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Quantitative Approaches for Fault Diagnosis Based on Bilinear Systems
7f-052
Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco. USA
QUANTITATIVE APPROACHES FOR FAULT DIAGNOSIS BASED ON BILINEAR SYSTEMS Derek N. Shields Maths Division,
Coventry University, Coventry, CVl 5FB, U.K.
Abstract: fault detection and isolation (FDI) of bilinear systems is considered. Two main quantitative methods are outlined. A bilinear fault detection observer is proposed which focuses on the problem of decoupling the unknown inputs from the residuals. Secondly, a parity space method is developed for bilinear systems. Robust issues are addressed throughout the paper. Related methods are briefly discussed and two applications are considered. Keywords: fault detection, bilinear systems, robustness
L INTRODUCTION Fault detection and isolation (FDI) based on analytic redundancy has recently received much attention in both research and applications due to an increasing demand for reliability in engineering systems (Frank, 1990; Gertler, 1991; Isermann, 1984). Many of these methods, are based on linear models. Because most practical systems display a degree of nonlinearity these methods generally only work well in small regions of the operating space. Although some authors have proposed nonlinear observer methods ( Seliger and Frank, 1991a,1991b) these methods are only effective for certain classes of systems. Recently, robust observer schemes have been developed whereby the linearisation errors are represented by an unknown input vector (Ge and Fang,1988; Frank and Wunnenberg,1989; Patton,1989). However, for highly non linear systems and a large operating region the unknown input distribution matrix is difficult to construct and approximations are necessary.
of processes and systems including the representations of nuclear reactor systems, suspension systems, fermentation processes, hydraulic drive systems, gas-burning furnace systems and heat exchange processes (Mohler, 1973). Although the control of bilinear systems is well developed ( Derese and Noldus, 1980; Burnham et al., 1987) , research progress for FDI ofbilinear systems has been limited. This paper proposes two main methods for FDI of bilinear systems.
2.SYSTEM SPECIFICATION
A discrete-time bilinear model is considered which is bilinear in terms of the state and control vectors but is otherwise non linear in terms of the control and output vectors. The model is more general than that used by Hac (1992) or Yu et al. U994).
Bilinear models form a more useable class of nonlinear models. They are used to represent a wide variety
6470
h
+
L
Aiuk(i)xk
+ Edk + Gfa(k)
(1)
A minimal-order state-function observer for the bilinear system (1)-(2) is considered of the form
i=1
(2)
Yk
Zk+1
=
A Zk
+ B °Yk + T J(un , Yn)
+
HUk
+ LBiuk(i)Yk
~o
~
h
where Xk E lRn , Uk E lRm and Yk E lRP are the state, input and output vectors, respectively. Here h < m is the number of the bilinear terms and the values uk(i), (i = 1"", h) are the components of Uk. Also, dk E 3?', fa(k) E lRq and f.(k) E lR s represent the unknown inputs, the component and actuator faults and sensor faults, respectively. AO, Ai, B, C, E, G, Q are constant matrices with compatible dimensions. It is assumed that rank{C} = p, rank{E} = I, rank{G} = q and rank{Q} = 8. Also, J(u n , Yn) , is any nonlinear function. Of relevance here are the concepts of observability and control ability for bilinear systems (Isidori, 1973). In this paper only systems without uncontrollable states are considered.
L1Zk + L 2 Yk
fk
(6)
where Zk E lR d is an estimation of a linear combination of state, Zk = TXk. The vector fk E lR
3. BILINEAR OBSERVER-BASED FAULT DETECTION METHODS
TE LIT+ L 2C A O is
The observer-based methods are of two kinds: (i) those based on full-order or minimal-order Luenberger-type observers where the main idea is state estimation and a function of the observation error is used as a residual for FDI; (ii) those developed from the unknown input observer for linear systems which is a more general observer in that what is observed is a linear function of the state which is decoupled from the unknown inputs, depending on how the latter is defined. The second kind is considered first.
