A Geometric Approach to Nonlinear Fault ... - Semantic Scholar
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A Geometric Approach to Nonlinear Fault Detection and Isolation Claudio De Persis and Alberto Isidori Abstract We present in this article a differential-geometric approach to the problem of fault detection and isolation for nonlinear systems. A necessary condition for the problem to be solvable is derived in terms of an unobservability distribution, which is computable by means of suitable algorithms. The existence and regularity of such a distribution implies the existence of changes of coordinates in the state and in the output space which induce an “observable” quotient subsystem unaffected by all fault signals but one. For this subsystem a fault detection filter is designed. Keywords
Fault detection and isolation, Nonlinear systems, Unobservability distributions, Nonlinear Observability, Observers. I. Introduction The problem of fault detection and isolation in dynamical systems is the problem of generating diagnostic signals sensitive to the occurrence of faults. Regarding a fault as an input acting on the system, a diagnostic signal must be able to “detect” its occurrence, as well as to “isolate” this particular input from all other inputs (disturbances, controls, other faults) affecting the system behavior. One specific diagnostic signal (also called residual) must be generated per each fault to be detected, each diagnostic signal being sensitive only to one particular fault. Set in these terms, the problem of fault detection and isolation has very much the connotation of a problem of designing a system which, processing all available information about the plant, yields a “noninteractive” map between faults (viewed as inputs) and residuals (viewed as outputs). This problem has attracted a good deal of attention, since its formulation in these terms by Beard [2] and Jones [15]. The original work of these authors addresses the problem in a fashion that corresponds to the solution of the dual version of a problem of noninteracting control by means of memoryless feedback. Later, Massoumnia et al. [18] have shown that the problem can be addressed and successfully solved in a more general setting, which turns out to correspond to the solution of the dual version of a problem of noninteracting control by means of dynamic feedback. In this way, a number of obstructions inherent in the Beard-Jones approach, namely the necessity of a vector relative degree and the stability of certain fixed modes were removed (see Section II). The core of the Beard-Jones fault detection filter is a Luenberger observer, designed in such a way that the resulting “error” system, in which the faults are viewed as inputs and the residuals are viewed as outputs, has a diagonal (and nontrivial) stable transfer function matrix. This “observerbased” approach to the problem of residual generation was later pursued by several authors seeking Claudio De Persis and Alberto Isidori are with the Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130, U.S.A. Alberto Isidori is also with the Dipartimento di Informatica e Sistemistica, Universit` a di Roma “La Sapienza”, 00184 Rome, ITALY. Address for Correspondence: Claudio De Persis, Department of Systems Science and Mathematics, Washington University, Campus Box 1040, One Brookings Dr., St. Louis, MO 63130-4899, U.S.A. Fax: 314-935-6121, E-mail: [email protected]. Research supported in part by ONR under grant N00014-99-1-0697, by AFOSR under grant F49620-95-1-0232, by DARPA, AFRL, AFMC, under grant F30602-99-2-0551, and by MURST. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA, of AFRL, or the U.S. Government. Submitted on December 14th, 1999. Revised on August 16th, 2000.
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extensions to more general classes of systems (as Seliger and Frank [22]), surveyed e.g. in [7], [8]. The most important issue of detecting and isolating faults in the presence of noise-corrupted measurements has been thoroughly investigated, in a framework corresponding to that considered by Beard and Jones in the noise-free case, by Speyer and co-authors in a series of recent papers (see [1] and [4]). In this paper, we approach the problem of fault detection and isolation for nonlinear systems, at a same level of generality as in Massoumnia et al. In essence, these authors have proven that a basic necessary and sufficient condition for the problem to be solvable is the existence of an unobservability subspace (a subspace that can be rendered “unobservable” via “output-injection” and “outputreduction”) leading to a quotient (observable) subsystem unaffected by all fault signals but one. The subspace in question can be determined in a straightforward manner, from the parameters that characterize the plant, by means of simple recursive algorithms which were introduced earlier in [23] to a similar purpose (disturbance decoupling, with internal stability, via measurement feedback) and which dualize those introduced by [19] for noninteracting control, with internal stability, via dynamic state feedback. Once this subspace has been determined, if the test of the necessary and sufficient condition is passed, the simple construction of an asymptotic observer for the quotient subsystem above yields the desired filter. A very similar program is carried out, for the case of nonlinear systems, in this paper. Looking at dual versions of the objects introduced in [14] (see also [16]) to solve the problem of noninteracting control for nonlinear systems, we approach here the problem of fault detection and isolation in differential geometric terms and we show that a solution to this problem can be characterized in terms of properties of certain distributions, which can be considered as the nonlinear analogue of the unobservability subspaces. As in the case of linear systems, it is shown (Section IV) that the problem is solvable only if one of such distributions exists leading to a quotient system which is unaffected by all fault signals but one. Unobservability distributions (recently introduced in [5] and reviewed here in Section III) can be computed by means of algorithms that extend to nonlinear systems those presented in [19] and [23]. Conversely, it is also shown (Section V) that if such a distribution exists, it is possible to perform changes of coordinates in the state and in the output spaces which highlight a special internal structure and in particular the existence of a locally weakly observable subsystem, which is not affected by all fault signals but one. Once these changes of coordinates have been performed, the problem of designing a fault detection filter can be reduced to the design of an observer for the quotient subsystem. As it is well-known, the local weak observability does not help in general to the construction of an observer. Motivated by this, we consider a suitable stronger property of observability introduced in [10], and design a fault detection filter accordingly. II. A geometric approach to the problem of fault detection and isolation A. Preliminaries We assume throughout the paper that the reader is familiar with the basic differential-geometric methods for the analysis and design of nonlinear control systems (see [13] and [21]). We consider systems modeled by equations of the form x˙ = f (x) + g(x)u + `(x)m1 + p(x)w y = h(x)
(1)
with state x defined in a neighborhood X of the origin in IRn , inputs u ∈ IRm , m1 ∈ IR, w ∈ IRd and output y ∈ IRp , in which f (x), the m columns g1 (x), . . . , gm (x) of g(x), `(x) and the d columns p1 (x), . . . , pd (x) of p(x) are smooth vector fields, h(x) is a smooth mapping and f (0) = 0, h(0) = 0. The three sets of components u, m1 , w of the input of (1) correspond, respectively, to an input channel u to be used for control purposes, to a fault signal m1 whose occurrence has to be “detected”, and to a disturbance signal w, whose components include actual disturbances as well as other fault signals from which the specific fault m1 has to be “isolated”.
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The main problem addressed in this paper is the design of a filter (which is throughout referred to as the “residual generator”), modeled by equations of the form x, y) + gˆ(ˆ x, y)u xˆ˙ = fˆ(ˆ ˆ r = h(ˆ x, y)
(2)
ˆ of the origin in IRν , inputs u, y and output r ∈ IRp˜, with with state xˆ defined in a neighborhood X ˆ x, y) is a smooth p ≤ p˜, in which fˆ(ˆ x, y) and the m columns of gˆ(ˆ x, y) are smooth vector fields, h(ˆ ˆ 0) = 0, such that the response r(·) of the cascaded system mapping, and fˆ(0, 0) = 0, h(0, d dt
x xˆ
=
f (x) ˆ f (ˆ x, h(x))
!
+
g(x) u+ x, h(x)) gˆ(ˆ
`(x) m1 + 0
p(x) w 0
(3)
ˆ x, h(x)) , r = h(ˆ depends “nontrivially” on (i.e is affected by) the input m1 (·), depends “trivially” on (i.e. is decoupled from) the inputs u and w and asymptotically converges to zero whenever m1 (·) is identically zero. B. The fundamental problem of residual generation In order to address the design problem outlined in the previous section, it is appropriate to recall first how the property that the output of a nonlinear system is “affected by” or “decoupled from” a fixed set of inputs can be expressed (see [13]). Consider a system x˙ = g0 (x) + y = h(x)
m X
gi (x)ui
(4)
i=1
in which x is defined in a neighborhood X of the origin in IRn , g0 (x), g1 (x), . . . , gm (x) are smooth vector fields and h(x) is a smooth function. Let ΩO denote the smallest codistribution invariant under g0 , g1 , . . . , gm which contains span{dh} (see [13, Section 1.9]). It can be shown (see [13, Section 3.3]) that the output y of (4) is decoupled from the input ui if gi ∈ Ω⊥ O (and, of course, that y of (4) is affected by the input ui if gi 6∈ Ω⊥ O ). Taking this characterization as point of departure, let x˙ e = g0e (xe ) + r
e
e
= h (x )
m X
gie (xe )ui + `e (xe )m1 +
d X
pie (xe )(xe )wi
i=1
i=1
(5)
denote the system (3), resulting from the composition of the plant (1) and the filter (2), in which e
x = e
e
x , xˆ
` (x ) =
g0e (xe )
`(x) , 0
=
!
f (x) , ˆ f (ˆ x, h(x))
pei (xe )
=
pi (x) 0
gie (xe )
=
gi (x) gˆi (ˆ x, h(x))
for i = 1, . . . , d,
for i = 1, . . . , m,
ˆ x, h(x)) . he (xe ) = h(ˆ
Moreover, let ΩeO denote the smallest codistribution which contains span{dhe } and is invariant under all gie , i = 0, . . . , m, and all pie , i = 1, . . . , d.
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In view of the above discussion, it can be asserted that the output r of (5) depends nontrivially on the input m1 if `e 6∈ (ΩeO )⊥ and depends trivially on the inputs u, w when m1 = 0 if
e ⊥ gie ∈ (ΩeO )⊥ , for i = 1, . . . , m and pei ∈ (ΩO ) , for i = 1, . . . , d.
(6)
Motivated by this, we address in the paper the following design problem. Problem. Given system (1), find – if possible – an integer ν and a residual generator (2) such that, in the cascade (5), the codistribution ΩeO satisfies e ⊥ e , pe1 , . . . , pde } ⊂ (ΩO ) , (i) span{g1e , . . . , gm e e ⊥ (ii) span{` } 6⊂ (ΩO ) , (iii) there is δ > 0 such that, if kxe (0)k ≤ δ, then m1 (t) = 0 for all t ≥ 0
⇒
lim kr(t)k = 0 .
t→∞
Henceforth, we denote this problem with the acronym `NLFPRG (where “`” underlines the fact that the fault detection and isolation features are required to hold only locally, and NLFPRG stands for Nonlinear Fundamental Problem of Residual Generation) and improperly refer to requirement (iii) as the stability requirement. C. Isolation of concurrent faults The formulation of the `NLFPRG just given lends itself to the possibility of designing a filter for the purpose of detecting as well as isolating multiple concurrent faults m1 , . . . , ms , in the case of a system of the form m s x˙ = f (x) +
X
gi (x)ui +
i=1
y = h(x) . To this end, set, for i = 1, . . . , s,
p˜i = (`1 . . . `i−1 `i+1 . . . `s p),
X
`i (x)mi + p(x)w
i=1
(7)
w ˜i = col(m1 , . . . , mi−1 , mi+1 , . . . , ms , w)
and rewrite system (7) as x˙ = f (x) + y = h(x) .
m X
˜i gi (x)ui + `i (x)mi + p˜i (x)w
i=1
(8)
Suppose that the residual generator xˆ˙ i = fˆi (ˆ xi , y) + gˆi (ˆ xi , y)u ˆ ri = hi (ˆ xi , y)
(9)
solves the `NLFPRG for system (8), i.e. it is a residual generator with respect to the fault mi . Then, it is immediate to see that the bank of residual generators xˆ˙ 1 = = ˙xˆs = r1 = rs
x1 , y) + gˆ1 (ˆ fˆ1 (ˆ x1 , y)u ... fˆs (ˆ xs , y) + gˆs (ˆ xs , y)u ˆh1 (ˆ x1 , y) ... ˆ s (ˆ = h xs , y)
(10)
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solves the problem of detecting and isolating each individual fault signal m1 , . . . , ms . Conversely, if there is a residual generator of the form xˆ˙ = fˆ(ˆ x, y) + gˆ(ˆ x, y)u ˆ 1 (ˆ r1 = h x, y) ... ˆ s (ˆ x, y) rs = h
(11)
such that, in cascaded system (7)–(11), for each i, the residual ri is affected by the input mi and decoupled from the inputs u and w˜i and asymptotically convergent to zero as mi (·) is identically zero, then necessarily the filter xˆ˙ = fˆ(ˆ x, y)u x, y) + gˆ(ˆ ˆ i (ˆ ri = h x, y) solves the `NLFPRG for (8). Remark. It is worth stressing that this manner of approaching the problem of detection and isolation of multiple faults, originally suggested in [18] in the case of linear systems, is substantially more general than the approach suggested in [2], [15]. In fact, for a linear system of the form x˙ = Ax + Bu + Lm + P w y = Cx , the so-called “Beard-Jones” approach to fault detection and isolation seeks a filter of the form xˆ˙ = Aˆ x + Bu + G(y − C xˆ) r = H xˆ , with G and H such that in the “error system” e˙ = (A + GC)e + Lm + P w r = HCe the transfer function between m and r is purely diagonal, the transfer function between w and r is zero, and (A + GC) is Hurwitz. The reader familiar with the theory of noninteracting control (see [20]) will easily recognize that determining a filter with this property is equivalent to solving a problem of noninteracting control via memoryless feedback, while determining a filter in the way indicated at the beginning (i.e. “stacking” together a set of filters each of which solves an FPRG for each individual fault signal) is equivalent to solving a problem of noninteracting control via dynamic feedback. The solvability of the former problem has severe obstructions, such as the requirement of “vector relative degree” and the stability of certain “fixed modes”, while the solvability of the latter problem only requires the transfer function between m and y to be “left-invertible”, which is a condition generically satisfied if the number of the components of y is larger than or equal to the number of components of m. / III. The observability codistributions and their properties A. Introduction In linear system theory, the concept of “complementary observability subspace” (or, in the terminology of [17], of “unobservability subspace”), which is the dual of the concept of “controllability subspace”, plays an important role in a number of important issues related to the design of filters for
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the observation of certain states in the presence of unknown (deterministic) disturbances. This concept, introduced in [23] to tackle problems of disturbance decoupling via output measurements with pole placement, was used in [18] to solve the fundamental problem of residual generation for linear systems (see also [7]). It was shown in [18] that the existence of a solution of the FPRG for x˙ = Ax + Bu + Lm1 + P w y = Cx ,
(12)
depends on a simple relation between the subspace L = span{L} and the “minimal unobservability subspace” which contains the subspace P = span{P }. Definition and properties of the unobservability subspaces are easily derived by duality from the corresponding notion and properties of the controllability subspaces, for which the reader is referred to [24]. Specifically, dualizing definition 5.1.1 of [24], a subspace Q ⊂ IRn is said to be an unobservability subspace of the pair (A, C) if there exist matrices H and G such that Q is the set of unobservable states of the system x˙ = (A + GC)x,
y = HCx .
