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A Nonlinear Observer Approach: Experience with a Hydraulic System
Copyright 0 IFAC Fault Detection, Supervision and Safety for Technical Processes, Kingston Upon Hull, UK, 1997
A NONLINEAR OBSERVER APPROACH: EXPERIENCE WITH A HYDRAULIC SYSTEM D.N.Shields* ,S. Ashton* ,S. Daley
t
t
GEC Alsthom, Engineering Research Centre, Whetstone, Leics, UK, LE82LP. Coventry University, Priory St, Coventry, CV15FB, U.K.
* Maths-MIS,
Abstract. A fault detection method is proposed for a fairly wide class of nonlinear systems based on a nonlinear observer. Other methods proposed in the field are related to this method or are special cases of it. The observer is designed such that the error dynamics are independent of state, input, output and unknown disturbances . Conditions are given for the observer to exist and to be robust with respect to an unknown input . Applications are made to a hydraulic system. Copyright 01998 IFAC Keywords: Nonlinear, Observer, Fault Detection, Application.
use of the Kronecker canonical form or by use of the eigenvalue assignment method and these observers are now seen as very effective in applications. These observers have also been developed for nonlinear systems ( Frank ,1994; Wunnenberg and Frank ,1990). In general, however, the existence conditions for such FDI observers are complicated and difficult to satisfy involving the solution of partial differential equations ( Frank,1994 ). This is a useful on-going area of research since, given a class of nonlinear models, a compromise must inevitably be made between model complexity and numerical tractability of the observer design. The choice of certain useful classes will ensure better designs , or in fact that designs actually exists and are useful for on-line FDI.
1.1NTRODUCTION Fault detection and isolation (FDI) is now an important part of an intelligent strategy for control and system monitoring. If a fault detection scheme can be developed that gives an early warning of component failures then the system can be maintained and kept reliable by means of this early warning, enabling repair or replacement to take place at the earliest or most convenient time with the minimum loss of time to productivity. The so called model-based fault diagnosis approach has received much attention. It is based on the concept of analytic redundancy, as opposed to physical redundancy, and uses signals generated by a mathematical model of the system being considered with the advantage that no additional hardware components are needed in order to realise a FDI algorithm. This paper deals with this approach and a quantitative, as opposed to a qualitative, observer-based method is considered for the design of a robust residual vector for FDI. Several problems have been solved for this approach for the linear case. The works of Frank ( 1990,1994), Gertler (1991) , Patton et. al. (1989) , for example, outline and solve some of these problems involved . Fault detection observers have been designed ( Patton, 1989 ) for unknown inputs for linear systems which can be designed by
Bilinear fault detection observers ( BDFO 's) have been developed for the class of bilinear systems with ' unknown inputs. These are more numerically tractable and also applicable to many industrial systems where linear models are inaccurate for the case of certain operating conditions ( Yu and Shields, 1996 ). These have been applied and tested on some hydraulic systems and a gas furnace systems ( Yu et al. , 1996a, 1996b). Some of these BFDO's only exist for a subclasss of bilinear models. The work on bilinear methods in Yu and Shields 489
( 1996) is extended in this paper applied to a real industrial system. A nonlinear fault detection observer ( FDO ) is considered for a system model which includes quadratic as well as bilinear nonlinear terms. The emphasis of the paper is on applications. Section two describes the theory behind the FDO . The main result on residual convergence, Result 1, is derived and conditions for detecability are given . Section three describes the application of the FDO to a hydraulic system.