0, (i=l,···,h) BOC
(7) (8)
TB
(9)
0
(10)
0 stable
(11) (12)
then the state-function estimation error ek, ek TXk, satisfies ~o
h
+
(BO
+L
uk(i)Bi)Qf.(k)
Definition of BFDO. The definition of the BFDO IS gIven as Definition 1: An observer is called a BFDO if for any Uk and dk , the output of the observer, fk, satisfies
(ii)
(13)
i=1
Furthermore, the residual vector then satisfies
(14)
A minimal-order BFDO method is proposed and is an extension of the unknown input observer method for linear systems of Frank and Wunnenberg (1989).
fk-+Oask-+oo, for fa(k) = 0 and f.(k) and for any Xo, Zo
= Zk -
ek+1 = A ek - TGfa(k)
3.1 Minimal-Order BDFO
(i)
(5)
i=1
where F* is defined as B*u·k
o
and where
Finally, if
= 0 Tt k,
rank{F*}
(3)
f:. 0, when fa(k) f:. 0 or f.(k) f:. 0 for k ~ ko; fa(k) = f.(k) = 0 for k < k o; Zko = TXkQ. (4)
q + 28
(15)
fk+l
then the observer (6)-(7) is a BFDO for the bilinear system (1)-(2) according to definition l. End of result.
6471
6472
6473
6474
Frank, P.M. Wunnenberg, J. (1989)., Robust fault diagnosis using unknown input schemes, Chapter 3, in book Fault diagnosis in dynamic systems. Patton et al, Prentice Hall. Frank, P.M.(1990)., Fault diagnosis in dynamic systems using analytical and knowledge-based redundancyA survey and some new results, Automatica. 26, No.3 , pp 459-474.
putation, July 18-22, Paris, Vol 2 , pp 431-434. Seliger, R. and Frank, P.M. (1991a). Robust component fault detection and isolation in nonlinear dynamic systems using nonlinear unknown input observers , Preprints of IFAC Symposium SAFEPROCESS'91, Sept. 10-13, Baden-Baden, FRG, Vol. 1 , pp 313-318.
Funahashi, Y. (1979). Stable state estimator for bilinear systems, Int. 1. Control, 29, No.2 , pp 181-188.
Seliger, R. and Frank, P.M. (1991b). Fault diagnosis by disturbance decoupled nonlinear observers. Proc. of 30th IEEE CDC, Dec 11-13, Brighton, U.K., Vol. 3 , pp 2248-2253.
Gertler, J.J.(1991). Analytical redundancy methods in failure detection and isolation, Preprints of IFAC Symposium SAFEPROCESS'91, Sept 10-13, Baden-Baden, FRG, VoU , pp 9-21.
Yu, D.L., Shields, D.N., Mahtani, J.L. (1994a). A nonlinear fault detection method for a hydraulic system. Proc. 4th lEE Inter. Con! CONTROL.94, Mar 21-24, Univ. of Warwick, U.K., Vol. 2 , pp 1318-1322.
Hac, A. (1992). Design of disturbance decoupled observer for bilinear systems, 1. Dynamic Syst. Measure. Control, 114 , No.12 , pp 556-562.
Yu, D.L., Shields, D.N. (1994). A fault detection method for a nonlinear system and its application on a hydraulic test rig. Preprints of IFAC Symposium on SAFEPROCESS'94, June 13-16, Espoo, Finland, Vol.2 , pp 305-310.