The set of all unobservability subspaces containing a given subspace P is closed with respect to subspace intersection, from which it results that this set has a unique minimal element, denoted U∗P , which can be determined – in a finite number of stages – by means of standard algorithms that dualize those presented in [24, Chapter 5] for the computation of the largest controllability subspace contained in a given subspace V. The algorithms in question involve, at each stage, only elementary linear-algebraic manipulations such as subspace additions and intersections. The subspace U∗P plays a crucial role in the solution of the problem outlined above, as shown in the following fundamental, yet very expressive, result proven in [18]. Theorem 1: Consider the system (12). There exists a solution of the FPRG if and only if L ∩ U∗P = {0}. It must be stressed that the proof of the sufficiency part of this theorem can be put in a constructive form, i.e. lends itself to the design of a filter that solves the problem in question. This can be easily seen as a special case of the general nonlinear results derived in this paper, which provide – as a byproduct – an alternative proof of Theorem 1. B. Conditioned Invariant Distributions In this and in the following section, we review some properties of the notions that provide the extension, to nonlinear systems, of the concept of “complementary observability subspace”. These notions have been recently introduced in [5], to which the reader is referred for further information, and are summarized here along with their most fundamental features. Consider a nonlinear system of the form (4). We recall that a distribution ∆ is said to be conditioned invariant (or, (h, f ) invariant, as in [14]) for (4) if it satisfies [gi , ∆ ∩ Ker{dh}] ⊂ ∆,
for all i = 0, . . . , m ,
(13)
where, for convenience, we have set g0 (x) = f (x), and Ker{dh} is the distribution annihilating the differentials of the rows of the mapping h(x). Let now p1 (x), . . . , pd (x) be a set of additional smooth vector fields, set P = span{p1 , . . . , pd } , and consider the non-decreasing sequence of distributions defined as follows S0 = P Sk+1 = S k +
m X i=0
[gi , S k ∩ Ker{dh}] ,
(14)
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where S denotes the involutive closure of S. Then, the following holds (see [14, page 341] and [13, Lemma 6.3.1]). Lemma 1: Suppose there exists an integer k ∗ such that Sk∗ +1 = S k∗
(15)
and set ΣP∗ = S k∗ . Then ΣP∗ is involutive, contains P and is conditioned invariant. Moreover, any other distribution ∆ which is involutive, contains P and is conditioned invariant satisfies ∆ ⊃ Σ∗P .
Remark. A codistribution Ω is said to be conditioned invariant if Lgi Ω ⊂ Ω + span{dh},
for all i = 0, . . . , m ,
(16)
where span{dh} is the codistribution spanned by the differentials of the rows of the mapping h(x). As shown in [14, page 341], if ∆∩Ker{dh} is a smooth distribution and Ω = ∆⊥ is a smooth codistribution, then Ω satisfies (16). If ΣP∗ is well-defined (i.e. condition (15) holds for some k ∗ ) and nonsingular, and ΣP∗ ∩ Ker{dh} is a smooth distribution, it can be asserted that (ΣP∗ )⊥ is the maximal conditioned invariant codistribution which is locally spanned by exact differentials and contained in P ⊥ . / C. The Observability Codistribution Algorithm and its Properties Consider again system (4), let Θ be a fixed codistribution and define the following non-decreasing sequence of codistributions Q0 = Θ ∩ span{dh} Qk+1 = Θ ∩
m X
Lgi Qk + span{dh} .
i=0
(17)
Suppose that all codistributions of this sequence are nonsingular, so that there is an integer k ∗ ≤ n−1 such that Qk = Qk∗ for all k > k ∗ , and set Ω∗ = Qk∗ . It is convenient to use the notation Ω∗ = o.c.a.(Θ) (where “o.c.a.” stands for “observability codistribution algorithm”) to stress the dependence on Θ of the codistribution Ω∗ = Qk∗ , at which the algorithm (17) has stopped. Then, the following properties hold (see [5]). Proposition 1: Suppose all codistributions generated by the algorithm (17) are nonsingular and let Ω∗ = o.c.a.(Θ) be defined as above. Then, Q0 = Ω∗ ∩ span{dh} ∗
Qk+1 = Ω ∩
m X i=0
Lgi Qk + span{dh} .
(18)
As a consequence o.c.a.(Ω∗ ) = Ω∗ . Moreover, if the codistribution Θ is conditioned invariant, so is the codistribution Ω∗ . Motivated by a well-known result which holds for the controllability subspaces of a linear system (see [24, Theorem 5.3]), we say that a codistribution Ω is an observability codistribution for (4) if Lgi Ω ⊂ Ω + span{dh}, o.c.a.(Ω) = Ω .
for all i = 0, . . . , m
(19)
Built in this definition is the implicit requirement that all codistributions of the sequence that defines o.c.a.(Ω) are nonsingular. Likewise, we may say that ∆ is an unobservability distribution if its annihilator Ω = ∆⊥ is an observability codistribution.
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Proposition 1 says that, if Θ is a conditioned invariant distribution, then o.c.a.(Θ) is an observability codistribution. If the algorithm (17) is initialized at (ΣP∗ )⊥ , then o.c.a.((ΣP∗ )⊥ ) is by construction an observability codistribution contained in P ⊥ . As a matter of fact, it can be readily seen that o.c.a.((ΣP∗ )⊥ ) is the largest codistribution having such property. Proposition 2: The codistribution o.c.a.(Θ) is the maximal (in the sense of codistribution inclusion) observability codistribution contained in Θ. Suppose that the distribution ΣP∗ is well-defined (i.e. condition (15) holds for some k ∗ ) and nonsingular, and that ΣP∗ ∩ Ker{dh} is a smooth distribution. Then o.c.a.((ΣP∗ )⊥ ) is the maximal (in the sense of codistribution inclusion) observability codistribution which is locally spanned by exact differentials and contained in P ⊥ . IV. The necessity The purpose of this section is to prove that, if the `NLFPRG has a solution, and an additional technical hypothesis holds, then (span{`})⊥ + o.c.a.((ΣP∗ )⊥ ) = T ∗ X ,
(20)
which is the nonlinear (and dual) version of the fundamental property L ∩ U∗P = {0} identified in Theorem 1 as necessary and sufficient condition for the solution of the FPRG for a linear system. Given a set {τ0 , . . . , τp } of smooth vector fields, and a smooth map h let (see [13, Section 1.9]) < τ0 , . . . , τp |span{dh} > denote the smallest smooth codistribution invariant with respect to τ0 , . . . , τp and containing span{dh} . Recall that the codistribution in question can be determined by means the non-decreasing sequence of smooth codistributions Ω0 = span{dh} Ωk+1 = Ωk +
p X
Lτi Ωk .
(21)
i=0
In fact, if Ωn−1 is nonsingular, then < τ0 , . . . , τp |span{dh} >= Ωn−1 . Moreover, < τ0 , . . . , τp |span{dh} > is locally spanned by exact differentials.
We prove first a general property, which is the nonlinear version of a property that trivially holds in the case of linear systems (namely the property that if U is an unobservability subspace of (A, C), then e if Ker{C} e ⊂ Ker{C}), but whose proof in the case U is also an unobservability subspace of (A, C) of nonlinear systems requires a little work, in view of the more elaborate definition of observability codistribution. Lemma 2: Let {τ0 , . . . , τp } be a set of smooth vector fields, let h be a smooth map, suppose the distribution Ωn−1 defined above is nonsingular, and set Ω = Ωn−1 . Let k be another smooth map, satisfying span{dk} ⊃ span{dh} and suppose Ω ∩ span{dk} is a smooth codistribution. Consider the non-decreasing sequence of codistributions Q0 = Ω ∩ span{dk} p X (22) Qj+1 = Ω ∩ Qj + Lτi Qj + span{dk} . i=0
Then, Lτi Ω ⊂ Ω + span{dk} for all i = 0, . . . , p
(23)
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and Qn−1 = Ω .
(24)
As a consequence, Ω is an observability codistribution for the system x˙ = τ0 (x) + y = k(x) .
p X
τi (x)ui
i=1
(25)
Proof: Consider the non-decreasing sequence of smooth codistributions e Ω 0 = Ω ∩ span{dk}
e e Ω j+1 = Ωj +
p X i=0
Since Ω ⊃ span{dh} and span{dk} ⊃ span{dh}, then
e . Lτi Ω j
(26)
e = Ω ∩ span{dk} ⊃ span{dh} = Ω . Ω 0 0
As a consequence, it is readily seen, by induction, that
This, for j = n − 1 yields
e ⊃ Ω for all j ≥ 0 . Ω j j e Ω n−1 ⊃ Ω .
But, on the other hand
e Ω n−1 ⊂< τ0 , . . . , τp |Ω ∩ span{dk} >⊂< τ0 , . . . , τp |Ω >= Ω ,
from which we conclude that
e Ω n−1 = Ω .
(27)
By construction, Lτi Ω ⊂ Ω and therefore (23) trivially holds. To prove (24) we show that, in the sequence (22), one has e (28) Qn−1 = Ω n−1
and this, in view of (27) will prove the claim. To this end, consider again the sequences of codistributions (22) and (26) and observe that, by definition, e . Q0 = Ω 0 e for some j. Then Suppose Qj = Ω j
Qj+1 = Ω ∩ (Qj + e + = Ω ∩ (Ω j
p X
Lτi Qj + span{dk})
i=0
e + span{dk}) Lτi Ω j
i=0 p X
e e e = Ω ∩ (Ω j+1 + span{dk}) = Ωj+1 + Ω ∩ span{dk} = Ωj+1 ,
e e e where, in the last passage, we have used the properties that Ω j+1 ⊂ Ω and Ω ∩span{dk} = Ω0 ⊂ Ωj+1 . Thus, (28) holds. To conclude the proof of the Lemma it suffices to observe that, if (24) holds, all codistributions of the sequence (22) with k ≥ n coincide with Ω . In other words, the observability codistribution algorithm for (25), initialized at Ω , converges to Ω .
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Suppose now that the `NLFPRG for (1) is solved by some filter (2) and consider the cascade plant/filter (3), rewritten in the form (5). Set also e
e
k (x ) =
h(x) xˆ
.
Define the non-decreasing sequence of codistributions Ωe0 = span{dhe }
e = Ωek + Ωk+1
m X i=0
Lgie Ωek ,
and recall (see [13, Section 1.9]) that, if Ωen+ˆn−1 is nonsingular, then e |span{dhe } > Ωen+ˆn−1 =< g0e , . . . , gm
i.e. Ωen+ˆn−1 is the smallest codistribution which contains span{dhe } and is invariant under gie for all e i = 0, . . . , m. Set Ωe = Ωn+ˆ n−1 and note that this codistribution, if nonsingular, is locally spanned by exact differentials. Specializing Lemma 2 to the case of the cascade plant/filter, we obtain the following. e e e Proposition 3: Suppose the distribution Ωn+ˆ n−1 . n−1 defined above is nonsingular and set Ω = Ωn+ˆ e e e Suppose Ω ∩ span{dk } is a smooth codistribution. Then Ω is an observability codistribution for the system x˙ e = g0e (xe ) + e
e
m X
gie (xe )ui
i=1
(29)
y = k (x ) Proof: It immediately follows from the Lemma 2 and from the fact that span{dk e } ⊃ span{dhe }. By definition, the distribution Ωe thus defined and the distribution ΩeO considered in the formulation of the `NLFPRG are such that Ωe ⊂ ΩeO . We deduce from this that, if the filter (2) solves the problem, necessarily pei ∈ (Ωe )⊥
for all i = 1, . . . , d.
If (Ωe )⊥ is involutive, this property shows that (Ωe )⊥ is automatically invariant under all pei ’s, and e coincide. therefore that Ωe and ΩO As a consequence, we see that if the filter (2) solves the problem and Ωe is nonsingular, we necessarily have span{pe1 , . . . , ped } ⊂ (Ωe )⊥ span{`e } 6⊂ (Ωe )⊥ . Now, we show the implications of these properties on the system (1). This will be done under the following hypothesis. Assumption I. The codistribution Ωe is nonsingular. Moreover, for each (x◦ , xˆ◦ ), there are ν smooth covector fields of the form ωje (xe ) = ( wj (x)
w ˆj (x, xˆ) )
j = 1, . . . ν ,
which span Ωe at any (x, xˆ) in a neighborhood U ◦ × Uˆ ◦ of (x◦ , xˆ◦ ).
(30)
11
Observe that w1 (x), . . . , wν (x), which are smooth covector fields defined on a neighborhood U ◦ ⊂ X, define on U ◦ a smooth distribution Ω† as follows Ω† (x) = span{w1 (x), . . . , wν (x)} . The codistribution defined in this way has a number of properties that help determine a necessary condition for the solution of the problem in question. Proposition 4: The following properties hold. (i) span{p1 , . . . , pd } ⊂ (Ω† )⊥ , (ii) span{`} 6⊂ (Ω† )⊥ , (iii) (Ω† )⊥ is involutive, (iv) Lgi Ω† ⊂ Ω† + span{dh}, for all i = 0, . . . , m. If all the codistributions of the sequence generated by the observability codistribution algorithm for (29), initialized at Ωe , are nonsingular and all the codistributions of the sequence generated by the observability codistribution algorithm for x˙ = f (x) + y = h(x) ,
m X
gi (x)ui
i=1
(31)
initialized at Ω† , are nonsingular, then (v) the observability codistribution algorithm for (31), initialized at Ω† , converges to Ω† . Thus, Ω† is an observability codistribution for the system (31). Proof: By hypothesis, pe (xe ) annihilates all covectors of Ωe (xe ) and, in particular, all vectors of (30). In view of the special form of pe (xe ), this yields wj (x)p(x) = 0 for all j = 0, . . . , ν , and proves (i). Similarly, since by hypothesis `e (xe ) cannot not annihilate all ν covectors of (30), `(x) cannot annihilate all the wj (x)’s and this proves (ii). To prove (iii), take two vector fields τ1 , τ2 ∈ (Ω† )⊥ and set τie
=
τi , 0
i = 1, 2 .
