Data Driven Fault Detection with Robustness to Uncertain Parameters

Data Driven Fault Detection with Robustness to Uncertain Parameters Identified in Closed Loop Jianfei Dong, Michel Verhaegen, and Fredrik Gustafsson Abstract— This paper presents a new robustified data-driven fault detection approach, connected to closed-loop subspace identification. Although data-driven detection methods have recently been reported in the literature, attention has not yet been given to a robust solution coping with identification errors. The key idea of this paper is to analytically quantify the effect of the identification errors on the residual generator of a new data-driven detection approach, namely FICSI. The comparisons of the proposed robust FICSI detection scheme with both its nominal counterpart and the nominal data-driven PSA solutions have verified the effectiveness of accounting the identification errors in improving the performance of the datadriven detection scheme.

I. I NTRODUCTION As model based FDI becoming a well-established literature [1], [2], [3], [4], numerous tools have been developed. With the help of these tools, the main efforts in implementing an FDI scheme may not exist in developing the algorithms themselves, but in modeling the dynamic process. Modeling by first principles is what users can rely on. However, such a modeling is often a time consuming and expensive task, especially for real processes of rather large scale [5], [6]. This has stimulated the development of system identification techniques, e.g. prediction error methods (PEM) [7] and subspace model identification (SMI) methods [8]. Taking SMI as an example, two steps need to be taken. First, either the range space of the extended observability matrix [8] or the unknown state sequence [9] is identified. Then, the state space matrices, e.g. (A, B, C, D), need to be estimated using standard realization theory. In fact, the recent literature shows that the state space matrices are not really needed in designing an FDI, e.g. [10], [11]. Based on SMIs [8], these approaches require computing the left null space of the identified range space of the extended observability matrix, directly identified from measurement data, without realizing (A, B, C, D). The projection of a residual vector onto the left null space aims at annihilating the influence of unknown initial states on the residual, which is known as parity space analysis (PSA) [12], [13], [14]. One of the drawbacks of computing the parity relation from data is that the residuals hence generated are sensitive to the approximation errors attributed to the model reduction step in identifying the range space of the extended observability matrix. We have recently developed in [15] a J. Dong and M. Verhaegen are with Delft Center for Systems and Control, Delft University of Technology, 2628CD, Delft, the Netherlands. [email protected] & [email protected]. F. Gustafsson is with Department of Electrical Engineering, Link¨opings universitet, SE-581 83 Link¨oping, Sweden. [email protected].

data-driven FdI scheme Connected to Subspace Identification (FICSI), whose parameters can be directly identified from data, with neither realizing (A, B, C, D), nor computing the range space of the extended observability matrix together with its left null space. A key step therein is to construct the product of the initial states with the extended observability matrix, from the past input and output (I/O) signals. Despite the large amount of existing literatures in FDI methods, the robustness of a detection scheme against model identification errors has not yet been investigated. The major challenge exists in the quantifying the uncertainties in the residual generator from the parameter identification errors. To overcome this difficulty, we consider the FICSI formulation in this paper, where we shall show that the uncertainties in the residual generator linearly depend on the parameter errors. It is hence possible to develop a robust data-driven fault detection scheme coping with the identification errors. The rest of the paper is organized as follows. In Section II, preliminaries and problem formulation are presented. Section III presents a new data-driven PSA approach connected to closed-loop subspace identification. The difficulty in analyzing the model error effect on its residual generator then motivates developing the robust FICSI solution in Section IV. Section V finally presents a simulation study on a VTOL model, where the nominal and robust solutions are compared. II. P ROBLEM

FORMULATION AND NOTATIONS

A. Predictor based subspace identification of a nominal plant In modern computer controlled systems, a model usually takes the following discrete-time state space form. x(k + 1) = y(k) =

Ax(k) + Bu(k) + F w(k),

(1)

Cx(k) + Du(k) + v(k).