Hara, S. and Furuta, K. (1986). Minimal order state observers for bilinear systems, Int. 1. Control, 24, No.5 , pp 705-718. Isermman, R. (1984). Process fault detection based on modelling and estimation methods: a survey, Automatica, 20 , No.4 , pp 387-404. Isidori, A. (1973). IEEE Trans. Automatic Control, 2 , pp 626-634. Kinnaert M., Peng Y., Hammouri H. (1995). Fault Diagnosis In Bilinear Systems- A Survey, Proc. of European Control Conference, ECC'95, Sept. 5-8, pp 3773782, Rome, Italy. Mechmeche, C, Nowakowski, S., Darouach, M. (1994). A failure detection procedure for bilinear systems based on a new formulation of unknown inputs bilinear observers ,Preprints of IFAC Symposium on SAFEPROCESS '94, June 13-16, Espoo, Finland, Vol. 2 , pp 305310. Mohler, R.R. (1973). Bilinear control processes , ( Mathematics in Science and Engineering), 2 , Academic Press.
Yu, D.L., Shields, D.N. Mahtani, J.L. (1994b). Fault detection for bilinear systems with application to a hydraulic system. Proc. 3nd IEEE Conf. on Control Applications, CCA '94, Aug. 24-26, Glasgow, U.K., Vol. 2 , pp 1379-1384. Yu D.L., Shields D.N. (1995a). A hybrid fault diagnosis approach using neural networks. Accepted for publication. 1nl. of Neural Computin.g and Applications. Yu, D.L., Shields, D.N. and Disdell, K.J., "Fault diagnosis for a gas-fired furnace using observer method", Proc. of American Control ence ACC'95, Vol. 2, pp1127-1131, June 21-23, USA.
(1995). bilinear ConferSeattle,
Yu, D.L., Shields, D.N., (1995b). "A modified parity space method for fault diagnosis for bilinear systems" , Proc. of American Control Conference ACC'95, Vol. 2 , pp1132-1133, June 21-23, Seattie, USA. Yu, D.L., Shields, D.N., (1995c). Fault Diagnosis In Bilinear Systems- A Survey, Proc. of European Control Conference, ECC'95, Sept. 5-8, pp 360-366, Rome, Italy.
Patton, R.J. (1989). Robust fault detection using eigenstructure assignment, Proc. 12th IMACS World Congress Mathematical modelling and Scientific Com-
6475
Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco. USA
QUANTITATIVE APPROACHES FOR FAULT DIAGNOSIS BASED ON BILINEAR SYSTEMS Derek N. Shields Maths Division,
Coventry University, Coventry, CVl 5FB, U.K.
Abstract: fault detection and isolation (FDI) of bilinear systems is considered. Two main quantitative methods are outlined. A bilinear fault detection observer is proposed which focuses on the problem of decoupling the unknown inputs from the residuals. Secondly, a parity space method is developed for bilinear systems. Robust issues are addressed throughout the paper. Related methods are briefly discussed and two applications are considered. Keywords: fault detection, bilinear systems, robustness
L INTRODUCTION Fault detection and isolation (FDI) based on analytic redundancy has recently received much attention in both research and applications due to an increasing demand for reliability in engineering systems (Frank, 1990; Gertler, 1991; Isermann, 1984). Many of these methods, are based on linear models. Because most practical systems display a degree of nonlinearity these methods generally only work well in small regions of the operating space. Although some authors have proposed nonlinear observer methods ( Seliger and Frank, 1991a,1991b) these methods are only effective for certain classes of systems. Recently, robust observer schemes have been developed whereby the linearisation errors are represented by an unknown input vector (Ge and Fang,1988; Frank and Wunnenberg,1989; Patton,1989). However, for highly non linear systems and a large operating region the unknown input distribution matrix is difficult to construct and approximations are necessary.
of processes and systems including the representations of nuclear reactor systems, suspension systems, fermentation processes, hydraulic drive systems, gas-burning furnace systems and heat exchange processes (Mohler, 1973). Although the control of bilinear systems is well developed ( Derese and Noldus, 1980; Burnham et al., 1987) , research progress for FDI ofbilinear systems has been limited. This paper proposes two main methods for FDI of bilinear systems.