Indeed, τ1e , τ2e ∈ (Ωe )⊥ , and so is [τ1e , τ2e ] because (Ωe )⊥ is involutive. Thus also [τ1 , τ2 ] ∈ (Ω† )⊥ . To prove (iv), take the derivative of any covector in (30) along gie (xe ), to obtain a covector of the form Lgie ωje (xe ) = ( Lgi wj (x) + κ(x, xˆ)dh(x) vˆ(x, xˆ) ) (32) in which
h ∂ˆ gi i
. ∂y y=h(x) By hypothesis, since Ωe (xe ) is an observability codistribution for (29), the covector field (32) is in Ωe + span{dk e }. Now, this codistribution is spanned by the rows of a matrix of the form κ(x, xˆ) = wˆj (x, xˆ)
w1 (x) .. .
wˆ1 (x, xˆ) . .. ˆν (x, xˆ) wν (x) w dh(x) 0 0 I
12
and therefore, the condition Lgie ωje (xe ) ∈ Ωe + span{dk e } implies Lgi wj (x) + κ(x, xˆ)dh(x) ∈ span{w1 (x), . . . , wν (x), dh(x)} . This yields Lgi wj (x) ⊂ span{w1 (x), . . . , wν (x), dh(x)} = Ω† (x) + span{dh}(x) and proves (iv). To prove (v), recall the that observability codistribution algorithm for (29), initialized at Ωe generates the sequence of codistributions Qe0 = Ωe ∩ span{dk e }
Qek+1 = Ωe ∩ Qek +
m X i=0
Lgie Qek + span{dk e } ,
(33)
while the observability codistribution algorithm for (31), initialized at Ω† generates the sequence of codistributions Q0 = Ω† ∩ span{dh} m X (34) Qk+1 = Ω† ∩ Qk + Lgi Qk + span{dh} . i=0
Moreover, by Proposition 3, Qen+ˆn−1 = Ωe . Since Ωe (xe ) is spanned by the rows of the matrix
w1 (x) wˆ1 (x, xˆ) . c ... W (x) W (x, xˆ) = .. wν (x) w ˆν (x, xˆ)
and span{dk e }(xe ) is spanned by the rows of the matrix
dh(x) 0 I 0
,
to determine the intersection (Ωe ∩ span{dk e })(xe ) we have to find, at each (x, xˆ), row vectors α, β, γ such that c (x, x W (x) W ˆ) (α β γ ) = (0 0) . dh(x) 0 0 I Thus, it is easily deduced that Q0e (xe ) is spanned by row vectors of the form
where α(x) is such that
c (x, x ˆ) α(x)W (x) α(x)W
α(x)W (x) = −β(x)dh(x)
(35)
for some β(x). Now, the vectors of the form α(x)W (x) with α(x) such that (35) holds for some β(x) are precisely the vectors which span (Ω† ∩ span{dh})(x). It can be concluded, therefore, that Qe0 (xe ) is spanned by vectors of the form ωj (xe ) = ( vj (x) vˆj (x, xˆ) ) , j = 1, . . . , ν0
13
in which the vj (x)’s span Q0 (x). We prove, by induction, that a similar property holds for all other codistributions of the sequence (33). For, suppose this is true at step k, namely that Qke (xe ) is spanned by the rows of a matrix
c (x, x Wk (x) W ˆ) k
in which the rows of Wk (x) span Qk (x). The derivative along gie of any of the rows of this matrix is a covector of the form (compare with (32)) Lgie ω e (xe ) = ( Lgi w(x) 0 ) + ( κ(x, xˆ) vˆ(x, xˆ) )
dh(x) 0 I 0
,
in which, by construction, Lgi w(x) is a covector in Lgi Qk (x), while the second addend in the right-hand side is a covector in span{dk e }. Therefore, we see that
Qek +
m X i=0
Lgi Qek + span{dk e } (xe )
is spanned by the rows of a matrix of the form
Wk+1 (x) 0 0 I
P
in which the rows of Wk+1 (x) span Qk + m i=0 Lgi Qk +span{dh} (x). From this, an argument identical e to the one used before for Q0 shows that Qek+1 (xe ) is spanned by vectors of the form ωj (xe ) = ( vj (x) vˆj (x, xˆ) ) , j = 1, . . . , νk+1 in which the vj (x)’s span Qk+1 (x). e e e e By hypothesis, Qn+ˆ n−1 = Ω . The arguments above show that Ω (x ) is spanned by the rows of a matrix of the form c ˆ) Wn+ˆn−1 (x) W n+ˆ n−1 (x, x in which the rows of Wn+ˆn−1 (x) span Qn+ˆn−1 (x). Thus, by definition of Ω† , we see that Ω† (x) = Qn+ˆn−1 (x) and this proves (v). This Proposition shows (under appropriate technical hypotheses) that, if the `NLFPRG has a solution, there is an observability codistribution, namely Ω† , which is locally spanned by exact differentials and satisfies properties (i) and (ii). Now, suppose that the hypotheses under which Proposition 2 is valid hold. Then, using property (i), it is seen that by definition Ω† is contained in the maximal observability codistribution which is locally spanned by exact differentials and contained in P ⊥ = span{p1 , . . . , pd }⊥ , i.e. that Ω† ⊂ o.c.a.((Σ∗P )⊥ ) .
As a consequence, property (ii) implies
span{`} 6⊂ [ o.c.a.((ΣP∗ )⊥ ) ]⊥ and this proves that, on some open subset of X, (20) necessarily holds. Remark. Note that the technical hypotheses used to derive the condition in question are trivially satisfied in the case of a linear system (with linear residual generator). Thus, the result proven in this section contains as a particular case the necessity part of Theorem 1. /
14
V. The sufficiency A. The construction of a locally weakly observable “quotient” system We begin by describing a useful change of coordinates, based on the properties of the observability codistribution algorithm, which is quite useful in addressing the problem of designing a residual generator. Proposition 5: Consider system (4). Let Ω be an observability codistribution. Let n1 denote the dimension of Ω. Suppose that Ω is locally spanned by exact differentials. Suppose that span{dh} is nonsingular. Let p − n2 denote the dimension of Ω ∩ span{dh} and suppose there exists a surjection Ψ1 : IRp → IRp−n2 such that Ω ∩ span{dh} = span{d(Ψ1 ◦ h)} Fix x◦ ∈ X and let y ◦ = h(x◦ ). Then, there exists a selection matrix H2 (i.e. a matrix in which any row has all 0 entries but one, which is equal to 1) such that Ψ(y) =
y1 y2
=
Ψ1 (y) H2 y
(36)
is a local diffeomorphism at y ◦ in IRp . Choose a neighborhood U ◦ of x◦ and a function Φ1 : U ◦ → IRn1 such that Ω = span{dΦ1 }
at any point of U ◦ . Then, there exists a function Φ3 : U ◦ → IRn−n1 −n2 such that
Φ1 (x) x1 H = Φ(x) = x 2 h(x) 2 Φ3 (x) x3
(37)
is a local diffeomorphism at x◦ in X. In the new local coordinates defined by (36)–(37), system (4) is described by equations of the form x˙ 1 x˙ 2 x˙ 3 y1 y2
= = = = =
f1 (x1 , x2 ) + g1 (x1 , x2 )u f2 (x1 , x2 , x3 ) + g2 (x1 , x2 , x3 )u f3 (x1 , x2 , x3 ) + g3 (x1 , x2 , x3 )u h1 (x1 ) x2 .
(38)
If p(x) is a vector field in the annihilator of Ω, then, trivially, in the new coordinates this vector field is expressed in the form T T (39) ( 0 pT 2 (x1 , x2 , x3 ) p3 (x1 , x2 , x3 ) ) . Proof: See [5]. The derivation of the special form (38) in the previous proposition was, actually, based only on the property that Ω is conditioned invariant. However, exploiting the fact that Ω is an observability codistribution, one can reach more interesting conclusions. As a matter of fact, the x1 -subsystem of (38), in which x2 can be identified with y2 (which of course is possible by virtue the special choice of the new coordinates) and viewed as an “independent input”, namely system x˙ 1 = f1 (x1 , y2 ) + g1 (x1 , y2 )u y1 = h(x1 )
(40)
can be shown, under a convenient additional assumption, to be locally weakly observable, in the sense of the classical definition of [12].
15
To this end, suppose that the hypotheses of Proposition 5 hold and (locally) change coordinates, in the state and output spaces, as described in this Proposition. This yields a system, of the form (4), in which, for i = 0, . . . , m, g1i (x1 , x2 ) gi (˜ x) = g2i (x1 , x2 , x3 ) g3i (x1 , x2 , x3 ) and
x) = h(˜
h1 (x1 ) x2
,
where x˜ = Φ(x). Define, for i = 0, . . . , m, and set
so that
ϕi (x1 , x2 ) = g1i (x1 , x2 ) − g1i (x1 , 0)
g1i (x1 , 0) g2i (x1 , x2 , x3 ) gi∗ (˜ x) = , g3i (x1 , x2 , x3 )
ϕi (x1 , x2 ) 0 ψi (˜ x) = , 0
x) . x) + ψi (˜ gi (˜ x) = gi∗ (˜ Assumption II. For every vector field τ ∈ Ker{dh} and every covector field ω ∈ Ω, < [ψi , τ ], ω >= 0. / Remark. Note that this assumption is trivially satisfied when ϕi (x1 , x2 ) is independent of x1 (i.e. when gi1 (x1 , x2 ) is the sum of a function of x1 and a function of x2 ). In fact, in this case, since Ker{dh} ⊂ span{
∂ ∂ , }, ∂x1 ∂x3
the vector field [ψi , τ ] is trivially zero. In particular, the assumption is satisfied if the gi (x)’s are linear vector fields. / Then, the following holds. Proposition 6: Consider system (4). Let Ω be an observability codistribution satisfying the hypotheses of Proposition 5 and choose a change of coordinates yielding the special form (38). Suppose also that the Assumption II above is satisfied. Then, the subsystem x˙ 1 = f1 (x1 , 0) + g1 (x1 , 0)u y1 = h1 (x1 ) ,
(41)
is locally weakly observable. Proof: See [5]. Remark. We recall (see again [12]) that the property of local weak observability implies that any point in the state space can be distinguished from any other point in a neighborhood, in the standard sense of state distinguishability (two states are distinguishable if they induce different input-output maps). Thus, if such a property holds for (41), it indeed holds – a fortiori – for the subsystem x˙ 1 = f1 (x1 , y2 ) + g1 (x1 , y2 )u y1 = h1 (x1 ) , viewed as a system with inputs u, y2 and output y1 . /
(42)
16
The result derived above carries very useful consequences for the solution of the problem of residual generation. To see why this is the case, consider a system of the form (1), determine the codistribution o.c.a.((ΣP∗ )⊥ ), which – as shown before in Proposition 2 – is the largest observability codistribution locally spanned by exact differentials and contained in P ⊥ , and suppose the condition (span{`})⊥ + o.c.a.((ΣP∗ )⊥ ) = T ∗ X ,
(43)
of which we have shown the necessity in the previous section, holds (note, trivially, that the maximality of o.c.a.((Σ∗P )⊥ ) maximizes the chances of having (43) fulfilled). Then, just changing the coordinates in the way described above, we see that the first set y1 of output variables is actually the output of a system of the form x˙ 1 = f1 (x1 , y2 ) + g1 (x1 , y2 )u + `1 (x1 , x2 , x3 )m1 y1 = h1 (x1 ) , which turns out to be locally weakly observable (viewing u and y2 as inputs) and in which the vector field `1 (x1 , x2 , x3 ) is by assumption nonzero. Hence, the occurrence of a fault m1 can be detected by appropriately processing the signals y1 , y2 and u. For example, in case system (1) is a linear system, if L ∩ U∗P = {0} (compare with Theorem 1) the decomposition described above yields x˙ 1 = A11 x1 + A12 y2 + B1 u + L1 m1 y1 = C1 x1 ,
(44)
in which L1 6= 0 and the pair (A11 , C1 ) is observable. To design a residual generator, set xˆ˙ = A11 xˆ + A12 y2 + B1 u + G(C1 xˆ − y1 ) r = C1 xˆ − y1 ,
(45)
with G such that the eigenvalues of (A11 + GC1 ) have negative real part. A trivial calculation, based on the observation that the difference e = x1 − xˆ satisfies e˙ = (A11 + GC1 )e + L1 m1 r = C1 e , shows that (45) solves the fundamental problem of residual generation. B. Example: Actuator fault detection for a point-mass satellite model As a simple application of the techniques illustrated in the latter subsection, we want to design a detection filter for revealing the occurrence of a fault on the tangential thrust in a point-mass planar model of a satellite. The model in question is provided by the following set of differential equations (see e.g. [3]) ρ˙ = v 1 v˙ = ρω 2 − θ1 2 + θ2 u1 + w ρ φ˙ = ω u2 m 2vω + θ2 + θ2 , ω˙ = − ρ ρ ρ in which (ρ, φ) denotes the position of the satellite in polar coordinates on the plane, v is the radial velocity, ω is the angular velocity and u1 , u2 are the radial and, respectively, tangential thrust, m is the fault signal (note, for instance, that m = −u2 models a total failure of the tangential thrust) and w represents the disturbance signal. The parameters θ1 , θ2 are supposed to be known, constant
17
and different from zero. Note that the disturbance w can make the detection of the fault m more difficult due to the presence of the coupling term 2vω in the equation of the angular acceleration ω. ˙ ρ The following quantities are assumed to be available for measurements ν1 = ρ,
ν2 = φ,
ν3 = ω .
In this example, the disturbance vector field is represented by the vector p = ( 0 1 0 0 )T . Using the algorithm (14), the minimal conditioned invariant distribution containing span{p} is found to be: 2ω T ) }. ΣP∗ = span{( 0 1 0 0 )T , ( 1 0 0 − ρ Note that ΣP∗ is smooth, nonsingular and involutive for any ρ = 6 0. Then, its annihilator (ΣP∗ )⊥ is locally spanned by exact differentials and can be expressed as follows (ΣP∗ )⊥ = span{dφ, d(ρ2 ω)} . We can now proceed to the computation of the maximal observability codistribution which is locally spanned by exact differentials and contained in P ⊥ (i.e. o.c.a.((ΣP∗ )⊥ ) via the algorithm (17). To this end, it is enough to note that span{dh} ⊃ (ΣP∗ )⊥ to conclude that o.c.a.((ΣP∗ )⊥ ) = (Σ∗P )⊥ . The change of coordinates in the state space induced by o.c.a.((ΣP∗ )⊥ ) is given by x1 =
!