The fault detection signal, or residual, is defined as
(4) where fk E ~r (r ~ d) . The observer error ek is defined as
(5) The following definition is appropriate to the FDO : Definition: The dynamic system (3)-(4) is a FDO for system (1)-(2) if for any Uk and dk
2. A FAULT DETECTION OBSERVER Consider a discrete state-space model of the form m
+ Edk + a(xk)
+
(i)
L uk(j)Atzxk j=1
limk-+oo fk = 0, for fa(k) = and fs(k) = 0, { for all k, any Xo, Zo
°
p
+ LYk(j)A~zXk+ Kfak
The FDO is strict if it also satisfies (ii) ;
(1)
j=1
(2)
(ii)all ko ~
where Uk(j) refers to the jth component of the input Uk and a(xk) = I:; aN(xk) , where aN(xk) is an n vector with homogeneous polynomials of degree N as its components ( see (8)) . Described in (1) are the state Xk E ~n , the input Uk E ~m, the output Yk E ~P , the disturbance, including both noise and modelling uncertainties, d k E ~l and the fault vector fak E ~q . Here A, B, C , E, K " A~z' (i = L .n) , A~z , (i = L .p) , and Atz, (j = L .m) are constant matrices of appropriate dimensions. Note that the model is more general than that considered elsewhere ( Yu and Shields, 1969, Yu et aI, 1996a,1996b, Daley,1987) . Sensor faults can be added to the right-hand side of (2) so that the form, Yk = CXk + Qf&k , results. The term Qfsk , where Q is a compatible matrix and fsk represents a vector of sensor faults , can be carried through in the folloewing analysis but is omitted here for brevity. Consider an observer of the form
°
° °
if fa(k ) = and fs(k) = 0, and Zko = TXko , for ~ k < ko , then (k+1 t 0 , for fa(k ) t or fs(k) to,k ~ ko
1
°
End of definit ion In this definition , condition (i) guarantees that the observer is asymptotically stable and the residual is de coupled from the system input and unknown input . Therefore, any non-zero steady-state response of the FDO will be caused by the faults . Condition (ii) implies that given any time, ko , and given that the state function estimation error, ek, and faults are zero before this time, a non-zero fault must result in a non-zero residual. A FDO may detect , robustly, some faults but cannot guarantee that all faults are detectable. A strict FDO is robust and can guarantee that all faults are detectable. From (1),(2) ,(3) and (5) m
+ Th(Yk , Uk)
+
L uk(j)HtzYk j=1
Zk+1
=
FZk
+
JUk
+
GYk
+
p
H(Yk)
+ LYk(j)HtzYk
p
j=1
+ LYk(j)HtzYk
+ H(Yk) - TAxk - TBuk
j=1 m
+ Th(Uk , Yk)
+
L uk(j)HtzYk
m
(3)
- TLuk(j)Atzxk - Th(uk,Yk)
j=1
j=1 p
where Zk is a linear estimate of TXk and H(Yk) is a function of Yk , Zk E ~d , and d is the order of the observer. The matrices F , J, G, T , Hl z , j = L.m, Htz , j = L .p and the function H(Yk) need to be chosen in the design to achieve a tractable observer error and a decoupling of the disturbance from the residual.
- TLYk(j)AizXk j=1
- Ta(xk) - TEdk - TKfak
490
(6)
m
GCXk
+
L: uk(j)HL,pXk j=l
2.1 Numerical Algorithm
P
The following numerical algorithm is given for solving the equations or conditions (11)-(18). The spirit of the approach is to obtain an observer error which is linear and use is made of the special form for C. Equation (1) is partitioned as
+ L:Yk(j)HtzCXk - TEdk j=l + H(Yk) - TAxk - TBuk m
- Ta(xk) - T
L: uk(j)A~zXk j=l
=
Xk+l
p
m
(7)
-TL: Yk(j)AtzXk - TKfak j=l
L:Uk(j) j=l
[A~za A~Zb]
n
n
+
a(xk) Kfak
+ L:Yk(j) [Atza Atzb] Xk i=l
(8) where A( i 1, .. , in) E !Rn xn and 2 ::; N ::; s. Let H(Yk) = L:~=2 HN(Yk) where
HN(Yk) =
+
Xk
p
L: ... L: A(i1 ' .. , in )Xk(i1)...Xk(iN)
p
Bduk +
+ Edk
The term aN can be written as
aN(Xk) =
+
[Aa Ab] Xk
(22)
Em(nxp) an dA b, Ajuzb E h A a, Aiuza' Ajyza:Jl were
p
, Ajyzb lRnx(n-p) .
L: ... L: A(i1, .. ,in)Yk(id···Yk(iN)
Matrix T is partitioned as
(9)
(23)
(10)
where T1 E lRdxp and T2 E lRdx(n-p). Equations (11) , (16) and (17) are then partitioned as
Using (8) and (9) , and the fact that
{k = L1ek + (L1T + L 2C)Xk
it can be shown that the following result holds true.