(2)

Here, we consider MIMO systems; i.e. x(k) ∈ Rn , y(k) ∈ Rℓ , and u(k) ∈ Rm . A, B, C, D, F are real bounded matrices with appropriate dimensions. The disturbances are represented by the process noise w(k) ∈ Rnw and the measurement noise v(k) ∈ Rnv . As standard assumptions in system identification, the measurement and the process noise, w(k), v(k), are white zero-mean Gaussian. The pair (C, A) is detectable. There are no uncontrollable modes of 1/2 1/2 1/2 (A, F Qw ) on the unit circle, where Qw · (Qw )T is the covariance matrix of w(k). As long as the input-output transfer function of the system is the main interest, the model (1, 2) can be reformulated in

the following innovation form [8].

by (see more detailed derivation in [16]):

x ˆ(k + 1) = Aˆ x(k) + Bu(k) + Ke(k), y(k) = C x ˆ(k) + Du(k) + e(k).

(3) (4)

Here, K is the Kalman gain. The innovation e(k) is white Gaussian due to the properties of w(k) and v(k). The covariance of e(k), denoted by Σe , is determined by those of w(k) and v(k) [7]. This type of model can be consistently identified from closed-loop data by the predictor-based subspace identification approach (PBSID) [9]. Since the innovation e(k) is unknown, it can be replaced by y(k) − C x ˆ(k) − Du(k). A closed-loop observer thus results from (3, 4); i.e. xˆ(k + 1) =

(A − KC) x ˆ(k) + (B − KD)u(k) | {z } Φ

y(k) =

+Ky(k),

(5)

Cx ˆ(k) + Du(k) + e(k).

(6)

We now define a data matrix for a signal s ∈ {u, y, e, z},  T where z(t) , uT (t) y T (t) , as St,p,N equals     

s(t) s(t + 1) .. . s(t + p − 1)

s(t + 1) s(t + 2) .. . s(t + p)

··· ···

···

s(t + N − 1) s(t + N )  . ..  . s(t + p + N − 2)



0

CΦp−1 B

. . . 0

. . . 0

···

···

···

CΦp−1 B

|

CB CΦB

···

. . . L−1 CΦ B

{z

HL,p z

·Zt−p,p,N .

Let T⋆L denote the (block lower triangular) Toeplitz matrix with L block columns and rows,   D⋆   CB⋆ D⋆   (7) ,  .. .. ..   . . . CΦL−2 B⋆ CΦL−3 B⋆ · · · D⋆

where with “⋆” standing for “z, u, y”, B⋆ respectively stands for “[B − KD K], B − KD, K”, and D⋆ for “[D 0], D, 0”. “0” is a zero matrix of suitable dimension. For brevity, denote in the sequel B = [B − KD K]. With these definitions, the model (5, 6) can be written on batch form as (8)

From (5,6), it can be derived that the product of the unknown state sequence, OL ·Xt,1,N , can be exactly replaced



···



(9)

}

Here, the symbol “⊗” stands for Kronecker product. IL is an L × L identity matrix. The block Hankel matrix, HzL,p , contains L block rows and p block columns (the past horizon), where the higher order terms, CΦp+j , j ≥ 0, are replaced with zeros. The benefit of this treatment is that HzL,p only depends on the sequence of Markov parameters, [CΦp−1 B · · · CB]. Now, substituting (9) into (8), one has Yt,L,N

= [IL ⊗ (CΦp )] · Xt−p,L,N + HzL,p · Zt−p,p,N (10) +TzL · Zt,L,N + Et,L,N

The Markov parameters in HzL,p and TzL can be identified by solving the following data equation in a least-squares (LS) sense [9]; i.e. setting L = 1.    1,p  Zt−p,p,N p D · Yt,1,N = CΦ Xt−p,1,N + Hz Ut,1,N {z } | {z } | | {z } bt−p,1,N Ξ Zi

+Et,1,N .

The subscript t indicates the time instant of the top left hand side element of St,p,N ; while p and N are respectively the number of block rows and columns of this matrix. The special case of a row block matrix St,1,N will occur frequently. Let OL denote the extended observability matrix of (5, 6) with L block elements; i.e. with Φ = A − KC,  T OL = C T (CΦ)T · · · (CΦL−1 )T .