φ ρ2 ω
,
x2 = ρ,
x3 = v .
As far as the change of coordinates in the output space is concerned, we observe that o.c.a.((Σ∗P )⊥ ) ∩ span{dh} = o.c.a.((ΣP∗ )⊥ ) = span{dy2 , d(y12 y3 )} . Hence the change of coordinates in the output space is given by y1 =
ν2 ν12 ν3
!
y2 = ν 1 .
,
In the new coordinates, the system is re-written as:
x˙ 11 x˙ 12
!
x˙ 2 x˙ 3
y11 y12
!
x12 x22 = θ2 x2 u2 + θ2 x2 m = x3 θ1 θ2 = 23 − 3 + θ2 u1 + w x2 x2
=
x11 x12
!
,
y2 = x2 .
(46)
18
Consider now a candidate residual generator of the form ξ˙ = θ2 y2 u2 + k(y12 − ξ) r = y12 − ξ ,
(47)
with k a positive real number. To see that (47) is actually a residual generator, it suffices to observe that the difference e = x12 − ξ satisfies e˙ = −ke + θ2 y2 m r = e, θ2 and hence, r(t) is the response of a linear filter with transfer function s+k to the input y2 (t)m(t). In this particular example, it is even possible to identify the value of m, if the latter is constant. To this end, consider the system ξ˙ = θ2 y2 u2 + k(y12 − ξ) + θ2 y2 m ˆ (48) ˙ m ˆ = λθ2 y2 (y12 − ξ) ,
with k, λ positive real numbers. The differences e = x12 − ξ and m ˆ satisfy ˜ =m−m ˜ e˙ = −ke + θ2 y2 m ˙ m ˜ = −λθ2 y2 e . Since in the present case, the function y2 = ρ remains always positive and away from zero, it is possible to apply the Persistency of Excitation Lemma which yields the globally exponentially stability of the equilibrium ξ = x12 and m ˆ = m, thus implying, in particular, the convergence of the estimate m ˆ to the true value of m. Fig. 1 shows a simulation corresponding to the following scenario: the actuators u1 , u2 are supposed to provide constant thrust equal to u¯1 and u¯2 = −1, respectively. At time t = 10 a total failure of the tangential actuator u2 occurs (Fig. 1 (a)). Moreover, a disturbance signal w is present (Fig. 1 (b)). From the observed variables (Fig. 1 (d,e,f)) no hint on the occurrence of the fault can be obtained. On the other hand, the output of the residual generator (Fig. 1 (c)) clearly shows the occurrence of the fault m and identify its actual value. C. The design of filters for fault detection To conclude the paper, we show in this section how to exploit the decomposition described in section V-A for the purpose of designing a residual generator in the case of a nonlinear system. For the sake of simplicity, we consider the case in which y1 is one-dimensional. Extension to the case in which y1 is a vector requires appropriate modification of the Assumption III below. Motivated by the previous discussion, we focus our attention on the system x˙ 1 = f1 (x1 , y2 ) + g1 (x1 , y2 )u y1 = h1 (x1 ) ,
(49)
viewed as a system with inputs (u, y2 ) and output y1 and we address the issue of designing an asymptotic observer for x1 . We have already shown that, under mild hypotheses, such a system is guaranteed to be locally weakly observable by construction, but we also know that this property may not suffice to the purposes of designing an asymptotic observer. To this end, a stronger observability property, introduced in [10], has to be assumed. Assumption III (Uniform Complete Observability). The maps f1 , g1 , h1 in (49) are analytic and globally defined. Moreover, there exists a globally defined analytic change of coordinates H : x1 7→ ξ
19 (a)
(d)
1.5
3
1
2
0.5
1
0
0
10
20
30
40
0
50
0
10
20
(b)
30
40
50
30
40
50
30
40
50
(e) 6
2 1
4
0 2
−1 −2 0
10
20
30
40
0
50
0
10
20 (f)
(c) 1.5
2 1.5
1
1 0.5 0
0.5 0
10
20
30
40
0
50
0
10
20
Fig. 1.
that transforms system (49) into a system of the form ξ˙1 = φ1 (ξ1 , ξ2 , y2 , u) ξ˙2 = φ2 (ξ1 , ξ2 , ξ3 , y2 , u) ··· ξ˙n˜ = φn˜ (ξ1 , ξ2 , . . . , ξn˜ , y2 , u) y1 = h0 (ξ1 ) and
(50)
∂h0 ∂φ1 ∂φn˜ −1 = 6 0, = 6 0, . . . , 6= 0 ∂ξ1 ∂ξ2 ∂ξn˜
for all ξ, y2 , u. Assumption IV (Boundedness). ku(·)k∞ < ∞ and kw(·)k∞ < ∞. Moreover, for each M0 > 0 and N0 > 0, there exists M > 0 such that, if kx(0)k ≤ M0 , ku(·)k∞ < N0 and kw(·)k∞ < N0 , then max{kξ(t)k, ky2 (t)k, ku(t)k} ≤ M for all t ≥ 0.
It can be shown (see [9]) that, for any choice of M > 0, there exist functions γ0 (ξ1 ), ϕ1 (ξ1 , ξ2 , y2 , u), ϕ2 (ξ1 , ξ2 , ξ3 , y2 , u), . . . , ϕn˜ (ξ1 , ξ2 , . . . , ξn˜ , y2 , u)
(51)
with the following properties: (i) γ0 (ξ1 ) agrees with h0 (ξ1 ) and, for each i = 1, . . . , n ˜ , ϕi (ξ1 , . . . , ξi+1 , y2 , u) agrees with fi (ξ1 , . . . , ξi , ξi+1 , y2 , u) on the compact set C = {(ξ, y2 , u) : kξk ≤ M, ky2 k ≤ M, kuk ≤ M } , (ii) each of the functions ϕi (ξ1 , . . . , ξi+1 , y2 , u), i = 1, . . . , n ˜ , is globally Lipschitz with respect to xi := col(ξ1 , . . . , ξi ), uniformly in ξi+1 , y2 , u,
20
(iii) there exist real numbers α, β, 0 < α < β such that ∂γ 0
α ≤
∂ξ1
≤ β,
α≤
∂ϕi ≤ β, i = 1, . . . , n ˜−1. ∂ξi+1
Let now M0 , N0 be any pair of positive numbers and let M be such that the condition in Assumption IV holds. Accordingly, fix the functions (51) and consider the system ξ˙1 = ϕ1 (ξ1 , ξ2 , y2 , u) ξ˙2 = ϕ2 (ξ1 , ξ2 , ξ3 , y2 , u) ··· ˙ξn˜ = ϕn˜ (ξ1 , ξ2 , . . . , ξn˜ , y2 , u) y1 = γ0 (ξ1 ) .
(52)
Clearly, if kx(0)k ≤ M0 and ku(·)k∞ ≤ N0 , kw(·)k∞ < N0 the state responses of (50) and of (52) agree for all t ≥ 0. Hence, to observe the state of (50) is the same thing as to observe the state of (52). Motivated by the results of Gauthier-Kupca [10], we consider a residual generator of the form x1 , xˆ2 , y2 , u) + Gk1 (y1 − γ0 (ˆ x1 )) xˆ˙ 1 = ϕ1 (ˆ x1 )) x1 , xˆ2 , xˆ3 , y2 , u) + G2 k2 (y1 − γ0 (ˆ xˆ˙ 2 = ϕ2 (ˆ ··· x1 )) x1 , xˆ2 , . . . , xˆn˜ , y2 , u) + Gn˜ kn˜ (y1 − γ0 (ˆ xˆ˙ n˜ = ϕn˜ (ˆ r = y1 − γ0 (ˆ x1 ) ,
(53)
in which G is a (sufficiently large) positive number and k1 , . . . , kn˜ are numbers depending on the parameters α and β in condition (iii) above, determined according to the following result of [9]. Lemma 3: Consider the pair of matrices
0 0 a2 (t) 0 0 a 3 (t) A(t) = · · · 0 0 0 0 0 0
0 ··· ··· 0 ··· · , · · · an˜ (t) ··· 0
C(t) = ( a1 (t) 0 · · · 0 )
and suppose that, for some pair of real numbers α, β, 0 < α < β, α < ai (t) < β,
i = 1, . . . , n ˜.
Then, there exist a vector K ∈ IRn˜ and a symmetric positive definite matrix S satisfying (A(t) − KC(t))T S + S(A(t) − KC(t)) ≤ −I . The following fundamental result, due to Gauthier-Kupca [10], [9], shows that the state of system (53) globally exponentially tracks the state of system (52). Theorem 2: For any a > 0 there is G > 1 (large enough) such that, for all pairs (ξ ◦ , xˆ◦ ) ∈ IRn˜ × IRn˜ , the responses ξ(t) of (52) with initial condition ξ(0) = ξ ◦ and xˆ(t) of (53) with initial condition xˆ(0) = xˆ◦ satisfy x(t) − ξ(t)k ≤ κ(a)e−at kˆ kˆ x(0) − ξ(0)k ˜. for some polynomial function κ(·) of degree n From this, it is immediate to conclude that system (53) is a residual generator for system (49).
21
Corollary 1: Consider system (49). For any pair M0 , N0 there is a residual generator of the form (53) and a number G > 1 (large enough) such that if kx(0)k ≤ M0 , ku(·)k∞ < N0 , kw(·)k∞ < N0 and m1 (t) = 0 for all t ≥ 0, then limt→∞ r(t) = 0. The “sensitivity” of the residual generated by the filter (53) with respect to the fault is a straightforward consequence of the observability property of the x1 -subsystem and of the property that the fault vector field is not identically zero. VI. Conclusions In this paper we have studied the problem of fault detection and isolation for nonlinear systems. Starting with a geometric characterization of the problem, we have derived under a mild hypothesis a basic necessary condition, which is expressed in terms of unobservability distributions. The fulfillment of this condition implies the existence of diffeomorphisms in the output space and the state space which highlight a special internal structure, particularly suitable for designing a residual generator which solves the problem of fault detection. The most significant feature of this structure is the existence of a disturbance-decoupled and “observable” subsystem which is affected by the fault. In the case of linear systems, this immediately leads to a solution of the problem, since for such a subsystem a standard observer can be used as a residual generator. In the case of nonlinear systems, the subsystem in question is guaranteed to be locally-weakly observable. If it is also “completely uniformly observable”, using some recent results from the theory of nonlinear observers, we have been able to construct an observer which plays the role of a residual generator. The extension to the case of multiple concurrent faults, although straightforward, has been briefly discussed. References [1] R.N. Banavar, J.L. Speyer, A linear-quadratic game approach to estimation and smoothing, in Proc. Amer. Contr. Conf., Boston, MA, 2818-2822, 1991. [2] R. V. Beard, Failure accomodation in linear systems through self-reorganization, Ph.D. dissertation, M.I.T., 1971. [3] R.W. Brockett, Finite-dimensional linear systems, New York, Wiley, 1970. [4] W.H. Chung, J.L. Speyer, A game theoretic fault detection filter, IEEE Trans. Aut. Contr., 43, 875-880, 1998. [5] C. De Persis, A. Isidori, On the observability codistributions of a nonlinear system, Syst. & Contr. Letters, 40, 297-304, 2000. [6] C. De Persis, A. Isidori, An addendum to the discussion on the paper “Fault detection and isolation for state affine systems”, European Journal of Control, 6, 2000. [7] P. M. Frank. Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy - a survey. Automatica, 26:459–474, 1990. [8] P. M. Frank. On-line fault detection in uncertain nonlinear systems using diagnostic observers: a survey. Int. J. Systems Sci., 25(12):2129–2154, 1994. [9] J.P. Gauthier and I.A.K. Kupca, Deterministic Observation Theory and Applications, Cambridge University Press, 2000 (to appear). [10] J.P. Gauthier and I.A.K. Kupca, Observability and observers for nonlinear systems, SIAM J. Contr. Optimiz., 32, pp. 975-994, 1994. [11] H. Hammouri, M. Kinnaert, E.H. El Yaagoubi, Observer-based approach to fault detection and isolation for nonlinear systems, IEEE Trans. Aut. Contr., 44, 1879-1884, 1999. [12] R. Hermann, A.J. Krener, Nonlinear controllability and observability, IEEE Trans. Aut. Contr., AC-22, 728-740, 1977. [13] A. Isidori, Nonlinear Control Systems, Springer Verlag, Third Edition, London, 1995. [14] A. Isidori, A.J. Krener, C. Gori-Giorgi, S. Monaco, Nonlinear decoupling via feedback: a differential geometric approach, IEEE Trans. Aut. Contr., AC-26, 331-345, 1981. [15] H. L. Jones, Failure detection in linear systems, Ph.D. dissertation, M.I.T., 1973. [16] A.J. Krener, Conditioned invariant and locally conditioned invariant distributions, Sys. Contr. Lett., 8, 69-74, 1986. [17] M.-A. Massoumnia, A geometric approach to the synthesis of failure detection filters, IEEE Trans. Aut. Contr., AC-31, 839-846, 1986. [18] M.-A. Massoumnia, G.C. Verghese, A.S. Willsky, Failure detection and identification, IEEE Trans. Aut. Contr., AC-34, 316-321, 1989. [19] A. S. Morse, W.M. Wonham, Decoupling and pole assignment by dynamic compensation, SIAM J. Control, 8, 317-337, 1970. [20] A. S. Morse, W.M. Wonham, Status of noninteracting control, IEEE Trans. Aut. Contr., AC-16, 568-581, 1971. [21] H. Nijmejier, A.J. van der Schaft, Nonlinear Dynamical Control Systems, Springer Verlag, 1990. [22] R. Seliger, P.M. Frank, Fault diagnosis by disturbance-decoupled nonlinear observers, 30th IEEE Conf. Dec. Contr., 22482253, 1991. [23] J.C. Willems, C. Commault, Disturbance decoupling by measurement feedback with stability or pole placement, SIAM J. Control, 19 (1981) 490-504. [24] W. M. Wohnam, Linear Multivariable Control: A geometric approach, Springer Verlag, 1985.
