An addendum to the discussion on the paper ... - Semantic Scholar

An addendum to the discussion on the paper “Fault Detection and Isolation for State Affine Systems” ∗ C. De Persis [ , A. Isidori

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Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130. ‡ Dipartimento di Informatica e Sistemistica, Universit`a di Roma “La Sapienza”, 00184 Rome, ITALY. Abstract In this note we show how to test whether or not certain algebraic equations derived in the paper Fault Detection and Isolation for State Affine Systems, by H. Hammouri, M. Kinnaert, E.H. El Yaagoubi, are solvable and to describe a solution method. Test and solution rely upon elementary linear algebraic techniques.

Keywords: Fault detection and isolation, State affine system, Observability subspace

1

Introduction

The purpose of this note is to further contribute to the discussion that followed the paper [3], and – specifically – to point out a simple solution to a design problem posed in that paper, within the context of the synthesis of a filter for fault detection and isolation for systems modeled by equations of the form x˙ = A(u)x + ψ(u, y) + e1 (x)ν1 + e2 (x)ν2 y = Cx

(1)

in which x ∈ IRn , u ∈ U ⊂ IRm , νi ∈ IR, i = 1, 2, y ∈ IRp , A(u) is a matrix of analytic functions, ψ(u, y), e1 (x) and e2 (x) are smooth maps, and C is a fixed matrix. These systems are said to be state-affine up to output injection. ∗

Research supported in part by ONR under grant N00014-99-1-0697, by AFOSR under grant F49620-95-1-0232, by ESF under program “Control of Complex Systems” and by MURST.

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It was shown in [3] that the so-called fundamental problem of residual generation for this kind of systems can be solved provided there is a solution to the following problem: find an integer p˜ < p, a constant p˜ × p matrix L and an n × p matrix D(u) of analytic functions such that the subspace V, defined as the largest subspace of Ker{LC} which is invariant under A(u) + D(u)C for all u ∈ U , satisfies: (i) (ii)

e1 (x) 6∈ V e2 (x) ∈ V ,

(2)

for all x (see Theorem 3 of [3], where V was denoted as ∆(L)). If matrices L and D(u) are known such that the conditions thus stated are fulfilled, then it is relatively straightforward to design a filter for the generation of the residual, which essentially exploits the property that, by construction, the “reduction” of x˙ = (A(u) + D(u)C)x + e1 (x)ν1 + e2 (x)ν2 y¯ = LCx ,

(3)

to the quotient IRn /V is independent of ν2 and is observable when ν1 = 0 (see again [3] for further details). Note also that (3) is constructed from (1) via output injection and (singular) change of coordinates in the output space. It must be stressed, however, that the existence conditions thus stated (namely the existence of L and D(u) having the properties indicated above) are not directly verifiable as such. They can be converted in algebraic equations involving L, D(u) and the system data A(u), C, e1 (x), e2 (x) but these, as also observed in [3], turn out to be nonlinear equations. The discussion that follows the paper, also, seems to suggest that the problem of determining this pair L and D(u) requires an advanced mathematical background. The purpose of this note is to show, on the contrary, that the fulfillment of the conditions in question (derived, as they are in [3], within the appropriate geometric approach) reduces to an easily verifiable test, which involves just linear-algebraic manipulations, and that if such test is passed, it is straightforward to construct L and D(u). An extension of some of these results presented here to more general classes of nonlinear system can be found in [2].

2

The Algorithms

The basic idea on which the result presented in [3] and summarized above reposes is the idea of using the output injection D(u)y − ψ(u, y), and the (singular) change of coordinates y¯ = Ly in the output space, in order to create an unobservable subspace which contains e2 (x) and does not contain e1 (x). For a linear system the solution to the corresponding problem is well-known since the work [5] and is based on the 2

notion of unobservability subspace (see also [4]). 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 [6] [1]. Specifically, dualizing definition (5.1.1) of [6], a subspace V ⊂ IRn is said to be an unobservability subspace of the pair (A, C) if there exist matrices H and G such that V is the set of unobservable states of the system x˙ = (A + GC)x,

y = HCx .

The unobservability subspaces have a number of interesting of properties. Among them we recall that 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. This minimal element can be determined with the help of simple recursive algorithms, dual versions of the algorithms needed to compute the maximal controllability subspace contained in a given subspace C = Ker{C} (see [6, Chapter 5]) It is not difficult to extend the algorithms in question to the case of systems modeled by equations of the form (1). • Set

P = span{e2 (x) : x ∈ IRn } .

(4)

• Consider the non-decreasing sequence of subspaces of IRn S0 = P X A(u)(Si ∩ Ker{C}) Si+1 = Si +

(5)

u∈U

with i = 0, . . . , n − 1, and set S∗P = Sn−1 .

1

• Consider the non-decreasing sequence of subspaces of the dual space of IRn Q0 = (S∗P )⊥ ∩ span{C} X Qi A(u) + span{C}) Qi+1 = (S∗P )⊥ ∩ (

(6)

u∈U

1

with i = 0, . . . , n − 1, and set QP∗ = Qn−1 .

