- Email: [email protected]
Inversion of LPV systems and its application to fault detection
IFAC
[:CC>
Copyright 0 IFAC Fault Detection, Supervision and Safety of Technical Processes, Washington, D.C., USA, 2003
Publications www.elsevier.comllocatelifac
INVERSION OF LPV SYSTEMS AND ITS APPLICATION TO FAULT DETECTION Z. Szaoo o• J. Bokoro.,l G. Balas O
• Department of Aerospace Engineering and Mechanics, University of Minnesota Computer and A utomation Research Institute, Budapest Hungarian Academy of Sciences Kende u. 19-17. H- llll Budapest, Hungary e-mail: [email protected]
Abstract: This paper investigates the problem of input reconstruction by means of dynamic system inversion for linear parameter varying (LPV) systems, where the system matrix depends affinely from the parameters. The paper presents a view on the properties of the inverse for LPV systyems from the point of view of the fault detection and isolation problem by using parameter varying invariant subspaces and the results of classical geometrical system theory. Copyright © 2003 IFAC Keywords: Failure detection, Input reconstruction, Inverse system, LPV systems.
1. PROBLEM FORMULATION
Throughout this paper the problem of fault detection and isolation for the class of linear parameter-varying (LPV) systems of which state matrix depends affinely on the parameter vector will be considered. This class of systems can be described as:
There are various approaches to residual generation, see e.g. the detection filter approach initiated by Massoumnia (1986) for LTI systems and used also by Edelmayer et al. (1997),Keviczky et al. (1993) for LTV systems and by Hammouri et al. (1999) for bilinear systems, the dedicated observers and the parity space approaches Gertler (1998) , the multiple model and the generalized likelihood ratio approaches, just to mention a few. These approaches are used in a number of situations differing in the assumptions on noise, disturbances, robustness properties and in the specific design methods, see some important representations in the literature,(Basseville, 1988), (Frank and Ding, 1997),(Isermann, 1997), (Chen and Patton, 1999), (Mangoubi, 1998).
m
x(t) = A(p)x(t)
+ B(p)u.(t) + :E L j (p)Vj (t) j=1
y(t) = Cx(t),
(1)
where Vj are the failures to be detected, C is right invertible,
+ PIA) + ... + PNA N , B(p) = Bo + p)BI + .. . + PNBN , Lj(p) = Lj,o + PILi,) + ... + PNLj,N, A(p) = Ao
(2) (3) (4)
and Pi are time varying parameters. It is assumed that each parameter Pi and its derivatives Pi ranges between known extremal values Pi(t) E [-p;,P.:1 and Pi(t) E [-Pi,PiJ, respectively. Let us denote this par3Dleter set by 'P.
1 This research was ijUpported in part by NASA Langley, NASA Grant NCC-1-337 Dr. Christine M. Belcastro technical monitor and in part by the Hungarian National Science Foundation (OTKA) under Grant T 030182.
233
The fault detector should satisfy a number of requirements. It should distinguish among different failure modes Vi, e.g., between two independent faults in two particular actuators. Moreover, it is aimed to completely decouple the faults from the effect of disturbances and also from the input signals. The problem of residual generation can be viewed as an input reconstruction process that addresses the problem of designing a filler which, on the basis of input and output measurements, returns the unknown inputs (failure modes and disturbance signals) by utilizing the inverse representation of the system, see (Szigeti et al. , 2001) and (Szigeti et al., 2002) . The solution of the problem of dynamic inversion of systems gave rise to considerable attention in the control literature in the past years. Silverman e.g., considered the properties and calculation of the inverse of LTI systems in his classical paper (Silverman, 1969) guaranteeing neither minimality (or observability, detect ability) nor stability properties of the resulting inverse system. This makes the approach useless from FDI perspective. The problem was also considered by Fliess (Fliess, 1986) for nonlinear input-output systems. For certain classes of nonlinear state space representations Isidori provided algorithms and also sufficient or necessary conditions of invertibility in (Isidori, 1989). Quite recently, algorithms for the inversion of systems from the FDI point of view have been given for linear and nonlinear systems respectively in (Szigeti et al., 2001) , (Edelmayer et al., 2002) and (Szigeti et al. , 2002) upon which the contribution of the present paper is based. The paper attempts to provide a better understanding of the inversion procedure for LPV systems. In the discussions the concepts of geometrical system theory is used. We derive a procedure based on the concept of invariant subspaces and on the related coordinate transforms that result in an inverse system supposed it is given in state space form and it is left invertible. A numerical example is presented which demonstrates the theoretical results and the procedure on which the inverse calculation is based.
(Basile and Marro, 1973; Basile and Marro, 1992; Wonham, 1985):
dim 1mB = m ,
V- n 1mB = O.
(5)
Let us observe, that if these conditions are fulfilled , one can always choose a basis of the state space as {A 1mB V"},
A C V· J. ,
that induces a coordinate transform of the form
x
= T- I z , where
T- I
= [A
1mB
V·l .