Yt,L,N = OL Xt,1,N + TzL · Zt,L,N + Et,L,N .

L O Xt,1,N = [IL ⊗ (CΦp )] · Xt−p,L,N + CΦ·p−1 B CΦp−2 B ··· ··· ···

(11)

Zi denotes the identification data matrix. Explicitly, Ξ is the following block row vector,   CΦp−1 (B − KD), CΦp−1 K, · · · , C(B − KD), CK, D . The LS estimate is then ˆ , min kYt,1,N − Ξ · Zi k2 . Ξ 2 Ξ

(12)

The term bt−p,1,N in the data equation will give rise to a bias that decays with p. As a standard assumption in system identification, the data matrix Zi has full row rank; i.e. ∃ρz > 0 such that Zi ZTi  ρ2z I.

(13)

Here, for two matrices M1 and M2 , M1  M2 means that M1 − M2 is positive semi-definite. Besides, we will use the following explicit bound on the covariance matrix of vec(Xt−p,1,N ) in the discussions to follow; i.e. ∃¯ σxx > 0 such that kCov(vec(Xt−p,1,N ))k2 ≤ σ ¯xx . (14) Lemma 1: The LS problem (12) for the data model (11) has the solution ˆ = Yt,1,N · Z† , Ξ (15) i where the superscript “†” stands for pseudo-inverse. The statistical bias and covariance are given by ˆ −Θ Θ

1/2

= δΘ + ΣΘ ˆ · ǫ, with

ΣΘ ˆ

=

δΘ

=

(16)

1/2 1/2 T (Zi ZTi )−1 ⊗ Σe = ΣΘ ˆ · (ΣΘ ˆ ) ,  T −1 p [(Zi Zi ) Zi ] ⊗ (CΦ ) · vec (Xt−p,1,N ) ,

ˆ , Θ, ˆ where expressed in the vectorized parameters, vec(Ξ) ΣΘ ˆ is the covariance matrix of this vector; while δΘ is unknown, but has a bounded 2 norm. ǫ ∈ R(p(m+ℓ)+m)ℓ×1 has zero mean and identity covariance matrix; and Θ represents the true parameters. Proof: See the proof of Lemma 1 in [17]. Treating Σe as known, or using the estimate,   ˆ e = Cov Yt,1,N − Ξ ˆ · Zi , Σ (17)

in its place, is a standard practice in the statistical signal detection literature [4], [18]. We shall hence not distinguish ˆ e and Σe in the rest of the paper. between Σ ˆ = Θ. However, when N and Asymptotically, limN,p→∞ Θ p are finite, the identification errors cannot be ignored. Denote, in the sequel, the identified Markov parameters and the Toeplitz and Hankel matrices consisting of them with a bar over their corresponding symbols, e.g. CΦK, T¯yL . With Σe ˆ the FICSI residual generator proposed and the identified Ξ, in [15] can already be fully parameterized, as reviewed in the next subsection. The PBSID method, however, goes one step further to identify (A, B, C, D, K). The details of this step is not of interest for this paper, and are thus omitted. B. Nominal data-driven FICSI detection We first review the link between fault detection and the PBSID subspace identification; i.e. the FICSI approach proposed in [15]. We shall use k to represent the time instant in fault detection, which is after the time instant t + p + N , when the identification experiment is over. In the formulation of FICSI, the integer L denotes the length of fault detection window. We shall always use p to denote the past horizon. For simplicity, we shall denote the single column vector collecting a sequence of output measurements as Yk−L+1,L,1 , yk,L =  T T y (k − L + 1) · · · y T (k) , and zk−L,p , uk,L , ek,L are denoted in the same fashion. Similar to (10), an L-step output equation up to the current time instant k can be constructed as yk,L

= bk,L + HzL,p zk−L,p + TuL uk,L + TyL yk,L +ek,L . (18)

Here, the noise covariance is denoted ΣL e = IL ⊗ Σe and the bias term becomes   bTk,L = (CΦp xˆ(k − L − p + 1))T , · · · , (CΦp x ˆ(k − p))T . The bias is negligible if p is large enough. The residual generator is then a direct consequence of comparing the measured and computed outputs in the sliding window; i.e. with the true system parameters, rk,L = (I − TyL )yk,L − HzL,p zk−L,p − TuL uk,L .

