suboptimal Fault Detection Filter for Bilinear Systems - Semantic Scholar

An H∞ -suboptimal Fault Detection Filter for Bilinear Systems? Claudio De Persis1 and Alberto Isidori1 1

2

2

Department of Systems Science and Mathematics, Washington University, St. Louis, MO 63130, USA Dipartimento di Informatica e Sistemistica, Universit` a di Roma “La Sapienza”, 00184 Rome, ITALY

Abstract. We address the problem of fault detection and isolation in presence of noisy observations for bilinear systems. Our solution rests on results derived in the geometric theory of fault detection.

1

Introduction

We consider in this paper the problem of fault detection and isolation for systems modeled by equations of the form x˙ = Ax +

m X

ui Ni x + `(x)m +

i=1

y = Cx ,

d X i=1

pi (x)wi

(1)

in which x ∈ IRn , y ∈ IRp , m ∈ IR, and P = span{p1 , . . . , pd } is “independent of x”. Note that we say that a distribution ∆ (codistribution Ω) on IRn is independent of x if there is a subspace D of IRn (a subspace W of the dual space of IRn ) such that, in the natural basis {∂/∂x1 , . . . , ∂/∂xn } of the tangent space to IRn at x, ∆(x) = D (in the natural basis {dx1 , . . . , dxn } of the cotangent space to IRn at x, Ω(x) = W). In these cases, it turns out quite useful to identify ∆ with a matrix D whose columns span D (and Ω with a matrix W whose rows span W). Solving the problem of fault detection and isolation for (1) means designing a filter (a residual generator) of the form x x, y)u ˆ˙ = fˆ(ˆ x, y) + gˆ(ˆ ˆ x, y) r = h(ˆ

(2)

? 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 F3060299-2-0551, and by MURST. To appear in: A. Isidori, F. Lamnabhi-Lagarrigue, W. Respondek (Eds.), Nonlinear Control in the Year 2000, Springer Verlag, 2000.

2

Claudio De Persis and Alberto Isidori

such that, in the cascaded system (1)-(2), the residual r(·) depends “nontrivially” on (i.e is affected by) the input m(·), depends “trivially” on (i.e. is decoupled from) the inputs u and w and asymptotically converges to zero whenever m(·) is identically zero (see [4]). Following Theorem 3 of [5], in which the study of the problem of fault detection and isolation for systems of the form x˙ = A(u)x + ψ(u, y) + e1 (x)ν1 + e2 (x)ν2 y = Cx

(3)

(νi ∈ IR, i = 1, 2, and A(u) is a matrix of analytic functions) was initiated, it is known that the problem of fault detection and isolation is solvable provided that a constant matrix L and a matrix D(u) of analytic functions exist 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: (a) e1 (x) 6∈ V and (b) e2 (x) ∈ V for all x. In [2], we introduced the notion of observability codistribution for systems of the form (3) and we proved that the existence of the matrices L, D(u) above can easily be characterized in terms of this notion. Here, we examine some details of this approach for systems of the form (1) and we use the results to address the problem of fault detection in the case in which the measurements are affected by noise v, i.e. y = Cx+v, proposing a filter in which the influence of the noise on the residual is attenuated. Related work on the problem of noise attenuation in fault detection, even though in a different setting, can be found in [1].

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Observability codistributions

The concept of observability codistributions for an input-affine nonlinear system and the related construction algorithms have been introduced in [3] (cf. formulas (8), (12) and Proposition 2). In the case of state-affine systems, the corresponding concept and algorithms are those presented in [2] (cf. formulas (4) and (5)). Bilinear systems (1) can be indifferently seen as a special subclass of the systems dealt with in either [2] or [3]. For these systems, the algorithms in question assume the special form described in what follows. First of all note that, for a system of the form (1), conditioned invariant distributions and observability codistributions are independent of x. Also, for notational convenience, set N0 = A. Given a fixed subspace P of IRn , consider the non-decreasing sequence of subspaces (of IRn ) S0 = P Si+1 = Si +

m X j=0

Nj (Si ∩ Ker{C})

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

(4)

An H∞ -suboptimal Fault Detection Filter for Bilinear Systems

3

Given a fixed subspace Q of the dual space of IRn , consider the nondecreasing sequence of subspaces (of the dual space of IRn ) Q0 = Q ∩ span{C} m X Qi+1 = Q ∩ ( Qi Nj + span{C}),

(5)

j=0

with i = 0, . . . , n − 1, and set o.s.a.(Q) = Qn−1 , where the acronym “o.s.a.” stands for “observability subspace algorithm”. Finally, set ∗ = o.s.a.((S∗P )⊥ ) . QP

The following results describe the main properties of the subspaces thus defined. Lemma 1. Consider the system (1). The subspace S∗P is the minimal element (with respect to subspace inclusion) of the family of all subspaces of IRn which satisfy S⊃P Ni (S ∩ KerC) ⊂ S,

i = 0, . . . , m .