Accordingly, the system will be decomposed to:
Xl = AllxI Xz
+ AIZXz + Blv
= AZlXl + Azzxz
y=
(6)
(7) (8)
CIX1'
It is clear that 1mB = 1mB l . Follows, that applying the feedback
(9)
such that V" is (A obtain the system:
Xl
+ BF, B)
invariant, one can
= AlIxI + Blv
Y=
C1XI'
(10) (11)
For the dynamical system (10) the subspace of unknown-input state unobservability by means of differentiators is exactly VI"' the maximal (All ,B)-invariant subspace contained in kerC l . Moreover, the system can be inverted for V belonging to the complementary subspace of B-1 Vt, see (Basile and Marro, 1973). By the maximality of v· follows that Vt = 0, Le ., both Xl and v can be expressed as functions of y and its derivatives. Follows, that (7) and (9) gives the inverse system equations, moreover, this realization is minimal. Denoting by
Ci
span{CI, '"
the rows of Cl let us consider , clAn ,···
,Cp ,'" ,CpAln
( 12)
where e;AiIBI = 0, for 1< 'Yi, and 'Yi are chosen such that the system to be linearly independent. By choosing a solution of the equation Al2 + BFz = 0 one can set FI = O. Follows, that choosing the basis (12) for V"J. , one has a particulary simple form of the decomposition (7) and feedback (9) with FI = 0, revealing immediately the structure of the minimal inverse system.
2. LINEAR TIME INVARIANT (LTI) SYSTEMS Let us recall first the results for LTI systems. An LTI system is strongly invertible if and only if n* = 0, where n* is the maximal controllability subspa.ce contained in kerC, see (Nijmeijer, 1991a). If V· denotes the maximal (A,B)-invariant subspace contained in kerC, then the invertibility conditions can be formulated as,
3. INVARIANT SUBSPACES These ideas can be also extended to the LPV case. To do this we have to introduce first the parameter varying counterparts of the LTI invariant subspaces.
234
Let us recall, first , some elementary definitions and facts from (Isidori, 1989) and (Nijmeijer and van der Schaft, 1991) stated for nonlinear input affine system E :
For the parameter varying case one can extend these notions, and introduce the parameter varying (A ,B)-invariant swspaces, as follows, (Bokor et al., 2002):
Definition 1. Let B(p) denote Im B(p) . Then a subspace V is called a parameter-varying (A,B)invariant subspace (or shortly (A, B)-invariant subspace) if for all pEP one has A(p)V
c V + B(p) .
(13)
As in the classical case one has the following characterization of the parameter varying (A,B)invariant subspaces:
Proposition 2. V is a parameter varying (A,B)invariant subspace if and only if for any pEP there exists a state feedback matrix F(p) such that (A(p)
+ B(p)F(p))V c
V.
Vo
= K,
(16)
= K, n nAil (Vk + B) .
A smooth connected submanifold M which contains the point Xo is said to be locally controlled invariant at Xo if their is a smooth feedback u(x) and a neighborhood Uo of Xo such that the vector field i(x) = f(x) + 9(X)U(X) is tangent to M for all x E M n Uo, i.e., M is locally invariant under
j.
Let us denote by Z· the locally maximal output zeroing submanifold. Then the invertability conditions can be stated as: dim span{ 9;(xo) I i
= 1, m} = m,
(24)
and dimspan{9i(X)
li = I , m} nT",Z· = O.
(25)
An algorithm for computing Z·, the zero dynamics algorithm, for a general case can be found in (Isidori, 1989) and (Nijmeijer and van der Schaft, 1991). However, in some cases Z· can be determined relative easily relating it to the maximal controlled invariant distribution Do· contained in kerdh.
N
Vk+l
(23)
;=1
(15)
In what follows the subscript p.v. will be dropped. This maximum which can be computed from the (A, B)-Invariant Subspace Algorithm:
ABISA :
+ L:9;(X)Ui
(14)
The set of all parameter varying (A,B)-invariant subspaces containing a given subspace C, is an upper semilattice with respect to the intersection of subspaces. This semilattice admits a maximum, denoted by V;.v.(C) := max V(A(p), B(p),C).
m
X = f(x)
It is not hard to figure out that if some technical conditions for the parameter functions (persistency) are fulfilled, then T",Z· = V·, where V· is the maximal (A, B)-invariant subspace contained in kerC.
(17)
;=0
The limit of this algorithm will be denoted by V· and its calculation needs a.t most n steps, for details see (Bokor et al., 2002) .
Conditions (24) and (25) reduce then to: dim/mB
4. INVERSION OF LPV SYSTEMS
Let us consider the class of linear parametervarying (LPV) systems of m inputs and p outputs that can be described as: x(t) = A(p(t»x(t)
yet) = Cx(t)
+ B(p(t»u(t)
V· n 1mB
= O.
Let us observe, that if these conditions are fulfilled , one can always choose a coordinate transform of the form z
(18)
= Tx , where
T- l
= [A
1mB V·
J,
A C V·J..