(19)

The distribution of rk,L belongs to the following parametric family [1] (with “E” standing for expectation):  N (Ebk,L , ΣL e
), fault free, rk,L ∼ (20) N Ebk,L + ϕf , ΣL e , faulty.

where N stands for normal distribution. More details of the bias effect on the residual distribution will be analyzed in Sec. IV. The vector ϕf depends on fault signals, and differs from zero when faults occur in the system, provided the faults are detectable. The definition of ϕf and the detectability conditions are not of the interest of this paper, which can instead be found in [15]. ˆ from If L ≤ p, then Σe and the identified parameters Ξ (15) can fully parameterize the residual generator and the statistical fault detection test, with the unknown bias bk,L ignored. We shall rewrite the residual generator built by the identified parameters as ¯ L,p zk−L,p − T¯ L uk,L . rk,L = (I − T¯yL )yk,L − H z u

(21)

¯ L,p , T¯ L , T¯ L contain errors, as The Markov parameters in H z y u discussed in Lemma 1. Since the residual is a linear function of the estimated matrices, the residual again becomes Gaussian with an additional covariance matrix denoted by ΣL Θ; i.e. ( no fault L L N (Er  k,L , Σe + ΣΘ), fault free, rk,L ∼ (22) faulty L L N Erk,L , Σe + ΣΘ , faulty. fault Here, Erno will be characterized later. This then brings k,L up the problem to be solved in this paper. Problem 1: Robust Residual Analysis and Evaluation 1) What is the additional covariance ΣL Θ in (22) by using the uncertain data model in (21), and is this distribution significantly different from the one in (20)? 2) What is the optimal test for the residual in (21)?

III. A N EW DATA - DRIVEN PSA

METHOD

In this section, we extend the data-driven PSA method in [10] to the PBSID identification approach. We shall show the difficulties in robustifying this approach against identification errors, which further motivates the advantage of the FICSI approach in terms of solving Problem 1. The data-driven PSA approach in [10] is based on the errors-in-variable MOESP subspace identification method [19], which can be used with closed-loop data, provided the excitation signals are known. In this subsection, we show that the parity relation can also be obtained following the LS solution (15) in the PBSID method, which does not require the information of the excitation signals. Due to (9), range(OL ) ≈ range(HzL,p · Zt−p,p,N ). Here, “≈” is due to the ignorance of [IL ⊗ (CΦp )] · Xt−p,L,N in (9). The matrix HzL,p can be built from the identified Markov parameters from the LS solution (15). Therefore, the left null ¯ L,p · Zt−p,p,N ; space of OL can be identified from that of H z i.e. via the following SVD,    T   0 VHZ ¯ zL,p ·Zt−p,p,N = UHZ U ⊥ · SHZ H · ⊥ ⊥ T . HZ 0 SHZ (VHZ ) ⊥ Then UHZ approximately spans the left null space of OL . Remark 1: In both the approach of [10] and the new extension in this subsection, SHZ is found by observing the ¯ zL,p · Zt−p,p,N . However, since dominant singular values of H

⊥ SHZ is usually not exactly a zero matrix as computed from ⊥ T L identification data, (UHZ ) O is usually not exactly zero. In other words, the data-driven PSA methods considered in this section cannot entirely annihilate the effect of the initial state. Remark 2: The statistics of the data-driven PSA residual in the presence of identification errors cannot be analyzed ¯ L,p , T¯ L , and T¯ L consist in an explicit form. To see this, H z y u of the identified Markov parameters from the LS solution ⊥ ˆ of (15). UHZ then has an error nonlinearly dependent on Ξ, ⊥ T ¯L due to the SVD. Furthermore, the errors in (UHZ ) · Tu and ⊥ T ¯L ˆ (UHZ ) · Ty are also nonlinearly dependent on those in Ξ. From Remarks 1 and 2, the SVD and the projection in the data-driven PSA methods complicate the analysis of the dependence of their residual generator on the identification errors. This motivates us to investigate Problem 1 in the formulation of FICSI, where neither SVD nor projection is required.