(6)

Remark 1. In the terminology of [3], the distribution ∆ : x 7→ S∗P is the minimal conditioned invariant distribution containing the distribution P : x 7→ P. ∗ 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⊥ QNi ⊂ Q + span{C}, Q = o.s.a.(Q).

i = 0, 1, . . . , m

(7)

Remark 2. In the terminology of [2], [3], a codistribution Q : x 7→ Q fulfilling the last two properties of (7) is an observability codistribution for (1), ∗ and hence the codistribution Q∗ : x 7→ QP is the maximal observability codistribution contained in the codistribution P ⊥ : x 7→ P ⊥ . /

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A natural candidate for residual generation

Specializing the results of [2] or [3] to system (1), we first observe that the following holds. Pm Proposition 1. Consider system (1), set A(u) := A + i=1 Ni ui and P := span{p1 , . . . , pd }. The following conditions are equivalent:

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(i) there exist a constant matrix L and a matrix D(u) of analytic functions such that the largest subspace V contained in Ker{LC} and invariant under A(u) + D(u)C, for all u ∈ IRm , satisfies: (a) `(x) 6∈ V and (b) P ⊂ V. (ii) `(x) ∈ / (Q∗P )⊥ ; (iii) there exists a change of coordinates x ˜ = col(x1 , x2 , x3 ) = T x on the state space and a change of coordinates y˜ = col(y1 , y2 ) = Hy on the output space which transform (1) into a system of the form: x˙ 1 = A11 x1 + A12 x2 + x˙ 2 = x˙ 3 =

3 X

k=1 3 X

m X i=1

A2k xk +

3 m X X

i=1 k=1

A3k xk +

m X 3 X

i=1 k=1

k=1

y1 = C1 x1 y2 = x2 ,

i i (N11 x1 + N12 x2 )ui + `1 (˜ x)m

N2k xk ui + `2 (˜ x)m + p2 (˜ x)w

(8)

x)w x)m + p3 (˜ N3k xk ui + `3 (˜

in which `1 is nonzero, and the subsystem x˙ 1 = A11 x1 +

m X

i ui N11 x1

i=1

y1 = C1 x1 ,

(9)

is observable. Proof. The proof of the various implications can be obtained by specializing the proofs of similar statements given in [2] and [3]. It is worth stressing that, in the implication (ii)⇒(iii), the form (8) can be obtained in the following way: define x1 = T1 x with T1 a matrix whose rows span Q∗P and define y1 = H1 y with H1 a matrix such that the rows of the matrix H1 C span ∗ QP ∩ span{C}. Then, the form (8) derives from the properties Q∗P Ni ⊂ Q∗P + span{C}, i = 0, 1, . . . , m ,

Q∗P ⊂ P ⊥ , `(x) ∈ / (Q∗P )⊥ , and from the fact that the chosen isomorphisms preserve the structure of the system. Note in particular that, since Q∗P is spanned by the rows of T1 , necessarily `1 (x) 6= 0, because otherwise `(x) ∈ (Q∗P )⊥ . The observability of the subsystem (9) derives from the fact that Q∗P is the maximal element of the family of all subspaces of the dual space of IRn which satisfy (7). PmThe implication (iii)⇒(i) is easily obtained by setting D(u) := D0 + i=1 Di ui , with   i 0 −N12 0  H, i = 0, 1, . . . , m, (10) Di = T −1  0 0 0


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and L = H1 .

