOs are shown in Fig. 5. It can be seen that the residuals of fault g2 observed by UIO2 fluctuate around zero, but the residuals of UIO1 and UIO3 are far from zero, which means UIO2 detected and located the fault. By comparing the residuals of the three UIOs, leakage fault was found.
The third group simulates the roll system fault. Ke is decreased by 30 % on the base of normal value. The residuals generated by the three UIOs are shown in Fig. 6. It can be seen that the residuals of fault g3
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observed by UIO3 fluctuate around zero, while the residuals of UIO1 and UIO2 are far from zero, which means UIO3 detected the fault. By comparing the residuals of the three UIOs, the fault in roll system can be diagnosed.
It can be seen from the above three groups of fault simulation that the unknown input observers have played an important role in rapid isolation and localization to the system faults.
0 0.5 1 1.5 2 2.5 3 3.5 40
50
100
The
Res
idua
lof
UIO
1 g2
0 0.5 1 1.5 2 2.5 3 3.5 4-2
0
2
4x 10-4
The
Res
idua
lof
UIO
2 g2
0 0.5 1 1.5 2 2.5 3 3.5 4
0
50
100
Time/s
The
Res
idua
lof
UIO
3 g2
Fig. 5. UIO output residuals with hydraulic system leakage.
0 0.5 1 1.5 2 2.5 3 3.5 4-10
-5
0
x 10-3
The
Res
idua
l o
f UIO
1 g3
0 0.5 1 1.5 2 2.5 3 3.5 4
0
10
20x 1010
The
Res
idua
l
of U
IO2
g3
0 0.5 1 1.5 2 2.5 3 3.5 4-4
-2
0
2
4
6x 10-4
Time/s
The
Res
idua
l of
UIO
3 g3
Fig. 6. UIO output residuals roll system fault. 5. Conclusions
Based on the model of hydraulic AGC system of magnesium sheet mill, UIO groups can be used to construct an effective scheme of fault diagnosis. With UIOs, faults occur in the magnesium sheet mill can be detected and isolated quickly.
Acknowledgements
This work was supported by the National Key Technology Research and Development Program of China under Grant 2012BAF09B01. References [1]. S. Khosravi, A. Afshar, F. Barazandeh, Design of a
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