IV. ROBUST

DATA - DRIVEN

FICSI

DETECTION

We shall now analyze and solve Problem 1. The key ideas L are to quantify the composite covariance matrix ΣL e + ΣΘ in (22). We shall also show that the error bound in approxiL mating Cov(rk,L ) by its computable components, ΣL e + ΣΘ , decays as the past horizon p increases. Finally, a statistically optimal test will be given for fault detection. A. Identification error effect on the residual generator The preliminary step before deriving the additional covariance ΣL Θ in (22), is to explicitly write the uncertainties in rk,L in terms of the statistical bias and covariance of the identified Markov parameters as in Lemma 1. The following lemma shows that the statistical uncertainties in rk,L due to the identification errors can be quantified by the two terms ζk,L and βk,L , defined therein. Lemma 2: If the Markov parameters contained in ¯ zL,p are identified by (15) from finitely many noisy T¯uL , T¯yL , H I/O samples, then the FICSI residual (21) computed at time instant k contains the following errors:
ζk,L = ZTp,ol ⊗ Iℓ · (Z†,T ⊗ Iℓ ) · vec(Et,1,N ), (23) i
βk,L = ZTp,ol ⊗ Iℓ · [Z†,T ⊗ (CΦp )] · vec(Xt−p,1,N ), (24) i where Zp,ol is defined by  zTk−L,p  zTk−L+1,p  ZTp,ol =  ..  . zTk−1,p

uT (k − L + 1) uT (k − L + 2) .. . uT (k)



  . 

The subscripts, “p, ol”, respectively remind that the elements of Zp,ol are the past I/Os measured online. Proof: Consider the first output element in (18); i.e. p

y(k− L + 1) = CΦ ˆ(k − p − L + 1) + e(k − L + 1)+  x  T 
z k−L,p ˆ· Ξ + zk−1,p uT (k) ⊗ Iℓ · u(k − L + 1)  †,T (Zi ⊗ Iℓ ) · vec(Et,1,N ) + [Z†,T ⊗ (CΦp )] · vec(Xt−p,1,N ) . i

On the right hand side, the last two terms are due to      zk−L,p zk−L,p ˆ + ∆Ξ) ˆ · Ξ· = vec (Ξ , u(k − L + 1) u(k − L + 1) ˆ = (Z†,T ⊗Iℓ )·vec(Et,1,N )+[Z†,T ⊗(CΦp )]· where vec(∆Ξ) i i vec(Xt−p,1,N ); and the property that vec(ABC) = (C T ⊗A)· vec(B), for A, B, C with appropriate dimensions.   Now, ∀i = 2, · · · , L, partition Ξ as Ξ1 Ξ2 , where Ξ1 , Ξ2 respectively have (p − i + 1)(m + ℓ) and (i − 1)(m + ℓ) + m columns. Then, y(k − L + i) can be written as in Eq. (25) on the next page, which is equivalent to y(k − L + i) = CΦp x ˆ(k − p − L +i) + e(k − L + i) zk−L+i−1,p +Ξ · . u(k − L + i) The last term of the above equation is equivalent to    
ˆ · zk−L+i−1,p + zTk−L+i−1,p uT (k − L + i) ⊗ Iℓ · Ξ u(k − L + i) n o (Z†,T ⊗ Iℓ ) · vec(Et,1,N ) + [Z†,T ⊗ (CΦp )] · vec(Xt−p,1,N )) i i

Now, assemble y(k − L + i), i = 1, · · · , L into yk,L , and arrange the terms, Eqs. (23) and (24) follow. The residual vector rk,L , as computed by (21), can now be characterized with the following structure, rk,L