Result 11fT, G, F, L 1, L 2, H~z (i = Lm) and H;z (i = Lp) can be found such that the following equations are satisfied
GC - TA Fstable J - TB TE
o
L1T + L 2C H~zC - TAtz O;j 1, .,m H{l,c - TAtz O;j 1, ., p TA(i1," , iN) i j ~ (p+ 1);j = 1, ., N;(all N)
o
FT
+
(21)
T A~za = H~z
j = Lm
TA~zb = 0
j = Lm
T Atza = Htz T A!Zb
(11)
=
j
= Lp
=
0 j Lp TAa - FT1 = G TAb - FT2 = 0
(24) (25) (26)
(27) (28) (29)
(12)
o o
= = = =
Define Z as the matrix Z = [Zl' Z2] where
(13) (14) (15)
Zl = [A(p+l , .. p+l), A(p+2, .. ,p+l) , .. , A(n , .. , n)] Z2 = [A~zb ' ., A~zb' A!zb , ., A~zb' E]
(16)
where the first set of matrices in Zl correspond to those in (18). Equations (25) , (27) and (14) can be combined to form
(17)
o
(18)
TZ =0 where HN(Yk) is defined in (9), then the error, ek , satisfies
(30)
T can be solved efficiently from (30) as
[T1 T 2] = MU;;'
(19)
(31)
where U;;' E lR(nxr) is the singular value decomposition of Z as
and the fault detection signal satisfies (20)
Hence, {k is a fault detection residual as defined in Patton et al (1989) ifrank(TK) = rank(K) . and M is an arbitrary d x r matrix with r, the order of the left null space of Z, where
A similar result holds true for system (1)-(2) when sensor faults are present although the residual vector then depends upon the input.
r
491
= n - rank(Z)
(33)
From (31) the maximum number of independent rows of T is r. The observer order is d
=
3.APPLICATION The method here is applied to a realistic industrial system: a hydraulic positioning system. This system has been used to test linear and bilinear fault detection methods in Yu et al (1996). Here, the model of the system is much more general, as included in the model are nonlinearities of bilinear and qudratic form . Also, nondynamic terms and nonlinear terms ( in either the input output or both) can be accommodated. The observer was applied to the a hydraulic test rig consisting of a stiff shaft which was driven by a hydraulic motor and loaded with a hydraulic pump . The oil flow to the motor was controlled by an electro-hydraulic servo-valve and the pressure differential across the pump could be changed to increase or decrease the loading on the shaft (Daley,1989). A reasonably accurate mathematical model of the system is
(34)
r
M has dimensions which maximize the rank of T.
u'f't, is partitioned as (35) where NI E ~(rxp ), N2 E (29) is equivalent to
~(rx(n-p) ).
Equation
(36)
(37) From (28) it can be seen that if either of the following conditions are satisfied T Ab (In - (T2)* (T2)) = 0 MU?;'Ab(In - (M N2)*(M N2)) = 0
(38) (39)
then (28) has a solution and the observer matrix F has the following form
+
F = MU?;, (M N 2 )* W(Id - M N2(M N 2)*) = A* + WC*
r
2[3 ---;u-
--;It
I
2
Id 8(t) dt 2
where W is d x d arbitrary matrix and (.)* represents the pseudo-inverse of (.). It is obvious if {A* , C*} is detectable the observer matrix can be designed to be stable by appropriate selection of W. Also if {A* , C*} is observable the eigenvalues of F can be arbitrarily assigned ( Yu and Shields, 1996) . There remains (15) to be solved. Partitioning equation (15) as
+ Dd8(t)
2
TT- d X.(t) 1 2 dt 2
dt
+
(48)
PC- 1
+ p rTJp
= Pm(t)CrTJm
(49)
T-)dX.(t) 2 dt X.(t) = I<.v(t)
(50)
(T
1
+
+
where 8(t) is the shaft position, X.(t) is the spool valve displacement, v(t) is the voltage input to the servo-valve.P.(t) , Pm(t) and Pp(t) are supply pressure, the pressure differential across the motor and the pressure differential across the pump, respectively. Also, I
(42) (43) a non-zero Ll exists, by (42) , iff d > n - p. Given M, both Ll and L2 can then be solved from (42 ) and (43) as Ll = W 1 UJ'b L2 -LITl
+ Vt dPm(t) + I
C d8(t)
(44) (45)
=
I
Qv(t)
where Unb is the SVD of T2
(51) (52)
where Qv(t) is the oil flow rate ( a more accurate approximation results if p.(t) is con at ant and known) . The input and the state were selected as
W 1 is an arbitrary q x rn
rn
u(t) = [v(t), Pp(t)
matrix where
= d - rank(T2 )
f
x(t) = [X.(t), X.(t), Pm(t), 8
(47)
r
Various models can be examined. Consider, for example, the state-space model
?d-(n-p)
Xk+l
=
AXk
+
BUk
+
X2kA..::r X k
+ Edk + 492
Kfak
(53)
cal solution algorithm has been given. The results show that the observer produces a residual signal capable of tracking even small faults (in terms of percentage error) and is robust with respect to the unknown disturbances . The observer has been successfully applied to a hydraulic system modelled in part by a quadratic nonlineararity. A fault-sensitivity analysis can be undertaken along the lines ofYu and Shields (1996).