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[25] W.M. Wonham, A. S. Morse, Decoupling and pole assignment in linear multivariable systems: a geometric approach, SIAM J. Control, 8, 1-18, 1970.
A Geometric Approach to Nonlinear Fault Detection and Isolation Claudio De Persis and Alberto Isidori Abstract We present in this article a differential-geometric approach to the problem of fault detection and isolation for nonlinear systems. A necessary condition for the problem to be solvable is derived in terms of an unobservability distribution, which is computable by means of suitable algorithms. The existence and regularity of such a distribution implies the existence of changes of coordinates in the state and in the output space which induce an “observable” quotient subsystem unaffected by all fault signals but one. For this subsystem a fault detection filter is designed. Keywords
Fault detection and isolation, Nonlinear systems, Unobservability distributions, Nonlinear Observability, Observers. I. Introduction The problem of fault detection and isolation in dynamical systems is the problem of generating diagnostic signals sensitive to the occurrence of faults. Regarding a fault as an input acting on the system, a diagnostic signal must be able to “detect” its occurrence, as well as to “isolate” this particular input from all other inputs (disturbances, controls, other faults) affecting the system behavior. One specific diagnostic signal (also called residual) must be generated per each fault to be detected, each diagnostic signal being sensitive only to one particular fault. Set in these terms, the problem of fault detection and isolation has very much the connotation of a problem of designing a system which, processing all available information about the plant, yields a “noninteractive” map between faults (viewed as inputs) and residuals (viewed as outputs). This problem has attracted a good deal of attention, since its formulation in these terms by Beard [2] and Jones [15]. The original work of these authors addresses the problem in a fashion that corresponds to the solution of the dual version of a problem of noninteracting control by means of memoryless feedback. Later, Massoumnia et al. [18] have shown that the problem can be addressed and successfully solved in a more general setting, which turns out to correspond to the solution of the dual version of a problem of noninteracting control by means of dynamic feedback. In this way, a number of obstructions inherent in the Beard-Jones approach, namely the necessity of a vector relative degree and the stability of certain fixed modes were removed (see Section II). The core of the Beard-Jones fault detection filter is a Luenberger observer, designed in such a way that the resulting “error” system, in which the faults are viewed as inputs and the residuals are viewed as outputs, has a diagonal (and nontrivial) stable transfer function matrix. This “observerbased” approach to the problem of residual generation was later pursued by several authors seeking Claudio De Persis and Alberto Isidori are with the Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130, U.S.A. Alberto Isidori is also with the Dipartimento di Informatica e Sistemistica, Universit` a di Roma “La Sapienza”, 00184 Rome, ITALY. Address for Correspondence: Claudio De Persis, Department of Systems Science and Mathematics, Washington University, Campus Box 1040, One Brookings Dr., St. Louis, MO 63130-4899, U.S.A. Fax: 314-935-6121, E-mail: [email protected]. Research supported in part by ONR under grant N00014-99-1-0697, by AFOSR under grant F49620-95-1-0232, by DARPA, AFRL, AFMC, under grant F30602-99-2-0551, and by MURST. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA, of AFRL, or the U.S. Government. Submitted on December 14th, 1999. Revised on August 16th, 2000.
2
extensions to more general classes of systems (as Seliger and Frank [22]), surveyed e.g. in [7], [8]. The most important issue of detecting and isolating faults in the presence of noise-corrupted measurements has been thoroughly investigated, in a framework corresponding to that considered by Beard and Jones in the noise-free case, by Speyer and co-authors in a series of recent papers (see [1] and [4]). In this paper, we approach the problem of fault detection and isolation for nonlinear systems, at a same level of generality as in Massoumnia et al. In essence, these authors have proven that a basic necessary and sufficient condition for the problem to be solvable is the existence of an unobservability subspace (a subspace that can be rendered “unobservable” via “output-injection” and “outputreduction”) leading to a quotient (observable) subsystem unaffected by all fault signals but one. The subspace in question can be determined in a straightforward manner, from the parameters that characterize the plant, by means of simple recursive algorithms which were introduced earlier in [23] to a similar purpose (disturbance decoupling, with internal stability, via measurement feedback) and which dualize those introduced by [19] for noninteracting control, with internal stability, via dynamic state feedback. Once this subspace has been determined, if the test of the necessary and sufficient condition is passed, the simple construction of an asymptotic observer for the quotient subsystem above yields the desired filter. A very similar program is carried out, for the case of nonlinear systems, in this paper. Looking at dual versions of the objects introduced in [14] (see also [16]) to solve the problem of noninteracting control for nonlinear systems, we approach here the problem of fault detection and isolation in differential geometric terms and we show that a solution to this problem can be characterized in terms of properties of certain distributions, which can be considered as the nonlinear analogue of the unobservability subspaces. As in the case of linear systems, it is shown (Section IV) that the problem is solvable only if one of such distributions exists leading to a quotient system which is unaffected by all fault signals but one. Unobservability distributions (recently introduced in [5] and reviewed here in Section III) can be computed by means of algorithms that extend to nonlinear systems those presented in [19] and [23]. Conversely, it is also shown (Section V) that if such a distribution exists, it is possible to perform changes of coordinates in the state and in the output spaces which highlight a special internal structure and in particular the existence of a locally weakly observable subsystem, which is not affected by all fault signals but one. Once these changes of coordinates have been performed, the problem of designing a fault detection filter can be reduced to the design of an observer for the quotient subsystem. As it is well-known, the local weak observability does not help in general to the construction of an observer. Motivated by this, we consider a suitable stronger property of observability introduced in [10], and design a fault detection filter accordingly. II. A geometric approach to the problem of fault detection and isolation A. Preliminaries We assume throughout the paper that the reader is familiar with the basic differential-geometric methods for the analysis and design of nonlinear control systems (see [13] and [21]). We consider systems modeled by equations of the form x˙ = f (x) + g(x)u + `(x)m1 + p(x)w y = h(x)
(1)
with state x defined in a neighborhood X of the origin in IRn , inputs u ∈ IRm , m1 ∈ IR, w ∈ IRd and output y ∈ IRp , in which f (x), the m columns g1 (x), . . . , gm (x) of g(x), `(x) and the d columns p1 (x), . . . , pd (x) of p(x) are smooth vector fields, h(x) is a smooth mapping and f (0) = 0, h(0) = 0. The three sets of components u, m1 , w of the input of (1) correspond, respectively, to an input channel u to be used for control purposes, to a fault signal m1 whose occurrence has to be “detected”, and to a disturbance signal w, whose components include actual disturbances as well as other fault signals from which the specific fault m1 has to be “isolated”.
3
The main problem addressed in this paper is the design of a filter (which is throughout referred to as the “residual generator”), modeled by equations of the form x, y) + gˆ(ˆ x, y)u xˆ˙ = fˆ(ˆ ˆ r = h(ˆ x, y)
(2)
ˆ of the origin in IRν , inputs u, y and output r ∈ IRp˜, with with state xˆ defined in a neighborhood X ˆ x, y) is a smooth p ≤ p˜, in which fˆ(ˆ x, y) and the m columns of gˆ(ˆ x, y) are smooth vector fields, h(ˆ ˆ 0) = 0, such that the response r(·) of the cascaded system mapping, and fˆ(0, 0) = 0, h(0, d dt
x xˆ
=
f (x) ˆ f (ˆ x, h(x))
!
+
g(x) u+ x, h(x)) gˆ(ˆ
`(x) m1 + 0
p(x) w 0
(3)
ˆ x, h(x)) , r = h(ˆ depends “nontrivially” on (i.e is affected by) the input m1 (·), depends “trivially” on (i.e. is decoupled from) the inputs u and w and asymptotically converges to zero whenever m1 (·) is identically zero. B. The fundamental problem of residual generation In order to address the design problem outlined in the previous section, it is appropriate to recall first how the property that the output of a nonlinear system is “affected by” or “decoupled from” a fixed set of inputs can be expressed (see [13]). Consider a system x˙ = g0 (x) + y = h(x)
m X
gi (x)ui
(4)
i=1
in which x is defined in a neighborhood X of the origin in IRn , g0 (x), g1 (x), . . . , gm (x) are smooth vector fields and h(x) is a smooth function. Let ΩO denote the smallest codistribution invariant under g0 , g1 , . . . , gm which contains span{dh} (see [13, Section 1.9]). It can be shown (see [13, Section 3.3]) that the output y of (4) is decoupled from the input ui if gi ∈ Ω⊥ O (and, of course, that y of (4) is affected by the input ui if gi 6∈ Ω⊥ O ). Taking this characterization as point of departure, let x˙ e = g0e (xe ) + r
e
e
= h (x )
m X
gie (xe )ui + `e (xe )m1 +
d X
pie (xe )(xe )wi
i=1
i=1
(5)
denote the system (3), resulting from the composition of the plant (1) and the filter (2), in which e
x = e
e
x , xˆ
` (x ) =
g0e (xe )
`(x) , 0
=
!
f (x) , ˆ f (ˆ x, h(x))
pei (xe )
=
pi (x) 0
gie (xe )
=
gi (x) gˆi (ˆ x, h(x))
for i = 1, . . . , d,
for i = 1, . . . , m,
ˆ x, h(x)) . he (xe ) = h(ˆ
Moreover, let ΩeO denote the smallest codistribution which contains span{dhe } and is invariant under all gie , i = 0, . . . , m, and all pie , i = 1, . . . , d.
4
In view of the above discussion, it can be asserted that the output r of (5) depends nontrivially on the input m1 if `e 6∈ (ΩeO )⊥ and depends trivially on the inputs u, w when m1 = 0 if
e ⊥ gie ∈ (ΩeO )⊥ , for i = 1, . . . , m and pei ∈ (ΩO ) , for i = 1, . . . , d.
(6)
Motivated by this, we address in the paper the following design problem. Problem. Given system (1), find – if possible – an integer ν and a residual generator (2) such that, in the cascade (5), the codistribution ΩeO satisfies e ⊥ e , pe1 , . . . , pde } ⊂ (ΩO ) , (i) span{g1e , . . . , gm e e ⊥ (ii) span{` } 6⊂ (ΩO ) , (iii) there is δ > 0 such that, if kxe (0)k ≤ δ, then m1 (t) = 0 for all t ≥ 0
⇒
lim kr(t)k = 0 .
t→∞
Henceforth, we denote this problem with the acronym `NLFPRG (where “`” underlines the fact that the fault detection and isolation features are required to hold only locally, and NLFPRG stands for Nonlinear Fundamental Problem of Residual Generation) and improperly refer to requirement (iii) as the stability requirement. C. Isolation of concurrent faults The formulation of the `NLFPRG just given lends itself to the possibility of designing a filter for the purpose of detecting as well as isolating multiple concurrent faults m1 , . . . , ms , in the case of a system of the form m s x˙ = f (x) +
X
gi (x)ui +
i=1
y = h(x) . To this end, set, for i = 1, . . . , s,
p˜i = (`1 . . . `i−1 `i+1 . . . `s p),
X
`i (x)mi + p(x)w
i=1
(7)
w ˜i = col(m1 , . . . , mi−1 , mi+1 , . . . , ms , w)
and rewrite system (7) as x˙ = f (x) + y = h(x) .
m X
˜i gi (x)ui + `i (x)mi + p˜i (x)w
i=1
(8)
Suppose that the residual generator xˆ˙ i = fˆi (ˆ xi , y) + gˆi (ˆ xi , y)u ˆ ri = hi (ˆ xi , y)
(9)
solves the `NLFPRG for system (8), i.e. it is a residual generator with respect to the fault mi . Then, it is immediate to see that the bank of residual generators xˆ˙ 1 = = ˙xˆs = r1 = rs
x1 , y) + gˆ1 (ˆ fˆ1 (ˆ x1 , y)u ... fˆs (ˆ xs , y) + gˆs (ˆ xs , y)u ˆh1 (ˆ x1 , y) ... ˆ s (ˆ = h xs , y)
(10)
5
solves the problem of detecting and isolating each individual fault signal m1 , . . . , ms . Conversely, if there is a residual generator of the form xˆ˙ = fˆ(ˆ x, y) + gˆ(ˆ x, y)u ˆ 1 (ˆ r1 = h x, y) ... ˆ s (ˆ x, y) rs = h
(11)
such that, in cascaded system (7)–(11), for each i, the residual ri is affected by the input mi and decoupled from the inputs u and w˜i and asymptotically convergent to zero as mi (·) is identically zero, then necessarily the filter xˆ˙ = fˆ(ˆ x, y)u x, y) + gˆ(ˆ ˆ i (ˆ ri = h x, y) solves the `NLFPRG for (8). Remark. It is worth stressing that this manner of approaching the problem of detection and isolation of multiple faults, originally suggested in [18] in the case of linear systems, is substantially more general than the approach suggested in [2], [15]. In fact, for a linear system of the form x˙ = Ax + Bu + Lm + P w y = Cx , the so-called “Beard-Jones” approach to fault detection and isolation seeks a filter of the form xˆ˙ = Aˆ x + Bu + G(y − C xˆ) r = H xˆ , with G and H such that in the “error system” e˙ = (A + GC)e + Lm + P w r = HCe the transfer function between m and r is purely diagonal, the transfer function between w and r is zero, and (A + GC) is Hurwitz. The reader familiar with the theory of noninteracting control (see [20]) will easily recognize that determining a filter with this property is equivalent to solving a problem of noninteracting control via memoryless feedback, while determining a filter in the way indicated at the beginning (i.e. “stacking” together a set of filters each of which solves an FPRG for each individual fault signal) is equivalent to solving a problem of noninteracting control via dynamic feedback. The solvability of the former problem has severe obstructions, such as the requirement of “vector relative degree” and the stability of certain “fixed modes”, while the solvability of the latter problem only requires the transfer function between m and y to be “left-invertible”, which is a condition generically satisfied if the number of the components of y is larger than or equal to the number of components of m. / III. The observability codistributions and their properties A. Introduction In linear system theory, the concept of “complementary observability subspace” (or, in the terminology of [17], of “unobservability subspace”), which is the dual of the concept of “controllability subspace”, plays an important role in a number of important issues related to the design of filters for
6
the observation of certain states in the presence of unknown (deterministic) disturbances. This concept, introduced in [23] to tackle problems of disturbance decoupling via output measurements with pole placement, was used in [18] to solve the fundamental problem of residual generation for linear systems (see also [7]). It was shown in [18] that the existence of a solution of the FPRG for x˙ = Ax + Bu + Lm1 + P w y = Cx ,
(12)
depends on a simple relation between the subspace L = span{L} and the “minimal unobservability subspace” which contains the subspace P = span{P }. Definition and properties of the unobservability subspaces are easily derived by duality from the corresponding notion and properties of the controllability subspaces, for which the reader is referred to [24]. Specifically, dualizing definition 5.1.1 of [24], a subspace Q ⊂ IRn is said to be an unobservability subspace of the pair (A, C) if there exist matrices H and G such that Q is the set of unobservable states of the system x˙ = (A + GC)x,
y = HCx .