Accordingly, the system will be decomposed to:
(19)
where
Xl
A(p(t» = Ao + PI (t)AI B(p(t» =
= m,
+ ... + PN(t)AN, (20) Bo + Pl(t)B I + ... + PN(t)B N , (21)
= All (t)Xl + A 12 (t)X2 + Bl v
X2 =
A21 (t)Xl
+ A22(t)X2
Y = CIXl .
(22)
(26) (27)
(28)
Follows, that applying the feedback
and the dimension of the state space is supposed to be n .
(29)
235
such that V· is (A obtain the system:
+ BF, B)
invariant, one can
XI =A l1 (t) x I +BIV y
= CIXI .
(30) (31)
If starting from the rows Ci of Cl, one can choose a linearly independent system such that the dual space of X t is spanned by
where S!{t)B t
= 0, for l < I;, and
S?(t) S;+1(t)
=Ci, =Sf(t) + Sf(t)An(t),
One can observe that to compute' the matrix S(t) one needs certain derivatives of the parameter functions Pi(Y), i.e., certain derivatives of the output y, but the order of these derivatives are bounded by maxi , •.
5. AN INVERSION BASED FD FILTER Let us consider now the system (1) and let us design an inverse system for the fault signals vi' Applying the steps described in the previous section, one can obtain the equations v = F2(t)X2+W and
= All (t)XI + BI (t)u + Lw = A 22 (t)X2 + B2(t)U + A2t(t)Xt y = CIXI '
(33)
i
l
(37)
(34)
i2
(38) (39)
see (Silverman and Medows, 1967; Bestle and Zeitz, 1983), then one can define a coordinate transform S(t) that maps Xl to ii, where
Since the additional term Bdt) is present the expression (35) for fj has to be modified to
(35)
y=jj- [u(1,O), '" ,u(l ,"l.)' '' · ,U(P,"lp)(,(40)
Since one can chose Ft (p) is given by:
= 0, the inverse system
where U(i,O)(t)
i7 = A 22 (t)1/ + A 21 (t)S-I(t)ii, U = F 2(p(t»1/ + Bl"S-I(t)(y- (S(t)Al1(t)S-I(t)
where B l
=0,
U{j,k)(t) =U(j,k_l)(t) - SJ(t)BI(t)u(t) . (42)
(36)
Then the inversion based FD filter can be obtained as:
+ S(t)S-I(t»y),
" is the right inverse of B I •
i7 = A 22 (t)1/ + B2(t)U + A 21 (t)S-I(t)y, (43)
S(t), i.e., the indices ,., can change during the time. From a practical point of view this is an unconvenience since one might prefer to work with a fix set of derivatives.
+ L -"S-l(t)(y (S(t)An (t)S-t (t) + S(t)S-1 (t»y) ,
v = F 2(p(t))1/
Remark 3. In general the structure of the matrix
-
where L -" is the right inverse of L .
Let us denote by Ak,n
6. EXAMPLE
= {A."l1Ai"l1 ·· · Ai.,n I ij
E {O, 1, · · · , N}}.
In order to determine a good matrix S(t), one can compute the sets
As an illustrative example for the (q)LPV inversion scheme let us consider the following linearized parameter varying model:
{e; , ... ,ciA"l~ ,l1} '
= A(p)x(t) + Lv(t) y(t) = Cx(t),
x(t)
where CiAI ,llBI = 0 for alll < If, and one has to determine the indices ,. = mink
,t.
where A(p) = Ao + PIAl + P2A2 and for sake of simplicity only fault inputs are considered. The state matrices are:
If the set
{Cl,' " ,SJ'(t), ...
(41)
,Cp, '"
,S;p(t)}
span the dual space of Xl, then the matrix S(t) will be a good choice in order to ensure that its structure remains unchanged, i.e., one can always use the same set of outputs and derivatives.
Ao
A2
Remark 4. It is clear that the method presented above can also be applied for quasi LPV systems.
236
=
=
-o1 -1 0 0 0 00 ] 0 0 -1 0 0 • o 0 0 1 0 [ o 0 0 0 -1
00000 00000] 1 0 0 0 0 , [ 00000 01000
L _
A1
01 [10] 0 1 , 00 00
=
[0 0 - 01 0 0 0 0 1 0
C
01 0 0 0
01 0 0 0
0] 1 0 • 0 0
00001 1 0 0 0] . 00011
= [0
The parameter varying subspace V· Applying the transformation Ti;
0 o0 0 1 0 0 I 0 0] 000 I I [ o I 0 0 0
= [0 0
1 0 0] .
0 o0 0 00 011 0] (. i.e .• )
°
I
0 0 0 I 0 0 0 -I I 0 0
o
[
1 0 0 0 0
o
PI
the system splits as
-
[
~ 2pI PI PI 1
0o 0I 0 0]
[o 1
0 0 0 1 0 0
0 1
0 0 0
0
7. CONCLUSION •
This paper investigated the problem of input reconstruction by means of dynamic system inversion for linear parameter varying (LPV) systems, where the system matrix depends affinely from the parameters. The fault detection and isolation problem is solved using a detection filter obtained by inverting the system. A procedure for the construction of the inverse, based on the geometric concept of parameter varying invariant subspaces and on the related coordinate transform was given.