= ζk,L + ek,L + βk,L + bk,L +ϕf . | {z }

(26)

Here, the underbraced terms are unknown, and will be treated as random variables in the analysis to follow. ϕf is assumed to be independent of the innovation e. For the boundedness of bk,L during the implementation of the fault detection scheme, we need the following condition. Assumption 1: Both the nominal system and the system under the influence of additive faults are internally stable, i.e. kZp,ol k2 ≤ ρ¯z,ol , ∀k. (27) Under this assumption, the state sequence,  T ˆT (k − p) , x ˆk−p,L = xˆT (k − p − L + 1) · · · x of the observer (5,6) during the implementation of FICSI has a bounded covariance as (for simplicity, with the same bound σ ¯xx as in (14), without loss of generality), kCov(ˆ xk−p,L )k2

≤

σ ¯xx , ∀k.

(28)

B. Statistically optimal fault detection test Equation (26) leads to the main result of this paper. Theorem 1: Let the signals in the identification experiment and fault detection respectively satisfy (13, 14) and (27, 28). If the past horizon p is chosen such that kCΦp k22 < ξ ·

kΣe k2 , 2 σ ¯xx

for an arbitrarily small positive number ξ; then h i
T T −1 ΣL Zp,ol ⊗ Σe + IL ⊗ Σe . Θ,e = Zp,ol Zi Zi

(29)

(30)

  y(k − L + i) = CΦp xˆ(k − p − L + i) + e(k − L + i) + 0ℓ×(i−1)(m+ℓ) Ξ1 · zk−L,p  T T +Ξ2 · u (k − L + 1) y T (k − L + 1) · · · uT (k − L + i − 1) y T (k − L + i − 1) uT (k − L + i) approximates the covariance matrix of the FICSI residual rk,L computed by (21) in the following sense: !

ρ¯2z,ol L

Cov(rk,L ) − ΣΘ,e < ξ · 1 + · kΣe k2 . 2 ρ2z Proof: The proof is lengthy and omitted due to page limitation, which can instead be found in [20]. According to Theorem 1, with p chosen satisfying (29) so that ΣL Θ,e approximates Cov(rk,L ) with an arbitrarily small error, the parametric family of the rk,L distribution can be represented by:
 N Eβk,L + Ebk,L , ΣL Θ,e
, fault free, rk,L ∼ N Eβk,L + Ebk,L + ϕf , ΣL Θ,e , faulty. (31)
− 12
− 12 L Whiten rk,L by ΣL , i.e. ˜ r , Σ r k,L k,L . Θ,e Θ,e The test statistic for the changing mean in the normal distribution can be written as

2
− 21

r (32) tk = ˜ rTk,L · ˜ rk,L = ΣL

. k,L Θ,e 2

2

Unfortunately, tk defines a noncentral χ distribution even in the fault free case, with a non-centrality parameter

2
− 12

λrk = ΣL · (Eβk,L + Ebk,L ) . (33) Θ,e 2

Denote this distribution by tk ∼ χ2Lℓ,λr , with Lℓ DoFs [4]. k The parameter λrk is again unknown. However, if the past horizon p also satisfies kCΦp k22 ≤

−1/2 2 k2

kΣe

ζ · Lℓ , (kE [vec(Xt−p,1,N )] k2 + kEˆ xk−p,L k2 )2 (34)

for an arbitrarily small ζ > 0; then the cumulative distribution function (cdf) of tk under the fault free case can be approximated by that of a central χ2 distribution. Detailed analysis of this approximation can be found in [20], and omitted here for brevity. The fault detection test can hence be derived based on the central χ2 distribution, i.e. χ2Lℓ , with a theoretical threshold, γχ2Lℓ ,α , determined by a pre-specified false alarm rate α. The robust data driven fault detection approach can now be summarized in Algorithm 1. V. S IMULATION EXAMPLE Consider the linearized VTOL (vertical takeoff and landing) aircraft model, in the form of (1, 2); with D = 0, F = I4 , and A, B, C as discretized at a sampling rate of 0.5 seconds, from the following continuous time model (the subscript “c” representing continuous time), x˙ c (t) =