(11)

In fact, by Proposition 2 in [2], the subspace V := Ker{T1 } is the largest subspace of Ker{LC} which is invariant under A(u) + D(u)C for all u ∈ IRm . Since the disturbance term does not appear in the x1 -subsystem, the inclusion P ⊂ V must hold. On the other hand, since `1 6= 0, the vector field `(x) cannot belong to V for all x. / The condition (i) is precisely the condition assumed in [5], Theorem 3, to establish the existence of a solution of the fundamental problem of residual generation for system (1). In this respect, the equivalences established in the Proposition above provide: a conclusive test – condition (ii) – to determine whether or not the subspace V and the pair of matrices D(u), L in (i) ever exist, and a straightforward procedure – based on the change of coordinates described in (iii) – to actually identify this subspace and to construct such matrices. Special form (8), and in particular subsystem (9), leads itself to an immediate interpretation of the construction proposed in [5]. As a matter of fact, it is easy to check that, if D(u), L are chosen as indicated in the proof of the Proposition above, the residual generator of [5] reduces to a system of the form x ˆ˙ = A11 x ˆ+

m X i=1

r = y1 − C1 x ˆ,

i x ˆ + A12 y2 + ui N11

m X i=1

i ui N12 y2 − GC1T (C1 x ˆ − y1 )

(12)

in which G is a suitable matrix of functions of time. Indeed, the latter is nothing else than an observer for x1 , yielding an “observation error” e = x1 −ˆ x obeying e˙ = (A11 +

m X i=1

r = C1 e .

i ui N11 − GC1T C1 )e + `1 (˜ x)m

(13)

If G is chosen, as suggested in [5], in such a way that the latter is exponentially stable for m = 0, then (12) solves the problem of residual generation.

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Fault detection with noisy observations

In the approach to residual generation summarized in the previous section the observed output y that feeds the residual generator is supposed to be noise-free. If this is not the case, as in any realistic setup, then the important problem of attenuating the effect that this noise may have on the residual r must be addressed. Following the fundamental results of [1], the problem

6


of noise attenuation in the design of residual generators can be cast as a problem of state estimation in a game-theoretic, or H∞ , setting. To this end, the special features of form (8) prove to be particularly helpful. Consider the case in which the observed output of system (1) is corrupted by a measurement noise v, i.e. x˙ = Ax +

m X

ui Ni x + `(x)m +

d X

pi (x)wi

(14)

i=1

i=1

y = Cx + v ,

and let v(·) be a function in L2 [0, t1 ), where [0, t1 ) is a fixed time interval (with t1 ≤ ∞). Proposition 1 shows, among the other things, that in the special new coordinates used to obtain (8) we can write x˙ 1 = A11 x1 + A12 x2 +

m X i i x2 )ui + `1 (˜ x)m x1 + N21 (N11 i=1

y1 = C1 x1 + v1 y2 = x2 + v2 .

(15)

Ideally, in the absence of the fault signal (the input m to (15)), the output of the residual residual generator must be identically zero, in spite of a possible nonzero disturbance w (whose role in (15) is taken by the input x2 ) and of the observation noise. In practice, in view of the way the noise affects the observations, both x2 and v are going to influence the residual when m = 0. As a matter of fact, if the second component y2 of the observed output is corrupted by noise, it is no longer possible to compensate – via output injection – the effect of x2 on the dynamics of the observation error, as it was done in the residual generator (12). Thus, in this case, the problem becomes a problem of attenuating the effect of both x2 and v on the residual signal, which for instance – as in [1] – can be cast in a game-theoretic setting. Motivated by the approach of [1], we address in what follows the problem of finding a filter with state variable x ˆ, driven by the observed outputs (y1 , y2 ), such that, for some fixed positive number γ and a given choice of weighting positive definite matrices Q, M, V, P0 , R t1 kC1 (x1 − x ˆ)k2Q dt 0 ≤ γ2 , (16) R t1 2 2 2 [kv k + kx − x k + kv ]dt (0) ˆ k 2 M −1 1 0 P0 1 V −1 0

for all the signals v1 and v2 and for all the initial conditions x1 (0), subject to the constraints x˙ 1 = A1 x1 + A2 x2 y1 = C1 x1 + v1 y2 = x2 + v2 , in which, we have set A1 := A11 + Pmfor convenience, i A12 + i=1 N12 ui .