100
a
0 00 00 01 00 0] 0 0 0 0 0 • [ 0 0 0 0 0
A~2] ~ 2
Ap [ Aa, Aaa
o
00]
. [c, 0]
[ 0' ] = [~! L
00
= 0,
F2
= 0 and
I 000] -I 0 0 0 0 0 0 I • I lOO
=[
maps
XI
~ gg~ Ig] . 10000
= [
S,
=
°° ° .
00-100] 0
= [0
Ft
= So + PISI + P2S2,
The transformation S(p) where So
0
= Fo + PIFI + P2F2 , is given by
The matrix F(p) Fo
0 I
[000 0 0 1 0J 0 0 0 0 0 • 0000
_ [YI.YI
to y =
52=
[
8. REFERENCES
o0 00 00 0] I 0000 ' o0 0 0
]T .
Y2 Y3
One can figure out that I
0
-1
0
o
1
-~
2..
S-'(t) =
[ PI
1
PI
PI
o
0
1
and
0] 0
0
0 0]
o _p,pa +P. 0 ~ PI
p,
o o
0
0
0
0
It follows, that S(t)All(t)S-l(t)
_ -
[
2.. 2.. -~ P2 p, -- ~
-I 0 0 0] 0 0 o
-I 0 0 0 -I 0 0 0 _I
=
- -
+ p,
[
p,
1
P't
p,
P'
0
0
2..
2.. -~
p,
p,
0]
[0 0 1 0]
0 0
+ Pa
0
P'
and I
L1rS- (t)
1'2] 0 PI .
11 -
= [ -PI
o
--
PI
0
1
0
For the inverse system one has
. = -Tj + (1'2 - PI )Yl + -YI 1'2 . -
Tj
and
PI
Pt
Balas, G. and J. Bokor (2000) . Detection filter design for LPV systems, In: Preprints of SAFEPROCESS'2000, Budapest, vo!. 2, 653-656. Basseville, M. (1988) . Detecting changes in signals and systems - a survey, Automatica 24(3), 309-326. Basile, G., and Marro (1973) . A new characterization of some struct ural properties of linear systems: unknown-input observability, invertability and functional controllability, Int. J. Control vol. 17, 931-943. Becker,G., and A. Packard (1994). Robust performance of linear, parametrically varying systems using parametrically-dependent linear dynamic feedback, Sys. and Contr. Letters 23, 205-215. Bhattacharyya, S.P., (1983) . Generalized controllability (A,B)-invariant subspaces and pa. 0 rameter mvanant control. S lAM J. Alg. Dr.sc. o· Meth vol 4 529-533 ° . . , . Chen, J. and R. J. Patton (1999). Robust ModelBed'" _.• D tagnoslS ' . fior D ynamlc . S ystems, as rawt Kluwer Academic Publishers. Edelmayer, A., Bokor, J., Szigeti, F. and Keviczky, L. (1997). Robust detection filter design in the presence of time-varying system perturbations, Automatica 33(3), 471--475. Fen W., X. H. Yang, A. Packard and G. Beeker, (1996). Induced £2 norm control for LPV systems with bounded parameter variation rates, International Journal of Nonlinear and Robust Control, vo!. 6, pp. 983-998
P~
-Y2, PI
0 0 -1 0 0 0 0 0 I
237
. .
tification, IEEE TIuns. Automat. Contr. 34, 316-321. Otsuka, N. (2000) . Generalized invariant subspaces and parameter insensitive disturbancerejection problems with static output feedback, IEEE TIuns. Automat. Contr. 45, 1691-1697. Packard,A., and G. Beeker (1992). Quadratic sta..bilization of parametrically dependent linear systems using parametrically dependent linear feedback, Advances in Robust and Nonlinear Control Systems, DSC-Vol. 43, 29-36, ASME Winter Annual Meeting. Persis, C. De, and A. Isidari (2000a). On the observability codistributions of a nonlinear systems, 8ys. Contr. Letters vol. 40, 297304. Persis,C. De, and A. Isidori (2000b) . A geometric approach to nonlinear fault detection and isolation, In Preprints of 8A FEPROCE88 '2000, Budapest, vo!. 1, 20!}-214. Silverman, L.M. (1969). Inversion of Multivariable Linear Systems, IEEE TIuns. Automat. Contr. 14, 270--276. Szigeti F ., C. E. Vera, J. Bakor and A. Edelmayer (2001) . Inversion based fault detection and isolation, In Proceedings of the 40th IEEE Conf. Dec. Cont, Orlando, FL, 1005-1010. Szigeti F ., J. Bokor and A. Edelmayer (2002) . Input reconstruction by means of system inversion: application to fault detection and isolation, In Proceedings of the 15th IFA C World Congress, Barcelona. Edelmayer, A., C. E. Vera and F . Szigeti (2002). A geometric view on inversion based fdi in linear systems, In Proceedings of the 10th Mediterranean Conf. on Contr. and Autom., Lisbon. Z. Szab6, J . Bokor and G. Balas (2002) . Detection filter design for LPV systems - a geometric approach, In Procudings of the 15th IFAC World Congress, Barcelona. Bokor, J., Z. Szab6 and G. Stikkel (2002). Invariant subspaces for LPV systems and their applications, In Proceedings of the 10th