The set of all unobservability subspaces containing a given subspace P is closed with respect to subspace intersection, from which it results that this set has a unique minimal element, denoted U∗P , which can be determined – in a finite number of stages – by means of standard algorithms that dualize those presented in [24, Chapter 5] for the computation of the largest controllability subspace contained in a given subspace V. The algorithms in question involve, at each stage, only elementary linear-algebraic manipulations such as subspace additions and intersections. The subspace U∗P plays a crucial role in the solution of the problem outlined above, as shown in the following fundamental, yet very expressive, result proven in [18]. Theorem 1: Consider the system (12). There exists a solution of the FPRG if and only if L ∩ U∗P = {0}. It must be stressed that the proof of the sufficiency part of this theorem can be put in a constructive form, i.e. lends itself to the design of a filter that solves the problem in question. This can be easily seen as a special case of the general nonlinear results derived in this paper, which provide – as a byproduct – an alternative proof of Theorem 1. B. Conditioned Invariant Distributions In this and in the following section, we review some properties of the notions that provide the extension, to nonlinear systems, of the concept of “complementary observability subspace”. These notions have been recently introduced in [5], to which the reader is referred for further information, and are summarized here along with their most fundamental features. Consider a nonlinear system of the form (4). We recall that a distribution ∆ is said to be conditioned invariant (or, (h, f ) invariant, as in [14]) for (4) if it satisfies [gi , ∆ ∩ Ker{dh}] ⊂ ∆,
for all i = 0, . . . , m ,
(13)
where, for convenience, we have set g0 (x) = f (x), and Ker{dh} is the distribution annihilating the differentials of the rows of the mapping h(x). Let now p1 (x), . . . , pd (x) be a set of additional smooth vector fields, set P = span{p1 , . . . , pd } , and consider the non-decreasing sequence of distributions defined as follows S0 = P Sk+1 = S k +
m X i=0
[gi , S k ∩ Ker{dh}] ,
(14)
7
where S denotes the involutive closure of S. Then, the following holds (see [14, page 341] and [13, Lemma 6.3.1]). Lemma 1: Suppose there exists an integer k ∗ such that Sk∗ +1 = S k∗
(15)
and set ΣP∗ = S k∗ . Then ΣP∗ is involutive, contains P and is conditioned invariant. Moreover, any other distribution ∆ which is involutive, contains P and is conditioned invariant satisfies ∆ ⊃ Σ∗P .
Remark. A codistribution Ω is said to be conditioned invariant if Lgi Ω ⊂ Ω + span{dh},
for all i = 0, . . . , m ,
(16)
where span{dh} is the codistribution spanned by the differentials of the rows of the mapping h(x). As shown in [14, page 341], if ∆∩Ker{dh} is a smooth distribution and Ω = ∆⊥ is a smooth codistribution, then Ω satisfies (16). If ΣP∗ is well-defined (i.e. condition (15) holds for some k ∗ ) and nonsingular, and ΣP∗ ∩ Ker{dh} is a smooth distribution, it can be asserted that (ΣP∗ )⊥ is the maximal conditioned invariant codistribution which is locally spanned by exact differentials and contained in P ⊥ . / C. The Observability Codistribution Algorithm and its Properties Consider again system (4), let Θ be a fixed codistribution and define the following non-decreasing sequence of codistributions Q0 = Θ ∩ span{dh} Qk+1 = Θ ∩
m X
Lgi Qk + span{dh} .
i=0
(17)
Suppose that all codistributions of this sequence are nonsingular, so that there is an integer k ∗ ≤ n−1 such that Qk = Qk∗ for all k > k ∗ , and set Ω∗ = Qk∗ . It is convenient to use the notation Ω∗ = o.c.a.(Θ) (where “o.c.a.” stands for “observability codistribution algorithm”) to stress the dependence on Θ of the codistribution Ω∗ = Qk∗ , at which the algorithm (17) has stopped. Then, the following properties hold (see [5]). Proposition 1: Suppose all codistributions generated by the algorithm (17) are nonsingular and let Ω∗ = o.c.a.(Θ) be defined as above. Then, Q0 = Ω∗ ∩ span{dh} ∗
Qk+1 = Ω ∩
m X i=0
Lgi Qk + span{dh} .
(18)
As a consequence o.c.a.(Ω∗ ) = Ω∗ . Moreover, if the codistribution Θ is conditioned invariant, so is the codistribution Ω∗ . Motivated by a well-known result which holds for the controllability subspaces of a linear system (see [24, Theorem 5.3]), we say that a codistribution Ω is an observability codistribution for (4) if Lgi Ω ⊂ Ω + span{dh}, o.c.a.(Ω) = Ω .
for all i = 0, . . . , m
(19)
Built in this definition is the implicit requirement that all codistributions of the sequence that defines o.c.a.(Ω) are nonsingular. Likewise, we may say that ∆ is an unobservability distribution if its annihilator Ω = ∆⊥ is an observability codistribution.
8
Proposition 1 says that, if Θ is a conditioned invariant distribution, then o.c.a.(Θ) is an observability codistribution. If the algorithm (17) is initialized at (ΣP∗ )⊥ , then o.c.a.((ΣP∗ )⊥ ) is by construction an observability codistribution contained in P ⊥ . As a matter of fact, it can be readily seen that o.c.a.((ΣP∗ )⊥ ) is the largest codistribution having such property. Proposition 2: The codistribution o.c.a.(Θ) is the maximal (in the sense of codistribution inclusion) observability codistribution contained in Θ. Suppose that the distribution ΣP∗ is well-defined (i.e. condition (15) holds for some k ∗ ) and nonsingular, and that ΣP∗ ∩ Ker{dh} is a smooth distribution. Then o.c.a.((ΣP∗ )⊥ ) is the maximal (in the sense of codistribution inclusion) observability codistribution which is locally spanned by exact differentials and contained in P ⊥ . IV. The necessity The purpose of this section is to prove that, if the `NLFPRG has a solution, and an additional technical hypothesis holds, then (span{`})⊥ + o.c.a.((ΣP∗ )⊥ ) = T ∗ X ,
(20)
which is the nonlinear (and dual) version of the fundamental property L ∩ U∗P = {0} identified in Theorem 1 as necessary and sufficient condition for the solution of the FPRG for a linear system. Given a set {τ0 , . . . , τp } of smooth vector fields, and a smooth map h let (see [13, Section 1.9]) < τ0 , . . . , τp |span{dh} > denote the smallest smooth codistribution invariant with respect to τ0 , . . . , τp and containing span{dh} . Recall that the codistribution in question can be determined by means the non-decreasing sequence of smooth codistributions Ω0 = span{dh} Ωk+1 = Ωk +
p X
Lτi Ωk .
(21)
i=0
In fact, if Ωn−1 is nonsingular, then < τ0 , . . . , τp |span{dh} >= Ωn−1 . Moreover, < τ0 , . . . , τp |span{dh} > is locally spanned by exact differentials.
We prove first a general property, which is the nonlinear version of a property that trivially holds in the case of linear systems (namely the property that if U is an unobservability subspace of (A, C), then e if Ker{C} e ⊂ Ker{C}), but whose proof in the case U is also an unobservability subspace of (A, C) of nonlinear systems requires a little work, in view of the more elaborate definition of observability codistribution. Lemma 2: Let {τ0 , . . . , τp } be a set of smooth vector fields, let h be a smooth map, suppose the distribution Ωn−1 defined above is nonsingular, and set Ω = Ωn−1 . Let k be another smooth map, satisfying span{dk} ⊃ span{dh} and suppose Ω ∩ span{dk} is a smooth codistribution. Consider the non-decreasing sequence of codistributions Q0 = Ω ∩ span{dk} p X (22) Qj+1 = Ω ∩ Qj + Lτi Qj + span{dk} . i=0
Then, Lτi Ω ⊂ Ω + span{dk} for all i = 0, . . . , p
(23)
9
and Qn−1 = Ω .
(24)
As a consequence, Ω is an observability codistribution for the system x˙ = τ0 (x) + y = k(x) .
p X
τi (x)ui
i=1
(25)
Proof: Consider the non-decreasing sequence of smooth codistributions e Ω 0 = Ω ∩ span{dk}
e e Ω j+1 = Ωj +
p X i=0
Since Ω ⊃ span{dh} and span{dk} ⊃ span{dh}, then
e . Lτi Ω j
(26)
e = Ω ∩ span{dk} ⊃ span{dh} = Ω . Ω 0 0
As a consequence, it is readily seen, by induction, that
This, for j = n − 1 yields
e ⊃ Ω for all j ≥ 0 . Ω j j e Ω n−1 ⊃ Ω .
But, on the other hand
e Ω n−1 ⊂< τ0 , . . . , τp |Ω ∩ span{dk} >⊂< τ0 , . . . , τp |Ω >= Ω ,
from which we conclude that
e Ω n−1 = Ω .
(27)
By construction, Lτi Ω ⊂ Ω and therefore (23) trivially holds. To prove (24) we show that, in the sequence (22), one has e (28) Qn−1 = Ω n−1
and this, in view of (27) will prove the claim. To this end, consider again the sequences of codistributions (22) and (26) and observe that, by definition, e . Q0 = Ω 0 e for some j. Then Suppose Qj = Ω j
Qj+1 = Ω ∩ (Qj + e + = Ω ∩ (Ω j
p X
Lτi Qj + span{dk})
i=0
e + span{dk}) Lτi Ω j
i=0 p X
e e e = Ω ∩ (Ω j+1 + span{dk}) = Ωj+1 + Ω ∩ span{dk} = Ωj+1 ,
e e e where, in the last passage, we have used the properties that Ω j+1 ⊂ Ω and Ω ∩span{dk} = Ω0 ⊂ Ωj+1 . Thus, (28) holds. To conclude the proof of the Lemma it suffices to observe that, if (24) holds, all codistributions of the sequence (22) with k ≥ n coincide with Ω . In other words, the observability codistribution algorithm for (25), initialized at Ω , converges to Ω .
10
Suppose now that the `NLFPRG for (1) is solved by some filter (2) and consider the cascade plant/filter (3), rewritten in the form (5). Set also e
e
k (x ) =
h(x) xˆ
.
Define the non-decreasing sequence of codistributions Ωe0 = span{dhe }
e = Ωek + Ωk+1
m X i=0
Lgie Ωek ,
and recall (see [13, Section 1.9]) that, if Ωen+ˆn−1 is nonsingular, then e |span{dhe } > Ωen+ˆn−1 =< g0e , . . . , gm
i.e. Ωen+ˆn−1 is the smallest codistribution which contains span{dhe } and is invariant under gie for all e i = 0, . . . , m. Set Ωe = Ωn+ˆ n−1 and note that this codistribution, if nonsingular, is locally spanned by exact differentials. Specializing Lemma 2 to the case of the cascade plant/filter, we obtain the following. e e e Proposition 3: Suppose the distribution Ωn+ˆ n−1 . n−1 defined above is nonsingular and set Ω = Ωn+ˆ e e e Suppose Ω ∩ span{dk } is a smooth codistribution. Then Ω is an observability codistribution for the system x˙ e = g0e (xe ) + e
e
m X
gie (xe )ui
i=1
(29)
y = k (x ) Proof: It immediately follows from the Lemma 2 and from the fact that span{dk e } ⊃ span{dhe }. By definition, the distribution Ωe thus defined and the distribution ΩeO considered in the formulation of the `NLFPRG are such that Ωe ⊂ ΩeO . We deduce from this that, if the filter (2) solves the problem, necessarily pei ∈ (Ωe )⊥
for all i = 1, . . . , d.
If (Ωe )⊥ is involutive, this property shows that (Ωe )⊥ is automatically invariant under all pei ’s, and e coincide. therefore that Ωe and ΩO As a consequence, we see that if the filter (2) solves the problem and Ωe is nonsingular, we necessarily have span{pe1 , . . . , ped } ⊂ (Ωe )⊥ span{`e } 6⊂ (Ωe )⊥ . Now, we show the implications of these properties on the system (1). This will be done under the following hypothesis. Assumption I. The codistribution Ωe is nonsingular. Moreover, for each (x◦ , xˆ◦ ), there are ν smooth covector fields of the form ωje (xe ) = ( wj (x)
w ˆj (x, xˆ) )
j = 1, . . . ν ,
which span Ωe at any (x, xˆ) in a neighborhood U ◦ × Uˆ ◦ of (x◦ , xˆ◦ ).
(30)
11
Observe that w1 (x), . . . , wν (x), which are smooth covector fields defined on a neighborhood U ◦ ⊂ X, define on U ◦ a smooth distribution Ω† as follows Ω† (x) = span{w1 (x), . . . , wν (x)} . The codistribution defined in this way has a number of properties that help determine a necessary condition for the solution of the problem in question. Proposition 4: The following properties hold. (i) span{p1 , . . . , pd } ⊂ (Ω† )⊥ , (ii) span{`} 6⊂ (Ω† )⊥ , (iii) (Ω† )⊥ is involutive, (iv) Lgi Ω† ⊂ Ω† + span{dh}, for all i = 0, . . . , m. If all the codistributions of the sequence generated by the observability codistribution algorithm for (29), initialized at Ωe , are nonsingular and all the codistributions of the sequence generated by the observability codistribution algorithm for x˙ = f (x) + y = h(x) ,
m X
gi (x)ui
i=1
(31)
initialized at Ω† , are nonsingular, then (v) the observability codistribution algorithm for (31), initialized at Ω† , converges to Ω† . Thus, Ω† is an observability codistribution for the system (31). Proof: By hypothesis, pe (xe ) annihilates all covectors of Ωe (xe ) and, in particular, all vectors of (30). In view of the special form of pe (xe ), this yields wj (x)p(x) = 0 for all j = 0, . . . , ν , and proves (i). Similarly, since by hypothesis `e (xe ) cannot not annihilate all ν covectors of (30), `(x) cannot annihilate all the wj (x)’s and this proves (ii). To prove (iii), take two vector fields τ1 , τ2 ∈ (Ω† )⊥ and set τie
=
τi , 0
i = 1, 2 .