(17) (18) (19) Pm

i=1

i ui and A2 := N11


To solve this problem, we consider the cost function Rt 2 2 J = 0 1 [kC1 (x1 − x ˆ)k2Q − γ 2 (ky2 − x2 kM −1 + ky1 − C1 x1 kV −1 )]dt −kx1 (0) − x ˆ0 k2Π0

7

(20)

(where Π0 := γ 2 P0 ) and the differential game min max max max J ≤ 0 x ˆ

(y1 ,y2 ) x2

x1 (0)

subject to the differential constraint (17). Standard methods show that, if the differential Riccati equation Π˙ + AT1 Π + ΠA1 + γ −2 ΠA2 M AT2 Π + C1T (Q − γ 2 V −1 )C1 = 0 ,

(21)

with initial condition Π(0) = Π0 , has a solution Π(t) defined and nonsingular for all t ∈ [0, t1 ], then the attenuation requirement (16) is fulfilled by an estimate x ˆ of x1 provided by the estimator ˆ), ˆ˙ = A1 x ˆ + A2 y2 + γ 2 Π −1 C1T V −1 (y1 − C1 x x

ˆ(0) = x ˆ0 . x

(22)

Comparing the dynamics of the estimator (22) with that of the residual generator (12) and bearing in mind the expressions of A1 and A2 , we see that (22) corresponds to choosing in (12) a “gain matrix” G of the form G = γ 2 Π −1 C1T V −1 , where Π is the solution of the differential Riccati equation (21). Indeed, this particular choice guarantees the attenuation properties expressed by (16). It is interesting to examine the special case in which t1 = ∞. Note, to ˆ obeys this end, that the estimation error e = x1 − x

e˙ = (A1 − γ 2 Π −1 C1T V −1 C1 )e + `1 (˜ x)m − A2 v2 − γ 2 Π −1 C1T V −1 v1 .(23)

In order to (16) make sense for t1 = ∞, we require e(·) to be in L2 [0, ∞). Assume that u(t) and x(t) are bounded on [0, ∞) and m(·) is in L2 [0, ∞). If also Π −1 (t) is bounded on [0, ∞), the “input” `1 (˜ x)m − A2 v2 − γ 2 Π −1 C1T V −1 v1

to (23) is in L2 [0, ∞) and hence e(·) is in L2 [0, ∞) if the homogeneous linear system e˙ = (A1 − γ 2 Π −1 C1T V −1 C1 )e

(24)

is exponentially stable. Conditions implying the exponential stability of the latter can be derived from the works [6] and [7]. In fact, recall that if Π(t) is defined and nonsingular for all t ∈ [0, ∞) the matrix P (t) := γ 2 Π −1 (t) satisfies the differential Riccati equation P˙ = AP + P AT + BB T + P [γ −2 LT L − C T C]P

(25)

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in which A := A1 ,

B := A2 M 1/2 ,

C := V −1/2 C1 ,

L := Q1/2 C1 .

The following result of [7] provides the requested condition. Lemma 3. Let P (t) be a symmetric solution of (25) satisfying 0 < β1 I ≤ P (t) ≤ β2 I for all t ∈ [0, ∞), for some β1 , β2 , and such that the system p˙ = (A + P [γ −2 LT L − C T C])p

is exponentially stable. Then, the system e˙ = (A − P C T C)e

(26)

is exponentially stable. In fact, by definition of A, C, P , system (26) coincides with the homogeneous linear system (24) and, hence, the hypotheses of the previous lemma guarantee that the estimator (22) renders (16) fulfilled over the infinite horizon.

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Conclusion

We have proposed, within a geometric framework, a solution to the problem of fault detection and isolation for systems of the form (1) in the presence of noisy observations.

References 1. Chung W.C., Speyer J. L. (1998) A game theoretic fault detection filter. IEEE Transactions on Automatic Control 43, 143-161 2. De Persis C., Isidori A. (2000) An addendum to the discussion on the paper “Fault Detection and Isolation for State Affine Systems”. European Journal of Control 3. De Persis C., Isidori A. (2000) On the observability codistributions of a nonlinear system. Systems & Control Letters 4. De Persis C., Isidori A. (1999) A differential-geometric approach to nonlinear fault detection and isolation. Submitted 5. Hammouri H., Kinnaert M., El Yaagoubi E.H. (1998) Fault detection and isolation for state affine systems. European Journal of Control 4, 2-16 6. Nagpal K.M., Khargonekar P.P. (1991) Filtering and Smoothing in an H ∞ Setting. IEEE Transactions on Automatic Control 36, 152-166 7. Ravi R., Pascoal A.M., Khargonekar P.P. (1992) Normalized coprime factorizations for linear time-varying systems. Systems & Control Letters 18, 455-465