Fliess, M. (1986). A note on the invertibility of nonlinear input-output differential systems, 8ys. Contr. Letters vol. 8, 147-15l. Frank, P. M. and Ding, X. (1997). Survey of robust residual generation and evaluation methods in observer-based fault detection systems, J. of Proce8S Control 7(6),403-424. Gertler, J . (1998). Fault Detection and Diagnoses in Engineering Systems, Marcel Dekker, New York. Gahinet,P., P. Apkarian and M. Chilali, (1996). Affine parameter dependent Lyapunov functions and real parameter uncertainity, IEEE 1h1ns. Automat. Contr. 41 , 436-442. Hammauri, H., Kinnaert, M. a.nd El Yaagoubi, E. H. (1999) . Observer-based approach to fault detection and isolation for nonJinear systems, IEEE 7hms. Automat. Contr. 44(10), 187!}1884. Isermann, R. (1997). Supervision, fault-detection and fault-diagnosis methods - an introduction, Contr. Eng. Practice 5(5), 63!Hi52. Isidori, A.,(1989). Nonlinear Control Systems, Springer-Verlag. Kabore, P., H. Hammouri and M. Kinnaert (2000) . A geometric approach to fa.ult detection and isolation for bilinear systems: application to heat exchangers, In Preprints of SA FEPR 0 CESS'2000, Budapest, vo!. 1, 244- 249. Keviczky, L., Bokor, J., Szigeti, F . and Edelma.yer, A. (1993). Modelling time varying system perturbations: Application to robust change detection and identification, Preprints of the 12th IFAC World Congress, Australia, pp. 517-520 (Vo!. 7.). Mangoubi, R. (1998). Robust Estimation and
Failure Detection - A Concise 7teatment, Springer-Verlag London Ltd. Basile, G. , and Marro (1987) . On the robust controlled invariant, 8ys. Contr. Letters vol. 9, 191-195. Basile, G., and Marro (1992). Controlled and
conditioned invariants in linear system theory, Prentice Hall, 1992. Conte, G., A.M. Perdon and G. Marro (1991). Computing the maximum robust controlled invariant subspace, 8ys. Contr. Letters vol. 17, 131-135. H. Nijmeijer and A.J. van der Schaft, (1991). Nonlinear Dynamical Control Systems, Springer-Verlag. H. Nijmeijer (1991a). Invertibility of affine nonlinear control systems: a geometric approach, Sys. Contr. Letters vol. 17, 163-168. Massoumnia, M.A. (1986) . A geometric approach to the synthesys of failure detection filters, IEEE 1h1ns. Automat. Contr. 31, 839-846. Massoumnia, M.A. ,G.C. Verghese and A.S. Willsky (1989). Failure detection and iden-
Mediterranron Conf. on Contr. and Autom., Lisbon. Wonham, W . M . (1985). Linear Multivariable Control - A Geometric Approach, SpringerVerlag, New York (Third Edition). Bestle, D. and M. Zeitz (1983). Canonical form observer design for non-linear tine-variable systems, Int. J. Control38, 41!}-431. Silverman, L.M. and H.E. Medows (1967). Controllability and observability in time -variable linear systems, J. SIAM Control 5, 64-73.
238
[:CC>
Copyright 0 IFAC Fault Detection, Supervision and Safety of Technical Processes, Washington, D.C., USA, 2003
Publications www.elsevier.comllocatelifac
INVERSION OF LPV SYSTEMS AND ITS APPLICATION TO FAULT DETECTION Z. Szaoo o• J. Bokoro.,l G. Balas O
• Department of Aerospace Engineering and Mechanics, University of Minnesota Computer and A utomation Research Institute, Budapest Hungarian Academy of Sciences Kende u. 19-17. H- llll Budapest, Hungary e-mail: [email protected]
Abstract: This paper investigates the problem of input reconstruction by means of dynamic system inversion for linear parameter varying (LPV) systems, where the system matrix depends affinely from the parameters. The paper presents a view on the properties of the inverse for LPV systyems from the point of view of the fault detection and isolation problem by using parameter varying invariant subspaces and the results of classical geometrical system theory. Copyright © 2003 IFAC Keywords: Failure detection, Input reconstruction, Inverse system, LPV systems.