Indeed, τ1e , τ2e ∈ (Ωe )⊥ , and so is [τ1e , τ2e ] because (Ωe )⊥ is involutive. Thus also [τ1 , τ2 ] ∈ (Ω† )⊥ . To prove (iv), take the derivative of any covector in (30) along gie (xe ), to obtain a covector of the form Lgie ωje (xe ) = ( Lgi wj (x) + κ(x, xˆ)dh(x) vˆ(x, xˆ) ) (32) in which
h ∂ˆ gi i
. ∂y y=h(x) By hypothesis, since Ωe (xe ) is an observability codistribution for (29), the covector field (32) is in Ωe + span{dk e }. Now, this codistribution is spanned by the rows of a matrix of the form κ(x, xˆ) = wˆj (x, xˆ)
w1 (x) .. .
wˆ1 (x, xˆ) . .. ˆν (x, xˆ) wν (x) w dh(x) 0 0 I
12
and therefore, the condition Lgie ωje (xe ) ∈ Ωe + span{dk e } implies Lgi wj (x) + κ(x, xˆ)dh(x) ∈ span{w1 (x), . . . , wν (x), dh(x)} . This yields Lgi wj (x) ⊂ span{w1 (x), . . . , wν (x), dh(x)} = Ω† (x) + span{dh}(x) and proves (iv). To prove (v), recall the that observability codistribution algorithm for (29), initialized at Ωe generates the sequence of codistributions Qe0 = Ωe ∩ span{dk e }
Qek+1 = Ωe ∩ Qek +
m X i=0
Lgie Qek + span{dk e } ,
(33)
while the observability codistribution algorithm for (31), initialized at Ω† generates the sequence of codistributions Q0 = Ω† ∩ span{dh} m X (34) Qk+1 = Ω† ∩ Qk + Lgi Qk + span{dh} . i=0
Moreover, by Proposition 3, Qen+ˆn−1 = Ωe . Since Ωe (xe ) is spanned by the rows of the matrix
w1 (x) wˆ1 (x, xˆ) . c ... W (x) W (x, xˆ) = .. wν (x) w ˆν (x, xˆ)
and span{dk e }(xe ) is spanned by the rows of the matrix
dh(x) 0 I 0
,
to determine the intersection (Ωe ∩ span{dk e })(xe ) we have to find, at each (x, xˆ), row vectors α, β, γ such that c (x, x W (x) W ˆ) (α β γ ) = (0 0) . dh(x) 0 0 I Thus, it is easily deduced that Q0e (xe ) is spanned by row vectors of the form
where α(x) is such that
c (x, x ˆ) α(x)W (x) α(x)W
α(x)W (x) = −β(x)dh(x)
(35)
for some β(x). Now, the vectors of the form α(x)W (x) with α(x) such that (35) holds for some β(x) are precisely the vectors which span (Ω† ∩ span{dh})(x). It can be concluded, therefore, that Qe0 (xe ) is spanned by vectors of the form ωj (xe ) = ( vj (x) vˆj (x, xˆ) ) , j = 1, . . . , ν0
13
in which the vj (x)’s span Q0 (x). We prove, by induction, that a similar property holds for all other codistributions of the sequence (33). For, suppose this is true at step k, namely that Qke (xe ) is spanned by the rows of a matrix
c (x, x Wk (x) W ˆ) k
in which the rows of Wk (x) span Qk (x). The derivative along gie of any of the rows of this matrix is a covector of the form (compare with (32)) Lgie ω e (xe ) = ( Lgi w(x) 0 ) + ( κ(x, xˆ) vˆ(x, xˆ) )
dh(x) 0 I 0
,
in which, by construction, Lgi w(x) is a covector in Lgi Qk (x), while the second addend in the right-hand side is a covector in span{dk e }. Therefore, we see that
Qek +
m X i=0
Lgi Qek + span{dk e } (xe )
is spanned by the rows of a matrix of the form
Wk+1 (x) 0 0 I
P
in which the rows of Wk+1 (x) span Qk + m i=0 Lgi Qk +span{dh} (x). From this, an argument identical e to the one used before for Q0 shows that Qek+1 (xe ) is spanned by vectors of the form ωj (xe ) = ( vj (x) vˆj (x, xˆ) ) , j = 1, . . . , νk+1 in which the vj (x)’s span Qk+1 (x). e e e e By hypothesis, Qn+ˆ n−1 = Ω . The arguments above show that Ω (x ) is spanned by the rows of a matrix of the form c ˆ) Wn+ˆn−1 (x) W n+ˆ n−1 (x, x in which the rows of Wn+ˆn−1 (x) span Qn+ˆn−1 (x). Thus, by definition of Ω† , we see that Ω† (x) = Qn+ˆn−1 (x) and this proves (v). This Proposition shows (under appropriate technical hypotheses) that, if the `NLFPRG has a solution, there is an observability codistribution, namely Ω† , which is locally spanned by exact differentials and satisfies properties (i) and (ii). Now, suppose that the hypotheses under which Proposition 2 is valid hold. Then, using property (i), it is seen that by definition Ω† is contained in the maximal observability codistribution which is locally spanned by exact differentials and contained in P ⊥ = span{p1 , . . . , pd }⊥ , i.e. that Ω† ⊂ o.c.a.((Σ∗P )⊥ ) .
As a consequence, property (ii) implies
span{`} 6⊂ [ o.c.a.((ΣP∗ )⊥ ) ]⊥ and this proves that, on some open subset of X, (20) necessarily holds. Remark. Note that the technical hypotheses used to derive the condition in question are trivially satisfied in the case of a linear system (with linear residual generator). Thus, the result proven in this section contains as a particular case the necessity part of Theorem 1. /
14
V. The sufficiency A. The construction of a locally weakly observable “quotient” system We begin by describing a useful change of coordinates, based on the properties of the observability codistribution algorithm, which is quite useful in addressing the problem of designing a residual generator. Proposition 5: Consider system (4). Let Ω be an observability codistribution. Let n1 denote the dimension of Ω. Suppose that Ω is locally spanned by exact differentials. Suppose that span{dh} is nonsingular. Let p − n2 denote the dimension of Ω ∩ span{dh} and suppose there exists a surjection Ψ1 : IRp → IRp−n2 such that Ω ∩ span{dh} = span{d(Ψ1 ◦ h)} Fix x◦ ∈ X and let y ◦ = h(x◦ ). Then, there exists a selection matrix H2 (i.e. a matrix in which any row has all 0 entries but one, which is equal to 1) such that Ψ(y) =
y1 y2
=
Ψ1 (y) H2 y
(36)
is a local diffeomorphism at y ◦ in IRp . Choose a neighborhood U ◦ of x◦ and a function Φ1 : U ◦ → IRn1 such that Ω = span{dΦ1 }
at any point of U ◦ . Then, there exists a function Φ3 : U ◦ → IRn−n1 −n2 such that
Φ1 (x) x1 H = Φ(x) = x 2 h(x) 2 Φ3 (x) x3
(37)
is a local diffeomorphism at x◦ in X. In the new local coordinates defined by (36)–(37), system (4) is described by equations of the form x˙ 1 x˙ 2 x˙ 3 y1 y2
= = = = =
f1 (x1 , x2 ) + g1 (x1 , x2 )u f2 (x1 , x2 , x3 ) + g2 (x1 , x2 , x3 )u f3 (x1 , x2 , x3 ) + g3 (x1 , x2 , x3 )u h1 (x1 ) x2 .
(38)
If p(x) is a vector field in the annihilator of Ω, then, trivially, in the new coordinates this vector field is expressed in the form T T (39) ( 0 pT 2 (x1 , x2 , x3 ) p3 (x1 , x2 , x3 ) ) . Proof: See [5]. The derivation of the special form (38) in the previous proposition was, actually, based only on the property that Ω is conditioned invariant. However, exploiting the fact that Ω is an observability codistribution, one can reach more interesting conclusions. As a matter of fact, the x1 -subsystem of (38), in which x2 can be identified with y2 (which of course is possible by virtue the special choice of the new coordinates) and viewed as an “independent input”, namely system x˙ 1 = f1 (x1 , y2 ) + g1 (x1 , y2 )u y1 = h(x1 )
(40)
can be shown, under a convenient additional assumption, to be locally weakly observable, in the sense of the classical definition of [12].
15
To this end, suppose that the hypotheses of Proposition 5 hold and (locally) change coordinates, in the state and output spaces, as described in this Proposition. This yields a system, of the form (4), in which, for i = 0, . . . , m, g1i (x1 , x2 ) gi (˜ x) = g2i (x1 , x2 , x3 ) g3i (x1 , x2 , x3 ) and
x) = h(˜
h1 (x1 ) x2
,
where x˜ = Φ(x). Define, for i = 0, . . . , m, and set
so that
ϕi (x1 , x2 ) = g1i (x1 , x2 ) − g1i (x1 , 0)
g1i (x1 , 0) g2i (x1 , x2 , x3 ) gi∗ (˜ x) = , g3i (x1 , x2 , x3 )
ϕi (x1 , x2 ) 0 ψi (˜ x) = , 0
x) . x) + ψi (˜ gi (˜ x) = gi∗ (˜ Assumption II. For every vector field τ ∈ Ker{dh} and every covector field ω ∈ Ω, < [ψi , τ ], ω >= 0. / Remark. Note that this assumption is trivially satisfied when ϕi (x1 , x2 ) is independent of x1 (i.e. when gi1 (x1 , x2 ) is the sum of a function of x1 and a function of x2 ). In fact, in this case, since Ker{dh} ⊂ span{
∂ ∂ , }, ∂x1 ∂x3
the vector field [ψi , τ ] is trivially zero. In particular, the assumption is satisfied if the gi (x)’s are linear vector fields. / Then, the following holds. Proposition 6: Consider system (4). Let Ω be an observability codistribution satisfying the hypotheses of Proposition 5 and choose a change of coordinates yielding the special form (38). Suppose also that the Assumption II above is satisfied. Then, the subsystem x˙ 1 = f1 (x1 , 0) + g1 (x1 , 0)u y1 = h1 (x1 ) ,
(41)
is locally weakly observable. Proof: See [5]. Remark. We recall (see again [12]) that the property of local weak observability implies that any point in the state space can be distinguished from any other point in a neighborhood, in the standard sense of state distinguishability (two states are distinguishable if they induce different input-output maps). Thus, if such a property holds for (41), it indeed holds – a fortiori – for the subsystem x˙ 1 = f1 (x1 , y2 ) + g1 (x1 , y2 )u y1 = h1 (x1 ) , viewed as a system with inputs u, y2 and output y1 . /
(42)
16
The result derived above carries very useful consequences for the solution of the problem of residual generation. To see why this is the case, consider a system of the form (1), determine the codistribution o.c.a.((ΣP∗ )⊥ ), which – as shown before in Proposition 2 – is the largest observability codistribution locally spanned by exact differentials and contained in P ⊥ , and suppose the condition (span{`})⊥ + o.c.a.((ΣP∗ )⊥ ) = T ∗ X ,
(43)
of which we have shown the necessity in the previous section, holds (note, trivially, that the maximality of o.c.a.((Σ∗P )⊥ ) maximizes the chances of having (43) fulfilled). Then, just changing the coordinates in the way described above, we see that the first set y1 of output variables is actually the output of a system of the form x˙ 1 = f1 (x1 , y2 ) + g1 (x1 , y2 )u + `1 (x1 , x2 , x3 )m1 y1 = h1 (x1 ) , which turns out to be locally weakly observable (viewing u and y2 as inputs) and in which the vector field `1 (x1 , x2 , x3 ) is by assumption nonzero. Hence, the occurrence of a fault m1 can be detected by appropriately processing the signals y1 , y2 and u. For example, in case system (1) is a linear system, if L ∩ U∗P = {0} (compare with Theorem 1) the decomposition described above yields x˙ 1 = A11 x1 + A12 y2 + B1 u + L1 m1 y1 = C1 x1 ,
(44)
in which L1 6= 0 and the pair (A11 , C1 ) is observable. To design a residual generator, set xˆ˙ = A11 xˆ + A12 y2 + B1 u + G(C1 xˆ − y1 ) r = C1 xˆ − y1 ,
(45)
with G such that the eigenvalues of (A11 + GC1 ) have negative real part. A trivial calculation, based on the observation that the difference e = x1 − xˆ satisfies e˙ = (A11 + GC1 )e + L1 m1 r = C1 e , shows that (45) solves the fundamental problem of residual generation. B. Example: Actuator fault detection for a point-mass satellite model As a simple application of the techniques illustrated in the latter subsection, we want to design a detection filter for revealing the occurrence of a fault on the tangential thrust in a point-mass planar model of a satellite. The model in question is provided by the following set of differential equations (see e.g. [3]) ρ˙ = v 1 v˙ = ρω 2 − θ1 2 + θ2 u1 + w ρ φ˙ = ω u2 m 2vω + θ2 + θ2 , ω˙ = − ρ ρ ρ in which (ρ, φ) denotes the position of the satellite in polar coordinates on the plane, v is the radial velocity, ω is the angular velocity and u1 , u2 are the radial and, respectively, tangential thrust, m is the fault signal (note, for instance, that m = −u2 models a total failure of the tangential thrust) and w represents the disturbance signal. The parameters θ1 , θ2 are supposed to be known, constant
17
and different from zero. Note that the disturbance w can make the detection of the fault m more difficult due to the presence of the coupling term 2vω in the equation of the angular acceleration ω. ˙ ρ The following quantities are assumed to be available for measurements ν1 = ρ,
ν2 = φ,
ν3 = ω .
In this example, the disturbance vector field is represented by the vector p = ( 0 1 0 0 )T . Using the algorithm (14), the minimal conditioned invariant distribution containing span{p} is found to be: 2ω T ) }. ΣP∗ = span{( 0 1 0 0 )T , ( 1 0 0 − ρ Note that ΣP∗ is smooth, nonsingular and involutive for any ρ = 6 0. Then, its annihilator (ΣP∗ )⊥ is locally spanned by exact differentials and can be expressed as follows (ΣP∗ )⊥ = span{dφ, d(ρ2 ω)} . We can now proceed to the computation of the maximal observability codistribution which is locally spanned by exact differentials and contained in P ⊥ (i.e. o.c.a.((ΣP∗ )⊥ ) via the algorithm (17). To this end, it is enough to note that span{dh} ⊃ (ΣP∗ )⊥ to conclude that o.c.a.((ΣP∗ )⊥ ) = (Σ∗P )⊥ . The change of coordinates in the state space induced by o.c.a.((ΣP∗ )⊥ ) is given by x1 =
!