2

Note that, in this and in the following algorithm, even though the sum appearing in the recursive formula extends in principle over an infinite index set, the summands are actually subspaces of IRn , thus a finite number of them is needed to determine the next subspace of the sequence. 2 Here, the notation span{C} denotes the subspace, of the dual space of IRn , spanned by the rows of the matrix C, and QA stands for the subspace consisting of all covectors of the form wA, where w is a covector in Q.

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In what follows, we will prove the following fact. Proposition 1 Consider the system (1). The problem indicated in the Introduction is solvable if and only if e1 (x) 6∈ (Q∗P )⊥ . Remark. The proof of the sufficiency part of this result is constructive, i.e. contains the explicit derivation of a pair L and D(u) having the required properties. / To prove Proposition 1 we need some properties of the two algorithms described above. These properties are stated hereafter without proof, since they consists in simple adaptations of the arguments which are used, e.g. in [6, Chapter 5], to derive corresponding properties for the controllability subspaces of a linear system. As far as the algorithm (5) is concerned, the property of interest is the following one. Lemma 1 The subspace (S∗P )⊥ is the maximal element (with respect to subspace inclusion) of the family of all subspaces of the dual space of IRn which satisfy Q ⊂ P⊥ QA(u) ⊂ Q + span{C} for all u . To illustrate a relevant feature of the second algorithm, is it is convenient to introduce some terminology. Let Q be a fixed subspace of the dual space of IRn , consider the sequence of subspaces of Q defined as Q0 = Q ∩ span{C} X Qi+1 = Q ∩ ( Qi A(u) + span{C}) ,

(7)

u∈U

and denote the last n-th element of the sequence, namely, Qn−1 , by o.s.a.(Q) (here, “o.s.a.” stands for “observability subspace algorithm”). Then, we say that the subspace Q passes the o.s.a. test if 3 o.s.a.(Q) = Q . Then, the following result holds. Lemma 2 The subspace QP∗ is the maximal element (with respect to subspace inclusion) of the family of all subspaces of the dual space of IRn which satisfy Q ⊂ P⊥ QA(u) ⊂ Q + span{C} for all u Q = o.s.a.(Q), i.e. Q passes the o.s.a. test. 3

From the definition, it only follows that o.s.a.(Q) is a proper subspace of Q.

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Note also that the sequence that defines o.s.a.(Q) is such that Qi+1 = Q ∩ (Qi + and also Qi+1 = Q ∩ (Qi +

X

u∈U

X

u∈U

Qi A(u) + span{C}) , ‘



Qi A(u) + D(u)C + span{C}) ,

(8)

(9)

for any matrix D(u).

3

The (Constructive) Sufficiency

We use now the properties indicated in the two Lemma above to show that is possible to perform a special change of coordinates in the state space, which provides an immediate solution to the problem indicated in the Introduction. Proposition 2 Consider the system (1) and suppose, without loss of generality, that the p rows of matrix C are linearly independent. Let the subspace P be defined as in (4). Compute the subspace Q∗P defined by means of the two algorithms (5) and (6). Assume that e1 (x) 6∈ (QP∗ )⊥ . Let n1 denote the dimension of Q∗P . Let p − n2 denote the dimension of Q∗P ∩ span{C} and choose a (p − n2 ) × p matrix H1 such that span{H1 C} = QP∗ ∩ span{C} . Choose a n2 × p selection matrix H2 (i.e. a matrix in which any row has all 0 entries but one, which is equal to 1) such that H= is nonsingular and define y˜ =

’

y1 y2

“

’

=

H1 H2 ’

“

H1 Cx H2 Cx

Choose an n1 × n matrix T1 such that QP∗ = span{T1 } , an (n − n1 − n2 ) × n matrix T3 such that





T1   T =  H2 C  T3 5

“

.

(10)

is nonsingular (note that such a T3 indeed exists because the first n1 + n2 rows of T are by construction linearly independent), and define 







T1 x x1     x˜ =  x2  =  H2 Cx  . T3 x x3

In the new coordinates thus defined, we have T A(u)T −1 Set





0 A11 (u) A12 (u)  =  A21 (u) A22 (u) A23 (u)  , A31 (u) A32 (u) A33 (u)

HCT −1 =

’

C1 0

0 I

0 0

“

.

(11)





0 −A12 (u)  −1  0 D(u) = T  0 H 0 0

and

L = H1 . The two matrices thus defined are such that the subspace 



0   −1 V = T span{ x2  : x2 ∈ IRn2 , x2 ∈ IRn−n1 −n2 } x3

has the following properties:

• it is the largest subspace of Ker{LC} which is invariant under A(u) + D(u)C, for all u • e1 (x) 6∈ V • e2 (x) ∈ V, for all x. Thus, the pair L and D(u) thus defined solves the problem indicated in the Introduction. Proof. Using Lemma 2, recall that the subspace Q∗P satisfies QP∗ A(u) ⊂ Q∗P + span{C} for all u ∈ U

(12)

and passes the o.s.a. test. By construction, in the new coordinates, the subspace QP∗ is spanned by vectors of the form ( xT 1

0 0) , 6

the subspace Q∗P + span{C} is spanned by vectors of the form ( xT 1

xT 2

0) .