1. PROBLEM FORMULATION
Throughout this paper the problem of fault detection and isolation for the class of linear parameter-varying (LPV) systems of which state matrix depends affinely on the parameter vector will be considered. This class of systems can be described as:
There are various approaches to residual generation, see e.g. the detection filter approach initiated by Massoumnia (1986) for LTI systems and used also by Edelmayer et al. (1997),Keviczky et al. (1993) for LTV systems and by Hammouri et al. (1999) for bilinear systems, the dedicated observers and the parity space approaches Gertler (1998) , the multiple model and the generalized likelihood ratio approaches, just to mention a few. These approaches are used in a number of situations differing in the assumptions on noise, disturbances, robustness properties and in the specific design methods, see some important representations in the literature,(Basseville, 1988), (Frank and Ding, 1997),(Isermann, 1997), (Chen and Patton, 1999), (Mangoubi, 1998).
m
x(t) = A(p)x(t)
+ B(p)u.(t) + :E L j (p)Vj (t) j=1
y(t) = Cx(t),
(1)
where Vj are the failures to be detected, C is right invertible,
+ PIA) + ... + PNA N , B(p) = Bo + p)BI + .. . + PNBN , Lj(p) = Lj,o + PILi,) + ... + PNLj,N, A(p) = Ao
(2) (3) (4)
and Pi are time varying parameters. It is assumed that each parameter Pi and its derivatives Pi ranges between known extremal values Pi(t) E [-p;,P.:1 and Pi(t) E [-Pi,PiJ, respectively. Let us denote this par3Dleter set by 'P.
1 This research was ijUpported in part by NASA Langley, NASA Grant NCC-1-337 Dr. Christine M. Belcastro technical monitor and in part by the Hungarian National Science Foundation (OTKA) under Grant T 030182.
233
The fault detector should satisfy a number of requirements. It should distinguish among different failure modes Vi, e.g., between two independent faults in two particular actuators. Moreover, it is aimed to completely decouple the faults from the effect of disturbances and also from the input signals. The problem of residual generation can be viewed as an input reconstruction process that addresses the problem of designing a filler which, on the basis of input and output measurements, returns the unknown inputs (failure modes and disturbance signals) by utilizing the inverse representation of the system, see (Szigeti et al. , 2001) and (Szigeti et al., 2002) . The solution of the problem of dynamic inversion of systems gave rise to considerable attention in the control literature in the past years. Silverman e.g., considered the properties and calculation of the inverse of LTI systems in his classical paper (Silverman, 1969) guaranteeing neither minimality (or observability, detect ability) nor stability properties of the resulting inverse system. This makes the approach useless from FDI perspective. The problem was also considered by Fliess (Fliess, 1986) for nonlinear input-output systems. For certain classes of nonlinear state space representations Isidori provided algorithms and also sufficient or necessary conditions of invertibility in (Isidori, 1989). Quite recently, algorithms for the inversion of systems from the FDI point of view have been given for linear and nonlinear systems respectively in (Szigeti et al., 2001) , (Edelmayer et al., 2002) and (Szigeti et al. , 2002) upon which the contribution of the present paper is based. The paper attempts to provide a better understanding of the inversion procedure for LPV systems. In the discussions the concepts of geometrical system theory is used. We derive a procedure based on the concept of invariant subspaces and on the related coordinate transforms that result in an inverse system supposed it is given in state space form and it is left invertible. A numerical example is presented which demonstrates the theoretical results and the procedure on which the inverse calculation is based.
(Basile and Marro, 1973; Basile and Marro, 1992; Wonham, 1985):
dim 1mB = m ,
V- n 1mB = O.
(5)
Let us observe, that if these conditions are fulfilled , one can always choose a basis of the state space as {A 1mB V"},
A C V· J. ,
that induces a coordinate transform of the form
x
= T- I z , where
T- I
= [A
1mB
V·l .
Accordingly, the system will be decomposed to:
Xl = AllxI Xz
+ AIZXz + Blv
= AZlXl + Azzxz
y=
(6)
(7) (8)
CIX1'
It is clear that 1mB = 1mB l . Follows, that applying the feedback
(9)
such that V" is (A obtain the system:
Xl
+ BF, B)
invariant, one can
= AlIxI + Blv
Y=
C1XI'
(10) (11)
For the dynamical system (10) the subspace of unknown-input state unobservability by means of differentiators is exactly VI"' the maximal (All ,B)-invariant subspace contained in kerC l . Moreover, the system can be inverted for V belonging to the complementary subspace of B-1 Vt, see (Basile and Marro, 1973). By the maximality of v· follows that Vt = 0, Le ., both Xl and v can be expressed as functions of y and its derivatives. Follows, that (7) and (9) gives the inverse system equations, moreover, this realization is minimal. Denoting by
Ci
span{CI, '"
the rows of Cl let us consider , clAn ,···
,Cp ,'" ,CpAln
( 12)
where e;AiIBI = 0, for 1< 'Yi, and 'Yi are chosen such that the system to be linearly independent. By choosing a solution of the equation Al2 + BFz = 0 one can set FI = O. Follows, that choosing the basis (12) for V"J. , one has a particulary simple form of the decomposition (7) and feedback (9) with FI = 0, revealing immediately the structure of the minimal inverse system.
2. LINEAR TIME INVARIANT (LTI) SYSTEMS Let us recall first the results for LTI systems. An LTI system is strongly invertible if and only if n* = 0, where n* is the maximal controllability subspa.ce contained in kerC, see (Nijmeijer, 1991a). If V· denotes the maximal (A,B)-invariant subspace contained in kerC, then the invertibility conditions can be formulated as,
3. INVARIANT SUBSPACES These ideas can be also extended to the LPV case. To do this we have to introduce first the parameter varying counterparts of the LTI invariant subspaces.