2.SYSTEM SPECIFICATION
A discrete-time bilinear model is considered which is bilinear in terms of the state and control vectors but is otherwise non linear in terms of the control and output vectors. The model is more general than that used by Hac (1992) or Yu et al. U994).
Bilinear models form a more useable class of nonlinear models. They are used to represent a wide variety
6470
h
+
L
Aiuk(i)xk
+ Edk + Gfa(k)
(1)
A minimal-order state-function observer for the bilinear system (1)-(2) is considered of the form
i=1
(2)
Yk
Zk+1
=
A Zk
+ B °Yk + T J(un , Yn)
+
HUk
+ LBiuk(i)Yk
~o
~
h
where Xk E lRn , Uk E lRm and Yk E lRP are the state, input and output vectors, respectively. Here h < m is the number of the bilinear terms and the values uk(i), (i = 1"", h) are the components of Uk. Also, dk E 3?', fa(k) E lRq and f.(k) E lR s represent the unknown inputs, the component and actuator faults and sensor faults, respectively. AO, Ai, B, C, E, G, Q are constant matrices with compatible dimensions. It is assumed that rank{C} = p, rank{E} = I, rank{G} = q and rank{Q} = 8. Also, J(u n , Yn) , is any nonlinear function. Of relevance here are the concepts of observability and control ability for bilinear systems (Isidori, 1973). In this paper only systems without uncontrollable states are considered.
L1Zk + L 2 Yk
fk
(6)
where Zk E lR d is an estimation of a linear combination of state, Zk = TXk. The vector fk E lR
3. BILINEAR OBSERVER-BASED FAULT DETECTION METHODS
TE LIT+ L 2C A O is
The observer-based methods are of two kinds: (i) those based on full-order or minimal-order Luenberger-type observers where the main idea is state estimation and a function of the observation error is used as a residual for FDI; (ii) those developed from the unknown input observer for linear systems which is a more general observer in that what is observed is a linear function of the state which is decoupled from the unknown inputs, depending on how the latter is defined. The second kind is considered first.
0, (i=l,···,h) BOC
(7) (8)
TB
(9)
0
(10)
0 stable
(11) (12)
then the state-function estimation error ek, ek TXk, satisfies ~o
h
+
(BO
+L
uk(i)Bi)Qf.(k)
Definition of BFDO. The definition of the BFDO IS gIven as Definition 1: An observer is called a BFDO if for any Uk and dk , the output of the observer, fk, satisfies
(ii)
(13)
i=1
Furthermore, the residual vector then satisfies
(14)
A minimal-order BFDO method is proposed and is an extension of the unknown input observer method for linear systems of Frank and Wunnenberg (1989).
fk-+Oask-+oo, for fa(k) = 0 and f.(k) and for any Xo, Zo
= Zk -
ek+1 = A ek - TGfa(k)
3.1 Minimal-Order BDFO
(i)
(5)
i=1
where F* is defined as B*u·k
o
and where
Finally, if
= 0 Tt k,
rank{F*}
(3)
f:. 0, when fa(k) f:. 0 or f.(k) f:. 0 for k ~ ko; fa(k) = f.(k) = 0 for k < k o; Zko = TXkQ. (4)
q + 28
(15)
fk+l
then the observer (6)-(7) is a BFDO for the bilinear system (1)-(2) according to definition l. End of result.
6471
6472
6473
6474
Frank, P.M. Wunnenberg, J. (1989)., Robust fault diagnosis using unknown input schemes, Chapter 3, in book Fault diagnosis in dynamic systems. Patton et al, Prentice Hall. Frank, P.M.(1990)., Fault diagnosis in dynamic systems using analytical and knowledge-based redundancyA survey and some new results, Automatica. 26, No.3 , pp 459-474.
putation, July 18-22, Paris, Vol 2 , pp 431-434. Seliger, R. and Frank, P.M. (1991a). Robust component fault detection and isolation in nonlinear dynamic systems using nonlinear unknown input observers , Preprints of IFAC Symposium SAFEPROCESS'91, Sept. 10-13, Baden-Baden, FRG, Vol. 1 , pp 313-318.