Using the definitions of y1 , y2 , x1 , x2 , we immediately find that HCT −1 has the form indicated (11). Moreover, using (12), we necessarily obtain that, for any x1 ∈ IRn1 , ( xT 1

0 0 ) T A(u)T −1 = ( ∗ ∗ 0 )

This proves the T A(u)T −1 has the the form indicated (11). Indeed, the suggested output injection renders the subspace V = (Q∗P )⊥ invariant under A(u) + D(u)C for all u ∈ U and, trivially, V ⊂ Ker{LC}. It remains to show that V is the largest subspace having these properties. Suppose there is a subspace W having these properties and such that W ⊃ V. Then, by duality, we have that span{LC} = Q∗P ∩ span{C} ⊂ W ⊥ W ⊥ A(u) ⊂ W ⊥ + span{C} for all u ∈ U W ⊥ ⊂ QP∗ .

(13)

Now, let Qi denote the i-th subspace of the sequence that defines o.s.a.(Q∗P ), i.e. of the sequence Q0 = QP∗ ∩ span{C} X Qi+1 = Q∗P ∩ ( Qi A(u) + span{C}) . u∈U

Using the first of (13) we see that W ⊥ ⊃ Q0 and, by induction, using the second of (13) we can prove that W ⊥ ⊃ Qi for all i ≥ 1. In fact Qi = QP∗ ∩ (

X

u∈U

Qi−1 A(u) + span{C}) ⊂ Q∗P ∩ (W ⊥ + span{C}) = W ⊥ + Q0 = W ⊥ .

Thus, W ⊥ ⊃ o.s.a.(Q∗P ). This, since Q∗P passes the o.s.a. test, yields W ⊥ ⊃ Q∗P . This, together with the third of (13) shows that W ⊥ = Q∗P and completes the proof. /

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4

The Necessity

To prove the necessity it suffices to show the following result. Lemma 3 Consider system (1). Pick any n × p matrix D(u) of functions of u and any p˜ × p matrix L. Let V be the largest subspace of Ker{LC} which is invariant under A(u) + D(u)C for all u ∈ U . Set Q = V ⊥ . Then Q satisfies QA(u) ⊂ Q + span{C} for all u

(14)

and passes the o.s.a. test. Proof. By duality, 

‘

A(u) + D(u)C V ⊂ V for all u ∈ U

⇒

V ⊥ A(u) ⊂ V ⊥ + span{C} for all u .

Define the sequence of subspaces (of the dual space of IRn )

O0 = span{LC}  ‘ X Oi A(u) + D(u)C Oi+1 = Oi + u∈U

and observe that, by construction, On−1 = V ⊥ .

Let Qi denote the i-th subspace of the sequence that defines o.s.a.(V ⊥ ), i.e. of the sequence Q0 = (V ⊥ ) ∩ span{C} X Qi+1 = (V ⊥ ) ∩ ( Qi A(u) + span{C}) . u∈U

It is seen that, by construction,

O0 ⊂ Q0

and, by induction, it can be easily proven that Oi ⊂ Qi

for all i ≥ 1. Thus, V ⊥ = On−1 ⊂ Qn−1 = o.s.a.(V ⊥ ). This, together with the fact that, always, o.s.a.(V ⊥ ) ⊂ V ⊥ proves that V ⊥ passes the o.s.a. test. /

Now, suppose there exists a solution pair L and D(u) of the problem indicated in the introduction. Then, by the previous Lemma, there must exists a subspace Q of the dual space of IRn which satisfies (14), passes the o.s.a. test, satisfies Q ⊂ P⊥

and, finally, is such that e1 (x) 6∈ Q⊥ . By Lemma 2, Q is necessarily a subspace of QP∗ . Thus, we conclude that e1 (x) 6∈ (Q∗P )⊥ . 8

References [1] G. Basile, G. Marro, Controlled and conditioned invariance in linear system theory, Prentice Hall, 1992. [2] C. De Persis, A. Isidori, “On the observability codistributions of a nonlinear system,” to appear in Systems and Control Letters, 2000. [3] H. Hammouri, M. Kinnaert, E.H. El Yaagoubi., Fault detection and isolation for state affine systems, European Journal of Control, 4, pp 2-16, 1998. [4] M.A. Massoumnia, “A geometric approach to the Synthesis of Failure Detection Filters,” IEEE Trans. Automatic Control, Vol. AC-31, No. 9, pp. 839-846, 1986. [5] J.C. Willems, C. Commault, “Disturbance Decoupling by Measurement Feedback with Stability or Pole Placement,” SIAM Journal of Control and Optimization, Vol. 19, No. 4, pp. 490-504, July 1981. [6] W.M. Wonham, Linear Multivariable Control: A geometric approach, Springer Verlag, 1985.

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