234
Let us recall, first , some elementary definitions and facts from (Isidori, 1989) and (Nijmeijer and van der Schaft, 1991) stated for nonlinear input affine system E :
For the parameter varying case one can extend these notions, and introduce the parameter varying (A ,B)-invariant swspaces, as follows, (Bokor et al., 2002):
Definition 1. Let B(p) denote Im B(p) . Then a subspace V is called a parameter-varying (A,B)invariant subspace (or shortly (A, B)-invariant subspace) if for all pEP one has A(p)V
c V + B(p) .
(13)
As in the classical case one has the following characterization of the parameter varying (A,B)invariant subspaces:
Proposition 2. V is a parameter varying (A,B)invariant subspace if and only if for any pEP there exists a state feedback matrix F(p) such that (A(p)
+ B(p)F(p))V c
V.
Vo
= K,
(16)
= K, n nAil (Vk + B) .
A smooth connected submanifold M which contains the point Xo is said to be locally controlled invariant at Xo if their is a smooth feedback u(x) and a neighborhood Uo of Xo such that the vector field i(x) = f(x) + 9(X)U(X) is tangent to M for all x E M n Uo, i.e., M is locally invariant under
j.
Let us denote by Z· the locally maximal output zeroing submanifold. Then the invertability conditions can be stated as: dim span{ 9;(xo) I i
= 1, m} = m,
(24)
and dimspan{9i(X)
li = I , m} nT",Z· = O.
(25)
An algorithm for computing Z·, the zero dynamics algorithm, for a general case can be found in (Isidori, 1989) and (Nijmeijer and van der Schaft, 1991). However, in some cases Z· can be determined relative easily relating it to the maximal controlled invariant distribution Do· contained in kerdh.
N
Vk+l
(23)
;=1
(15)
In what follows the subscript p.v. will be dropped. This maximum which can be computed from the (A, B)-Invariant Subspace Algorithm:
ABISA :
+ L:9;(X)Ui
(14)
The set of all parameter varying (A,B)-invariant subspaces containing a given subspace C, is an upper semilattice with respect to the intersection of subspaces. This semilattice admits a maximum, denoted by V;.v.(C) := max V(A(p), B(p),C).
m
X = f(x)
It is not hard to figure out that if some technical conditions for the parameter functions (persistency) are fulfilled, then T",Z· = V·, where V· is the maximal (A, B)-invariant subspace contained in kerC.
(17)
;=0
The limit of this algorithm will be denoted by V· and its calculation needs a.t most n steps, for details see (Bokor et al., 2002) .
Conditions (24) and (25) reduce then to: dim/mB
4. INVERSION OF LPV SYSTEMS
Let us consider the class of linear parametervarying (LPV) systems of m inputs and p outputs that can be described as: x(t) = A(p(t»x(t)
yet) = Cx(t)
+ B(p(t»u(t)
V· n 1mB
= O.
Let us observe, that if these conditions are fulfilled , one can always choose a coordinate transform of the form z
(18)
= Tx , where
T- l
= [A
1mB V·
J,
A C V·J..
Accordingly, the system will be decomposed to:
(19)
where
Xl
A(p(t» = Ao + PI (t)AI B(p(t» =
= m,
+ ... + PN(t)AN, (20) Bo + Pl(t)B I + ... + PN(t)B N , (21)
= All (t)Xl + A 12 (t)X2 + Bl v
X2 =
A21 (t)Xl
+ A22(t)X2
Y = CIXl .
(22)
(26) (27)
(28)
Follows, that applying the feedback
and the dimension of the state space is supposed to be n .
(29)
235
such that V· is (A obtain the system:
+ BF, B)
invariant, one can
XI =A l1 (t) x I +BIV y
= CIXI .
(30) (31)
If starting from the rows Ci of Cl, one can choose a linearly independent system such that the dual space of X t is spanned by
where S!{t)B t
= 0, for l < I;, and
S?(t) S;+1(t)
=Ci, =Sf(t) + Sf(t)An(t),
One can observe that to compute' the matrix S(t) one needs certain derivatives of the parameter functions Pi(Y), i.e., certain derivatives of the output y, but the order of these derivatives are bounded by maxi , •.
5. AN INVERSION BASED FD FILTER Let us consider now the system (1) and let us design an inverse system for the fault signals vi' Applying the steps described in the previous section, one can obtain the equations v = F2(t)X2+W and
= All (t)XI + BI (t)u + Lw = A 22 (t)X2 + B2(t)U + A2t(t)Xt y = CIXI '
(33)
i
l
(37)
(34)
i2
(38) (39)
see (Silverman and Medows, 1967; Bestle and Zeitz, 1983), then one can define a coordinate transform S(t) that maps Xl to ii, where
Since the additional term Bdt) is present the expression (35) for fj has to be modified to
(35)
y=jj- [u(1,O), '" ,u(l ,"l.)' '' · ,U(P,"lp)(,(40)
Since one can chose Ft (p) is given by:
= 0, the inverse system
where U(i,O)(t)
i7 = A 22 (t)1/ + A 21 (t)S-I(t)ii, U = F 2(p(t»1/ + Bl"S-I(t)(y- (S(t)Al1(t)S-I(t)
where B l
=0,
U{j,k)(t) =U(j,k_l)(t) - SJ(t)BI(t)u(t) . (42)
(36)
Then the inversion based FD filter can be obtained as:
+ S(t)S-I(t»y),
" is the right inverse of B I •
i7 = A 22 (t)1/ + B2(t)U + A 21 (t)S-I(t)y, (43)
S(t), i.e., the indices ,., can change during the time. From a practical point of view this is an unconvenience since one might prefer to work with a fix set of derivatives.