(54)
Yk
where
A
B
=
=
0
1
-1
(T1+T2 ) TIT2
TIT2 2fl.K s P .!
v, 0
0 0 0
[ J£..... 0
Tf
-C r ,?pI
0 0 A.,.,
= [ -(3K8!
0
-2fl.KI
0
Crl'/'"
c = [:
o o o
0 0 0 0 0 0 0 0
0 1 0
v, I
~l' V,P,
-D I
REFERENCES
l
V,P,
0
0 0
S. Daley. (1987) Application of a Fast Self-tuning Algorithm to a Hydraulic Test Rig, Proc. Instn. M echo Engrs., 201 , No.C4, 285-294. Gertler, J.( 1991) Analytical Redundancy Methods in Fault Detection and Isolation, Proc. IF AC/ I M AC S Symposium SAF EP ROCESS'91, Baden-Baden, Sept. 10-13. Frank, P.M.(1990) Fault Diagnosis in Dynamic Systems Using Analytical and Knowledge-based RedundancyA Survey and Some New Results. Automatica,26, 3,459-474. Patton, R. , Frank, P. and Clark, R. (1989)Fault Diagnosis in Dynamic Systems, Theory and Applications. Prentice Hall Frank, P.M. (1994) On-line fault detection in uncertain nonlinearsystems using diagnostic observers : a survey. Int .J.Systems Sci, 25 , 12, 2129-2154. Wunnenberg, J . and Frank, P.M. (1990) Dynamic model based incipient fault detection concepts for robots. Proc of 11th IF AC World Congress 13-17 Aug, Tallimm, pp76-81. Yu D, Shields D. N., Distell K. (1996a) Fault diagnosis for a gas-fired furnace using a bilinear observer method. Control Engineering Practice ( IFAC ) , 4 , 12, 1681-1691. Yu D, Shields D.N ., Daley S. (1996b) A bilinear fault detection observer and its application to a hydraulic system, Int. Jnl. of Control, 64 ,6 , 1023-1047. Yu D., Shields D. N. (1996. ) A bilinear fault detection observer ,Automatica ( IFAG ) , 32 , 11, 1597-1602.
n
n
In practice E was approximated by the method used in Patton et al (1989) using real data. Firstly, several unknown model parameters were identified using real data. Then simulated faults were superimposed on the model indicative of realistic fault situations. With real input and output data the following plots were obtained : 1) The estimation error for the fault-free case, figure 1. 2) The fault detection signal for the fault-free case, figure 2. 3) The fault detection signal for a one percent error in Xs, figure 3. 4) The fault detection signal for a one percent error in Pm, figure 4. 5) The .fault detection signal for a one percent error in () , figure 5. 6) The fault detection signal for a ten percent change in TJ ,figure 6. These faults were simulated to occur after a thousand time steps and arranged to last for one hundred time steps before being made to return to their original values. The fault detection signal responded to each of the four faults. The problem of isolation was more difficult as the first and third faults had the same fault direction (fig. 3 and fig . 5). The second and fourth faults had the same fault direction but in the opposite direction to the first and third fault(fig . 4 and fig. 6) . 4. CONCLUSIONS The FDO proposed here is applicable to a wide range of nonlinear systems. An efficient numeri493
xl 0-11 estimation error 0.5 ,------,---r-----,---,----,,----,
Pm+1%
10,--~--,----~--,----~
o
Sf-
-
01----' 1 _ - - - - - - - -
-0.5 -1~-~--~-~-~~-~
o
1000
2000
3000
4000
-5L--~--~-~---~--~
o
5000
Figure 1: No faults
1000
2000
3000
4000
Figure 4: Sensor fault in Pm
X10-12 fault detection signal
5,-----,---r-----,---r--~
20r-----r---~-~--~----,
o
01-----,1---------
-5
-20 f-
1000
2000
3000
4000
5000
Figure 2: No faults
Figure 5: Sensor fault in ()
Xs +1% 0.5,-----,---,___----,---r---,
o
f---~
1_--------
-0.5 -1 0
100r----r-----,--,---,---, 50
-
Ol---~I---------
-
1000
2000
3000
-
-40 ~----'---~----'---~---' o 1000 2000 3000 4000 5000
-10~-~--~-~-~~-~
o
5000
4000
-50 '------'----'-------'----'-------' o 1000 2000 3000 4000 5000
5000
Figure 3: Sensor fault in Xs
Figure 6: Pump fault (17)
494
A NONLINEAR OBSERVER APPROACH: EXPERIENCE WITH A HYDRAULIC SYSTEM D.N.Shields* ,S. Ashton* ,S. Daley
t
t
GEC Alsthom, Engineering Research Centre, Whetstone, Leics, UK, LE82LP. Coventry University, Priory St, Coventry, CV15FB, U.K.