φ ρ2 ω
,
x2 = ρ,
x3 = v .
As far as the change of coordinates in the output space is concerned, we observe that o.c.a.((Σ∗P )⊥ ) ∩ span{dh} = o.c.a.((ΣP∗ )⊥ ) = span{dy2 , d(y12 y3 )} . Hence the change of coordinates in the output space is given by y1 =
ν2 ν12 ν3
!
y2 = ν 1 .
,
In the new coordinates, the system is re-written as:
x˙ 11 x˙ 12
!
x˙ 2 x˙ 3
y11 y12
!
x12 x22 = θ2 x2 u2 + θ2 x2 m = x3 θ1 θ2 = 23 − 3 + θ2 u1 + w x2 x2
=
x11 x12
!
,
y2 = x2 .
(46)
18
Consider now a candidate residual generator of the form ξ˙ = θ2 y2 u2 + k(y12 − ξ) r = y12 − ξ ,
(47)
with k a positive real number. To see that (47) is actually a residual generator, it suffices to observe that the difference e = x12 − ξ satisfies e˙ = −ke + θ2 y2 m r = e, θ2 and hence, r(t) is the response of a linear filter with transfer function s+k to the input y2 (t)m(t). In this particular example, it is even possible to identify the value of m, if the latter is constant. To this end, consider the system ξ˙ = θ2 y2 u2 + k(y12 − ξ) + θ2 y2 m ˆ (48) ˙ m ˆ = λθ2 y2 (y12 − ξ) ,
with k, λ positive real numbers. The differences e = x12 − ξ and m ˆ satisfy ˜ =m−m ˜ e˙ = −ke + θ2 y2 m ˙ m ˜ = −λθ2 y2 e . Since in the present case, the function y2 = ρ remains always positive and away from zero, it is possible to apply the Persistency of Excitation Lemma which yields the globally exponentially stability of the equilibrium ξ = x12 and m ˆ = m, thus implying, in particular, the convergence of the estimate m ˆ to the true value of m. Fig. 1 shows a simulation corresponding to the following scenario: the actuators u1 , u2 are supposed to provide constant thrust equal to u¯1 and u¯2 = −1, respectively. At time t = 10 a total failure of the tangential actuator u2 occurs (Fig. 1 (a)). Moreover, a disturbance signal w is present (Fig. 1 (b)). From the observed variables (Fig. 1 (d,e,f)) no hint on the occurrence of the fault can be obtained. On the other hand, the output of the residual generator (Fig. 1 (c)) clearly shows the occurrence of the fault m and identify its actual value. C. The design of filters for fault detection To conclude the paper, we show in this section how to exploit the decomposition described in section V-A for the purpose of designing a residual generator in the case of a nonlinear system. For the sake of simplicity, we consider the case in which y1 is one-dimensional. Extension to the case in which y1 is a vector requires appropriate modification of the Assumption III below. Motivated by the previous discussion, we focus our attention on the system x˙ 1 = f1 (x1 , y2 ) + g1 (x1 , y2 )u y1 = h1 (x1 ) ,
(49)
viewed as a system with inputs (u, y2 ) and output y1 and we address the issue of designing an asymptotic observer for x1 . We have already shown that, under mild hypotheses, such a system is guaranteed to be locally weakly observable by construction, but we also know that this property may not suffice to the purposes of designing an asymptotic observer. To this end, a stronger observability property, introduced in [10], has to be assumed. Assumption III (Uniform Complete Observability). The maps f1 , g1 , h1 in (49) are analytic and globally defined. Moreover, there exists a globally defined analytic change of coordinates H : x1 7→ ξ
19 (a)
(d)
1.5
3
1
2
0.5
1
0
0
10
20
30
40
0
50
0
10
20
(b)
30
40
50
30
40
50
30
40
50
(e) 6
2 1
4
0 2
−1 −2 0
10
20
30
40
0
50
0
10
20 (f)
(c) 1.5
2 1.5
1
1 0.5 0
0.5 0
10
20
30
40
0
50
0
10
20
Fig. 1.
that transforms system (49) into a system of the form ξ˙1 = φ1 (ξ1 , ξ2 , y2 , u) ξ˙2 = φ2 (ξ1 , ξ2 , ξ3 , y2 , u) ··· ξ˙n˜ = φn˜ (ξ1 , ξ2 , . . . , ξn˜ , y2 , u) y1 = h0 (ξ1 ) and
(50)
∂h0 ∂φ1 ∂φn˜ −1 = 6 0, = 6 0, . . . , 6= 0 ∂ξ1 ∂ξ2 ∂ξn˜
for all ξ, y2 , u. Assumption IV (Boundedness). ku(·)k∞ < ∞ and kw(·)k∞ < ∞. Moreover, for each M0 > 0 and N0 > 0, there exists M > 0 such that, if kx(0)k ≤ M0 , ku(·)k∞ < N0 and kw(·)k∞ < N0 , then max{kξ(t)k, ky2 (t)k, ku(t)k} ≤ M for all t ≥ 0.
It can be shown (see [9]) that, for any choice of M > 0, there exist functions γ0 (ξ1 ), ϕ1 (ξ1 , ξ2 , y2 , u), ϕ2 (ξ1 , ξ2 , ξ3 , y2 , u), . . . , ϕn˜ (ξ1 , ξ2 , . . . , ξn˜ , y2 , u)
(51)
with the following properties: (i) γ0 (ξ1 ) agrees with h0 (ξ1 ) and, for each i = 1, . . . , n ˜ , ϕi (ξ1 , . . . , ξi+1 , y2 , u) agrees with fi (ξ1 , . . . , ξi , ξi+1 , y2 , u) on the compact set C = {(ξ, y2 , u) : kξk ≤ M, ky2 k ≤ M, kuk ≤ M } , (ii) each of the functions ϕi (ξ1 , . . . , ξi+1 , y2 , u), i = 1, . . . , n ˜ , is globally Lipschitz with respect to xi := col(ξ1 , . . . , ξi ), uniformly in ξi+1 , y2 , u,
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(iii) there exist real numbers α, β, 0 < α < β such that ∂γ 0
α ≤
∂ξ1
≤ β,
α≤
∂ϕi ≤ β, i = 1, . . . , n ˜−1. ∂ξi+1
Let now M0 , N0 be any pair of positive numbers and let M be such that the condition in Assumption IV holds. Accordingly, fix the functions (51) and consider the system ξ˙1 = ϕ1 (ξ1 , ξ2 , y2 , u) ξ˙2 = ϕ2 (ξ1 , ξ2 , ξ3 , y2 , u) ··· ˙ξn˜ = ϕn˜ (ξ1 , ξ2 , . . . , ξn˜ , y2 , u) y1 = γ0 (ξ1 ) .
(52)
Clearly, if kx(0)k ≤ M0 and ku(·)k∞ ≤ N0 , kw(·)k∞ < N0 the state responses of (50) and of (52) agree for all t ≥ 0. Hence, to observe the state of (50) is the same thing as to observe the state of (52). Motivated by the results of Gauthier-Kupca [10], we consider a residual generator of the form x1 , xˆ2 , y2 , u) + Gk1 (y1 − γ0 (ˆ x1 )) xˆ˙ 1 = ϕ1 (ˆ x1 )) x1 , xˆ2 , xˆ3 , y2 , u) + G2 k2 (y1 − γ0 (ˆ xˆ˙ 2 = ϕ2 (ˆ ··· x1 )) x1 , xˆ2 , . . . , xˆn˜ , y2 , u) + Gn˜ kn˜ (y1 − γ0 (ˆ xˆ˙ n˜ = ϕn˜ (ˆ r = y1 − γ0 (ˆ x1 ) ,
(53)
in which G is a (sufficiently large) positive number and k1 , . . . , kn˜ are numbers depending on the parameters α and β in condition (iii) above, determined according to the following result of [9]. Lemma 3: Consider the pair of matrices
0 0 a2 (t) 0 0 a 3 (t) A(t) = · · · 0 0 0 0 0 0
0 ··· ··· 0 ··· · , · · · an˜ (t) ··· 0
C(t) = ( a1 (t) 0 · · · 0 )
and suppose that, for some pair of real numbers α, β, 0 < α < β, α < ai (t) < β,
i = 1, . . . , n ˜.
Then, there exist a vector K ∈ IRn˜ and a symmetric positive definite matrix S satisfying (A(t) − KC(t))T S + S(A(t) − KC(t)) ≤ −I . The following fundamental result, due to Gauthier-Kupca [10], [9], shows that the state of system (53) globally exponentially tracks the state of system (52). Theorem 2: For any a > 0 there is G > 1 (large enough) such that, for all pairs (ξ ◦ , xˆ◦ ) ∈ IRn˜ × IRn˜ , the responses ξ(t) of (52) with initial condition ξ(0) = ξ ◦ and xˆ(t) of (53) with initial condition xˆ(0) = xˆ◦ satisfy x(t) − ξ(t)k ≤ κ(a)e−at kˆ kˆ x(0) − ξ(0)k ˜. for some polynomial function κ(·) of degree n From this, it is immediate to conclude that system (53) is a residual generator for system (49).
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Corollary 1: Consider system (49). For any pair M0 , N0 there is a residual generator of the form (53) and a number G > 1 (large enough) such that if kx(0)k ≤ M0 , ku(·)k∞ < N0 , kw(·)k∞ < N0 and m1 (t) = 0 for all t ≥ 0, then limt→∞ r(t) = 0. The “sensitivity” of the residual generated by the filter (53) with respect to the fault is a straightforward consequence of the observability property of the x1 -subsystem and of the property that the fault vector field is not identically zero. VI. Conclusions In this paper we have studied the problem of fault detection and isolation for nonlinear systems. Starting with a geometric characterization of the problem, we have derived under a mild hypothesis a basic necessary condition, which is expressed in terms of unobservability distributions. The fulfillment of this condition implies the existence of diffeomorphisms in the output space and the state space which highlight a special internal structure, particularly suitable for designing a residual generator which solves the problem of fault detection. The most significant feature of this structure is the existence of a disturbance-decoupled and “observable” subsystem which is affected by the fault. In the case of linear systems, this immediately leads to a solution of the problem, since for such a subsystem a standard observer can be used as a residual generator. In the case of nonlinear systems, the subsystem in question is guaranteed to be locally-weakly observable. If it is also “completely uniformly observable”, using some recent results from the theory of nonlinear observers, we have been able to construct an observer which plays the role of a residual generator. The extension to the case of multiple concurrent faults, although straightforward, has been briefly discussed. References [1] R.N. Banavar, J.L. Speyer, A linear-quadratic game approach to estimation and smoothing, in Proc. Amer. Contr. Conf., Boston, MA, 2818-2822, 1991. [2] R. V. Beard, Failure accomodation in linear systems through self-reorganization, Ph.D. dissertation, M.I.T., 1971. [3] R.W. Brockett, Finite-dimensional linear systems, New York, Wiley, 1970. [4] W.H. Chung, J.L. Speyer, A game theoretic fault detection filter, IEEE Trans. Aut. Contr., 43, 875-880, 1998. [5] C. De Persis, A. Isidori, On the observability codistributions of a nonlinear system, Syst. & Contr. Letters, 40, 297-304, 2000. [6] C. De Persis, A. Isidori, An addendum to the discussion on the paper “Fault detection and isolation for state affine systems”, European Journal of Control, 6, 2000. [7] P. M. Frank. Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy - a survey. Automatica, 26:459–474, 1990. [8] P. M. Frank. On-line fault detection in uncertain nonlinear systems using diagnostic observers: a survey. Int. J. Systems Sci., 25(12):2129–2154, 1994. [9] J.P. Gauthier and I.A.K. Kupca, Deterministic Observation Theory and Applications, Cambridge University Press, 2000 (to appear). [10] J.P. Gauthier and I.A.K. Kupca, Observability and observers for nonlinear systems, SIAM J. Contr. Optimiz., 32, pp. 975-994, 1994. [11] H. Hammouri, M. Kinnaert, E.H. El Yaagoubi, Observer-based approach to fault detection and isolation for nonlinear systems, IEEE Trans. Aut. Contr., 44, 1879-1884, 1999. [12] R. Hermann, A.J. Krener, Nonlinear controllability and observability, IEEE Trans. Aut. Contr., AC-22, 728-740, 1977. [13] A. Isidori, Nonlinear Control Systems, Springer Verlag, Third Edition, London, 1995. [14] A. Isidori, A.J. Krener, C. Gori-Giorgi, S. Monaco, Nonlinear decoupling via feedback: a differential geometric approach, IEEE Trans. Aut. Contr., AC-26, 331-345, 1981. [15] H. L. Jones, Failure detection in linear systems, Ph.D. dissertation, M.I.T., 1973. [16] A.J. Krener, Conditioned invariant and locally conditioned invariant distributions, Sys. Contr. Lett., 8, 69-74, 1986. [17] M.-A. Massoumnia, A geometric approach to the synthesis of failure detection filters, IEEE Trans. Aut. Contr., AC-31, 839-846, 1986. [18] M.-A. Massoumnia, G.C. Verghese, A.S. Willsky, Failure detection and identification, IEEE Trans. Aut. Contr., AC-34, 316-321, 1989. [19] A. S. Morse, W.M. Wonham, Decoupling and pole assignment by dynamic compensation, SIAM J. Control, 8, 317-337, 1970. [20] A. S. Morse, W.M. Wonham, Status of noninteracting control, IEEE Trans. Aut. Contr., AC-16, 568-581, 1971. [21] H. Nijmejier, A.J. van der Schaft, Nonlinear Dynamical Control Systems, Springer Verlag, 1990. [22] R. Seliger, P.M. Frank, Fault diagnosis by disturbance-decoupled nonlinear observers, 30th IEEE Conf. Dec. Contr., 22482253, 1991. [23] J.C. Willems, C. Commault, Disturbance decoupling by measurement feedback with stability or pole placement, SIAM J. Control, 19 (1981) 490-504. [24] W. M. Wohnam, Linear Multivariable Control: A geometric approach, Springer Verlag, 1985.
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