+ L -"S-l(t)(y (S(t)An (t)S-t (t) + S(t)S-1 (t»y) ,
v = F 2(p(t))1/
Remark 3. In general the structure of the matrix
-
where L -" is the right inverse of L .
Let us denote by Ak,n
6. EXAMPLE
= {A."l1Ai"l1 ·· · Ai.,n I ij
E {O, 1, · · · , N}}.
In order to determine a good matrix S(t), one can compute the sets
As an illustrative example for the (q)LPV inversion scheme let us consider the following linearized parameter varying model:
{e; , ... ,ciA"l~ ,l1} '
= A(p)x(t) + Lv(t) y(t) = Cx(t),
x(t)
where CiAI ,llBI = 0 for alll < If, and one has to determine the indices ,. = mink
,t.
where A(p) = Ao + PIAl + P2A2 and for sake of simplicity only fault inputs are considered. The state matrices are:
If the set
{Cl,' " ,SJ'(t), ...
(41)
,Cp, '"
,S;p(t)}
span the dual space of Xl, then the matrix S(t) will be a good choice in order to ensure that its structure remains unchanged, i.e., one can always use the same set of outputs and derivatives.
Ao
A2
Remark 4. It is clear that the method presented above can also be applied for quasi LPV systems.
236
=
=
-o1 -1 0 0 0 00 ] 0 0 -1 0 0 • o 0 0 1 0 [ o 0 0 0 -1
00000 00000] 1 0 0 0 0 , [ 00000 01000
L _
A1
01 [10] 0 1 , 00 00
=
[0 0 - 01 0 0 0 0 1 0
C
01 0 0 0
01 0 0 0
0] 1 0 • 0 0
00001 1 0 0 0] . 00011
= [0
The parameter varying subspace V· Applying the transformation Ti;
0 o0 0 1 0 0 I 0 0] 000 I I [ o I 0 0 0
= [0 0
1 0 0] .
0 o0 0 00 011 0] (. i.e .• )
°
I
0 0 0 I 0 0 0 -I I 0 0
o
[
1 0 0 0 0
o
PI
the system splits as
-
[
~ 2pI PI PI 1
0o 0I 0 0]
[o 1
0 0 0 1 0 0
0 1
0 0 0
0
7. CONCLUSION •
This paper investigated the problem of input reconstruction by means of dynamic system inversion for linear parameter varying (LPV) systems, where the system matrix depends affinely from the parameters. The fault detection and isolation problem is solved using a detection filter obtained by inverting the system. A procedure for the construction of the inverse, based on the geometric concept of parameter varying invariant subspaces and on the related coordinate transform was given.
100
a
0 00 00 01 00 0] 0 0 0 0 0 • [ 0 0 0 0 0
A~2] ~ 2
Ap [ Aa, Aaa
o
00]
. [c, 0]
[ 0' ] = [~! L
00
= 0,
F2
= 0 and
I 000] -I 0 0 0 0 0 0 I • I lOO
=[
maps
XI
~ gg~ Ig] . 10000
= [
S,
=
°° ° .
00-100] 0
= [0
Ft
= So + PISI + P2S2,
The transformation S(p) where So
0
= Fo + PIFI + P2F2 , is given by
The matrix F(p) Fo
0 I
[000 0 0 1 0J 0 0 0 0 0 • 0000
_ [YI.YI
to y =
52=
[
8. REFERENCES
o0 00 00 0] I 0000 ' o0 0 0
]T .
Y2 Y3
One can figure out that I
0
-1
0
o
1
-~
2..
S-'(t) =
[ PI
1
PI
PI
o
0
1
and
0] 0
0
0 0]
o _p,pa +P. 0 ~ PI
p,
o o
0
0
0
0
It follows, that S(t)All(t)S-l(t)
_ -
[
2.. 2.. -~ P2 p, -- ~
-I 0 0 0] 0 0 o
-I 0 0 0 -I 0 0 0 _I
=
- -
+ p,
[
p,
1
P't
p,
P'
0
0
2..
2.. -~
p,
p,
0]
[0 0 1 0]
0 0
+ Pa
0
P'
and I
L1rS- (t)
1'2] 0 PI .
11 -
= [ -PI
o
--
PI
0
1
0
For the inverse system one has
. = -Tj + (1'2 - PI )Yl + -YI 1'2 . -
Tj
and
PI
Pt
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