* Maths-MIS,
Abstract. A fault detection method is proposed for a fairly wide class of nonlinear systems based on a nonlinear observer. Other methods proposed in the field are related to this method or are special cases of it. The observer is designed such that the error dynamics are independent of state, input, output and unknown disturbances . Conditions are given for the observer to exist and to be robust with respect to an unknown input . Applications are made to a hydraulic system. Copyright 01998 IFAC Keywords: Nonlinear, Observer, Fault Detection, Application.
use of the Kronecker canonical form or by use of the eigenvalue assignment method and these observers are now seen as very effective in applications. These observers have also been developed for nonlinear systems ( Frank ,1994; Wunnenberg and Frank ,1990). In general, however, the existence conditions for such FDI observers are complicated and difficult to satisfy involving the solution of partial differential equations ( Frank,1994 ). This is a useful on-going area of research since, given a class of nonlinear models, a compromise must inevitably be made between model complexity and numerical tractability of the observer design. The choice of certain useful classes will ensure better designs , or in fact that designs actually exists and are useful for on-line FDI.
1.1NTRODUCTION Fault detection and isolation (FDI) is now an important part of an intelligent strategy for control and system monitoring. If a fault detection scheme can be developed that gives an early warning of component failures then the system can be maintained and kept reliable by means of this early warning, enabling repair or replacement to take place at the earliest or most convenient time with the minimum loss of time to productivity. The so called model-based fault diagnosis approach has received much attention. It is based on the concept of analytic redundancy, as opposed to physical redundancy, and uses signals generated by a mathematical model of the system being considered with the advantage that no additional hardware components are needed in order to realise a FDI algorithm. This paper deals with this approach and a quantitative, as opposed to a qualitative, observer-based method is considered for the design of a robust residual vector for FDI. Several problems have been solved for this approach for the linear case. The works of Frank ( 1990,1994), Gertler (1991) , Patton et. al. (1989) , for example, outline and solve some of these problems involved . Fault detection observers have been designed ( Patton, 1989 ) for unknown inputs for linear systems which can be designed by
Bilinear fault detection observers ( BDFO 's) have been developed for the class of bilinear systems with ' unknown inputs. These are more numerically tractable and also applicable to many industrial systems where linear models are inaccurate for the case of certain operating conditions ( Yu and Shields, 1996 ). These have been applied and tested on some hydraulic systems and a gas furnace systems ( Yu et al. , 1996a, 1996b). Some of these BFDO's only exist for a subclasss of bilinear models. The work on bilinear methods in Yu and Shields 489
( 1996) is extended in this paper applied to a real industrial system. A nonlinear fault detection observer ( FDO ) is considered for a system model which includes quadratic as well as bilinear nonlinear terms. The emphasis of the paper is on applications. Section two describes the theory behind the FDO . The main result on residual convergence, Result 1, is derived and conditions for detecability are given . Section three describes the application of the FDO to a hydraulic system.