Funahashi, Y. (1979). Stable state estimator for bilinear systems, Int. 1. Control, 29, No.2 , pp 181-188.
Seliger, R. and Frank, P.M. (1991b). Fault diagnosis by disturbance decoupled nonlinear observers. Proc. of 30th IEEE CDC, Dec 11-13, Brighton, U.K., Vol. 3 , pp 2248-2253.
Gertler, J.J.(1991). Analytical redundancy methods in failure detection and isolation, Preprints of IFAC Symposium SAFEPROCESS'91, Sept 10-13, Baden-Baden, FRG, VoU , pp 9-21.
Yu, D.L., Shields, D.N., Mahtani, J.L. (1994a). A nonlinear fault detection method for a hydraulic system. Proc. 4th lEE Inter. Con! CONTROL.94, Mar 21-24, Univ. of Warwick, U.K., Vol. 2 , pp 1318-1322.
Hac, A. (1992). Design of disturbance decoupled observer for bilinear systems, 1. Dynamic Syst. Measure. Control, 114 , No.12 , pp 556-562.
Yu, D.L., Shields, D.N. (1994). A fault detection method for a nonlinear system and its application on a hydraulic test rig. Preprints of IFAC Symposium on SAFEPROCESS'94, June 13-16, Espoo, Finland, Vol.2 , pp 305-310.
Hara, S. and Furuta, K. (1986). Minimal order state observers for bilinear systems, Int. 1. Control, 24, No.5 , pp 705-718. Isermman, R. (1984). Process fault detection based on modelling and estimation methods: a survey, Automatica, 20 , No.4 , pp 387-404. Isidori, A. (1973). IEEE Trans. Automatic Control, 2 , pp 626-634. Kinnaert M., Peng Y., Hammouri H. (1995). Fault Diagnosis In Bilinear Systems- A Survey, Proc. of European Control Conference, ECC'95, Sept. 5-8, pp 3773782, Rome, Italy. Mechmeche, C, Nowakowski, S., Darouach, M. (1994). A failure detection procedure for bilinear systems based on a new formulation of unknown inputs bilinear observers ,Preprints of IFAC Symposium on SAFEPROCESS '94, June 13-16, Espoo, Finland, Vol. 2 , pp 305310. Mohler, R.R. (1973). Bilinear control processes , ( Mathematics in Science and Engineering), 2 , Academic Press.
Yu, D.L., Shields, D.N. Mahtani, J.L. (1994b). Fault detection for bilinear systems with application to a hydraulic system. Proc. 3nd IEEE Conf. on Control Applications, CCA '94, Aug. 24-26, Glasgow, U.K., Vol. 2 , pp 1379-1384. Yu D.L., Shields D.N. (1995a). A hybrid fault diagnosis approach using neural networks. Accepted for publication. 1nl. of Neural Computin.g and Applications. Yu, D.L., Shields, D.N. and Disdell, K.J., "Fault diagnosis for a gas-fired furnace using observer method", Proc. of American Control ence ACC'95, Vol. 2, pp1127-1131, June 21-23, USA.
(1995). bilinear ConferSeattle,
Yu, D.L., Shields, D.N., (1995b). "A modified parity space method for fault diagnosis for bilinear systems" , Proc. of American Control Conference ACC'95, Vol. 2 , pp1132-1133, June 21-23, Seattie, USA. Yu, D.L., Shields, D.N., (1995c). Fault Diagnosis In Bilinear Systems- A Survey, Proc. of European Control Conference, ECC'95, Sept. 5-8, pp 360-366, Rome, Italy.
Patton, R.J. (1989). Robust fault detection using eigenstructure assignment, Proc. 12th IMACS World Congress Mathematical modelling and Scientific Com-
6475