The fault detection signal, or residual, is defined as
(4) where fk E ~r (r ~ d) . The observer error ek is defined as
(5) The following definition is appropriate to the FDO : Definition: The dynamic system (3)-(4) is a FDO for system (1)-(2) if for any Uk and dk
2. A FAULT DETECTION OBSERVER Consider a discrete state-space model of the form m
+ Edk + a(xk)
+
(i)
L uk(j)Atzxk j=1
limk-+oo fk = 0, for fa(k) = and fs(k) = 0, { for all k, any Xo, Zo
°
p
+ LYk(j)A~zXk+ Kfak
The FDO is strict if it also satisfies (ii) ;
(1)
j=1
(2)
(ii)all ko ~
where Uk(j) refers to the jth component of the input Uk and a(xk) = I:; aN(xk) , where aN(xk) is an n vector with homogeneous polynomials of degree N as its components ( see (8)) . Described in (1) are the state Xk E ~n , the input Uk E ~m, the output Yk E ~P , the disturbance, including both noise and modelling uncertainties, d k E ~l and the fault vector fak E ~q . Here A, B, C , E, K " A~z' (i = L .n) , A~z , (i = L .p) , and Atz, (j = L .m) are constant matrices of appropriate dimensions. Note that the model is more general than that considered elsewhere ( Yu and Shields, 1969, Yu et aI, 1996a,1996b, Daley,1987) . Sensor faults can be added to the right-hand side of (2) so that the form, Yk = CXk + Qf&k , results. The term Qfsk , where Q is a compatible matrix and fsk represents a vector of sensor faults , can be carried through in the folloewing analysis but is omitted here for brevity. Consider an observer of the form
°
° °
if fa(k ) = and fs(k) = 0, and Zko = TXko , for ~ k < ko , then (k+1 t 0 , for fa(k ) t or fs(k) to,k ~ ko
1
°
End of definit ion In this definition , condition (i) guarantees that the observer is asymptotically stable and the residual is de coupled from the system input and unknown input . Therefore, any non-zero steady-state response of the FDO will be caused by the faults . Condition (ii) implies that given any time, ko , and given that the state function estimation error, ek, and faults are zero before this time, a non-zero fault must result in a non-zero residual. A FDO may detect , robustly, some faults but cannot guarantee that all faults are detectable. A strict FDO is robust and can guarantee that all faults are detectable. From (1),(2) ,(3) and (5) m
+ Th(Yk , Uk)
+
L uk(j)HtzYk j=1
Zk+1
=
FZk
+
JUk
+
GYk
+
p
H(Yk)
+ LYk(j)HtzYk
p
j=1
+ LYk(j)HtzYk
+ H(Yk) - TAxk - TBuk
j=1 m
+ Th(Uk , Yk)
+
L uk(j)HtzYk
m
(3)
- TLuk(j)Atzxk - Th(uk,Yk)
j=1
j=1 p
where Zk is a linear estimate of TXk and H(Yk) is a function of Yk , Zk E ~d , and d is the order of the observer. The matrices F , J, G, T , Hl z , j = L.m, Htz , j = L .p and the function H(Yk) need to be chosen in the design to achieve a tractable observer error and a decoupling of the disturbance from the residual.
- TLYk(j)AizXk j=1
- Ta(xk) - TEdk - TKfak
490
(6)
m
GCXk
+
L: uk(j)HL,pXk j=l
2.1 Numerical Algorithm
P
The following numerical algorithm is given for solving the equations or conditions (11)-(18). The spirit of the approach is to obtain an observer error which is linear and use is made of the special form for C. Equation (1) is partitioned as
+ L:Yk(j)HtzCXk - TEdk j=l + H(Yk) - TAxk - TBuk m
- Ta(xk) - T
L: uk(j)A~zXk j=l
=
Xk+l
p
m
(7)
-TL: Yk(j)AtzXk - TKfak j=l
L:Uk(j) j=l
[A~za A~Zb]
n
n
+
a(xk) Kfak
+ L:Yk(j) [Atza Atzb] Xk i=l
(8) where A( i 1, .. , in) E !Rn xn and 2 ::; N ::; s. Let H(Yk) = L:~=2 HN(Yk) where
HN(Yk) =
+
Xk
p
L: ... L: A(i1 ' .. , in )Xk(i1)...Xk(iN)
p
Bduk +
+ Edk
The term aN can be written as
aN(Xk) =
+
[Aa Ab] Xk
(22)
Em(nxp) an dA b, Ajuzb E h A a, Aiuza' Ajyza:Jl were
p
, Ajyzb lRnx(n-p) .
L: ... L: A(i1, .. ,in)Yk(id···Yk(iN)
Matrix T is partitioned as
(9)
(23)
(10)
where T1 E lRdxp and T2 E lRdx(n-p). Equations (11) , (16) and (17) are then partitioned as
Using (8) and (9) , and the fact that
{k = L1ek + (L1T + L 2C)Xk
it can be shown that the following result holds true.
Result 11fT, G, F, L 1, L 2, H~z (i = Lm) and H;z (i = Lp) can be found such that the following equations are satisfied
GC - TA Fstable J - TB TE
o
L1T + L 2C H~zC - TAtz O;j 1, .,m H{l,c - TAtz O;j 1, ., p TA(i1," , iN) i j ~ (p+ 1);j = 1, ., N;(all N)
o
FT
+
(21)
T A~za = H~z
j = Lm
TA~zb = 0
j = Lm
T Atza = Htz T A!Zb
(11)
=
j
= Lp
=
0 j Lp TAa - FT1 = G TAb - FT2 = 0
(24) (25) (26)
(27) (28) (29)
(12)
o o
= = = =
Define Z as the matrix Z = [Zl' Z2] where
(13) (14) (15)
Zl = [A(p+l , .. p+l), A(p+2, .. ,p+l) , .. , A(n , .. , n)] Z2 = [A~zb ' ., A~zb' A!zb , ., A~zb' E]
(16)
where the first set of matrices in Zl correspond to those in (18). Equations (25) , (27) and (14) can be combined to form
(17)
o
(18)
TZ =0 where HN(Yk) is defined in (9), then the error, ek , satisfies
(30)
T can be solved efficiently from (30) as
[T1 T 2] = MU;;'
(19)
(31)
where U;;' E lR(nxr) is the singular value decomposition of Z as
and the fault detection signal satisfies (20)
Hence, {k is a fault detection residual as defined in Patton et al (1989) ifrank(TK) = rank(K) . and M is an arbitrary d x r matrix with r, the order of the left null space of Z, where
A similar result holds true for system (1)-(2) when sensor faults are present although the residual vector then depends upon the input.
r
491
= n - rank(Z)
(33)
From (31) the maximum number of independent rows of T is r. The observer order is d
=
3.APPLICATION The method here is applied to a realistic industrial system: a hydraulic positioning system. This system has been used to test linear and bilinear fault detection methods in Yu et al (1996). Here, the model of the system is much more general, as included in the model are nonlinearities of bilinear and qudratic form . Also, nondynamic terms and nonlinear terms ( in either the input output or both) can be accommodated. The observer was applied to the a hydraulic test rig consisting of a stiff shaft which was driven by a hydraulic motor and loaded with a hydraulic pump . The oil flow to the motor was controlled by an electro-hydraulic servo-valve and the pressure differential across the pump could be changed to increase or decrease the loading on the shaft (Daley,1989). A reasonably accurate mathematical model of the system is
(34)
r
M has dimensions which maximize the rank of T.
u'f't, is partitioned as (35) where NI E ~(rxp ), N2 E (29) is equivalent to
~(rx(n-p) ).
Equation
(36)
(37) From (28) it can be seen that if either of the following conditions are satisfied T Ab (In - (T2)* (T2)) = 0 MU?;'Ab(In - (M N2)*(M N2)) = 0
(38) (39)
then (28) has a solution and the observer matrix F has the following form
+
F = MU?;, (M N 2 )* W(Id - M N2(M N 2)*) = A* + WC*
r
2[3 ---;u-
--;It
I
2
Id 8(t) dt 2
where W is d x d arbitrary matrix and (.)* represents the pseudo-inverse of (.). It is obvious if {A* , C*} is detectable the observer matrix can be designed to be stable by appropriate selection of W. Also if {A* , C*} is observable the eigenvalues of F can be arbitrarily assigned ( Yu and Shields, 1996) . There remains (15) to be solved. Partitioning equation (15) as
+ Dd8(t)
2
TT- d X.(t) 1 2 dt 2
dt
+
(48)
PC- 1
+ p rTJp
= Pm(t)CrTJm
(49)
T-)dX.(t) 2 dt X.(t) = I<.v(t)
(50)
(T
1
+
+
where 8(t) is the shaft position, X.(t) is the spool valve displacement, v(t) is the voltage input to the servo-valve.P.(t) , Pm(t) and Pp(t) are supply pressure, the pressure differential across the motor and the pressure differential across the pump, respectively. Also, I
(42) (43) a non-zero Ll exists, by (42) , iff d > n - p. Given M, both Ll and L2 can then be solved from (42 ) and (43) as Ll = W 1 UJ'b L2 -LITl
+ Vt dPm(t) + I
C d8(t)
(44) (45)
=
I
Qv(t)
where Unb is the SVD of T2
(51) (52)
where Qv(t) is the oil flow rate ( a more accurate approximation results if p.(t) is con at ant and known) . The input and the state were selected as
W 1 is an arbitrary q x rn
rn
u(t) = [v(t), Pp(t)
matrix where
= d - rank(T2 )
f
x(t) = [X.(t), X.(t), Pm(t), 8
(47)
r
Various models can be examined. Consider, for example, the state-space model
?d-(n-p)
Xk+l
=
AXk
+
BUk
+
X2kA..::r X k
+ Edk + 492
Kfak
(53)
cal solution algorithm has been given. The results show that the observer produces a residual signal capable of tracking even small faults (in terms of percentage error) and is robust with respect to the unknown disturbances . The observer has been successfully applied to a hydraulic system modelled in part by a quadratic nonlineararity. A fault-sensitivity analysis can be undertaken along the lines ofYu and Shields (1996).
(54)
Yk
where
A
B
=
=
0
1
-1
(T1+T2 ) TIT2
TIT2 2fl.K s P .!
v, 0
0 0 0
[ J£..... 0
Tf
-C r ,?pI
0 0 A.,.,
= [ -(3K8!
0
-2fl.KI
0
Crl'/'"
c = [:
o o o
0 0 0 0 0 0 0 0
0 1 0
v, I
~l' V,P,
-D I
REFERENCES
l
V,P,
0
0 0
S. Daley. (1987) Application of a Fast Self-tuning Algorithm to a Hydraulic Test Rig, Proc. Instn. M echo Engrs., 201 , No.C4, 285-294. Gertler, J.( 1991) Analytical Redundancy Methods in Fault Detection and Isolation, Proc. IF AC/ I M AC S Symposium SAF EP ROCESS'91, Baden-Baden, Sept. 10-13. Frank, P.M.(1990) Fault Diagnosis in Dynamic Systems Using Analytical and Knowledge-based RedundancyA Survey and Some New Results. Automatica,26, 3,459-474. Patton, R. , Frank, P. and Clark, R. (1989)Fault Diagnosis in Dynamic Systems, Theory and Applications. Prentice Hall Frank, P.M. (1994) On-line fault detection in uncertain nonlinearsystems using diagnostic observers : a survey. Int .J.Systems Sci, 25 , 12, 2129-2154. Wunnenberg, J . and Frank, P.M. (1990) Dynamic model based incipient fault detection concepts for robots. Proc of 11th IF AC World Congress 13-17 Aug, Tallimm, pp76-81. Yu D, Shields D. N., Distell K. (1996a) Fault diagnosis for a gas-fired furnace using a bilinear observer method. Control Engineering Practice ( IFAC ) , 4 , 12, 1681-1691. Yu D, Shields D.N ., Daley S. (1996b) A bilinear fault detection observer and its application to a hydraulic system, Int. Jnl. of Control, 64 ,6 , 1023-1047. Yu D., Shields D. N. (1996. ) A bilinear fault detection observer ,Automatica ( IFAG ) , 32 , 11, 1597-1602.
n
n
In practice E was approximated by the method used in Patton et al (1989) using real data. Firstly, several unknown model parameters were identified using real data. Then simulated faults were superimposed on the model indicative of realistic fault situations. With real input and output data the following plots were obtained : 1) The estimation error for the fault-free case, figure 1. 2) The fault detection signal for the fault-free case, figure 2. 3) The fault detection signal for a one percent error in Xs, figure 3. 4) The fault detection signal for a one percent error in Pm, figure 4. 5) The .fault detection signal for a one percent error in () , figure 5. 6) The fault detection signal for a ten percent change in TJ ,figure 6. These faults were simulated to occur after a thousand time steps and arranged to last for one hundred time steps before being made to return to their original values. The fault detection signal responded to each of the four faults. The problem of isolation was more difficult as the first and third faults had the same fault direction (fig. 3 and fig . 5). The second and fourth faults had the same fault direction but in the opposite direction to the first and third fault(fig . 4 and fig. 6) . 4. CONCLUSIONS The FDO proposed here is applicable to a wide range of nonlinear systems. An efficient numeri493
xl 0-11 estimation error 0.5 ,------,---r-----,---,----,,----,
Pm+1%
10,--~--,----~--,----~
o
Sf-
-
01----' 1 _ - - - - - - - -
-0.5 -1~-~--~-~-~~-~
o
1000
2000
3000
4000
-5L--~--~-~---~--~
o
5000
Figure 1: No faults
1000
2000
3000
4000
Figure 4: Sensor fault in Pm
X10-12 fault detection signal
5,-----,---r-----,---r--~
20r-----r---~-~--~----,
o
01-----,1---------
-5
-20 f-
1000
2000
3000
4000
5000
Figure 2: No faults
Figure 5: Sensor fault in ()
Xs +1% 0.5,-----,---,___----,---r---,
o
f---~
1_--------
-0.5 -1 0
100r----r-----,--,---,---, 50
-
Ol---~I---------
-
1000
2000
3000
-
-40 ~----'---~----'---~---' o 1000 2000 3000 4000 5000
-10~-~--~-~-~~-~
o
5000
4000
-50 '------'----'-------'----'-------' o 1000 2000 3000 4000 5000
5000
Figure 3: Sensor fault in Xs
Figure 6: Pump fault (17)
494