Ac xc (t) + Bc uc (t)

yc (t) =

Cc xc (t) " −0.0366

Ac

=

0.0482 0.1002 0

0.0271 −1.01 0.3681 0

0.0188 0.0024 −0.707 1

−0.4555 −4.0208 1.42 0

#

(25)

Algorithm 1 (Robust Data-Driven FICSI): Design Phase: 1) choose the horizons L ≤ p and the false alarm rate α, and determine the threshold γχ2Lℓ ,α ; 2) measure the I/O signals from a plant, and formulate the data matrix Zi ; ˆ by (15), Σ ˆ e by (17), and (Zi ZT )−1 ; 3) compute Ξ i ¯ L,p , T¯ L , T¯ L . 4) formulate the matrices, H z u y Implementation Phase at the time instant k: 1) measure the past p + L I/Os and u(k) from the plant, and formulate the data matrix Zp,ol ; 2) compute rk,L by (21) and its covariance ΣL Θ,e by (30); 3) compute the test statistic tk by (32) and compare it to the threshold γχ2Lℓ ,α .

Bc

=

"

0.4422 3.5446 −5.52 0

#

0.1761 −7.5922 4.49 0

, Cc =

"

1 0 0 0

0 1 0 1

0 0 1 1

0 0 0 1

#

.

The process and measurement noise, w(k), v(k), are assumed to be zero mean white noise, respectively with a covariance of Qw = 0.25 · I4 and Qv = 2 · I2 . In the identification experiment, an empirical stabilizing output feedback controller was used; i.e.   0 0 −0.5 0 u(k) = − 0 0 −0.1 −0.1 · y(k) + η(k). η(k) is a zero-mean white noise with a covariance of diag(1, 1), which ensures that the system is persistently excited to any order. 2000 data samples were collected from the simulation. The past horizon was chosen as p = 15. The covariance of the identified parameters, i.e. ΣΘ ˆ , has a maximum singular value of 0.0525, corresponding to an SNR of 10 log10 (1/N ·1/kΣΘ ˆ k∞ ) = −20.2dB. We checked the condition (29) based on the steady-state Kalman filter [21] of the discrete-time VTOL model. Indeed, kCΦp k22 = 2 2.1×10−5 ≪ kΣe k2 /¯ σxx = 0.245. The design phase of Alg. ˆ Σ ˆ e , (Zi ZT )−1 , H ¯ zL,p , T¯uL , T¯yL . 1 produced the parameters, Ξ, i The online detection experiment is designed as follows. T The initial states of the system were set as [10 10 1 1] . An LQG trajectory-tracking controller was designed for the VTOL system, to maintain a vertical velocity of 60. The purpose of this closed-loop experiment is to clearly show the advantage of the robust FICSI detection method, when the I/O signals in the system are large enough for ΣL Θ,e to be significantly different from ΣL e. Consider the actuator fault, i.e. a biased collective pitch control, happened at the 300-th sampling instant; i.e.   T 0 0 , k < 300,  T f (k) = 5

0

,

300 ≤ k < 600,

test statistics

4

test statistics

test statistics

4

10

10 threshold Frb Fno

Pss

Pmp

threshold

threshold

3

10

3

3

10

10

2

2

10

10 2

10

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Left: Test statistics of Frb (solid) and Fno (dash-dotted). Middle: Test statistics of Pss (solid). Right: Test statistics of Pmp (solid).

We checked (34) based on the steady-state Kalman filter, Lℓ = 7 × 10−5. We used −1/2 2 kΣe k2 (k[vec(Xt−p,1,N )]k2 +kˆ xk−p,L k2 )2 x ˆk−p,L , instead of Eˆ xk−p,L , since the latter is difficult to evaluate. The parameters of the data-driven PSA and FISCI algorithms were chosen as p = 15, L = 10, α = 0.003. We tested and compared four data-driven detection solutions: Pss . Classical model-based PSA, with (A, B, C, D, K) identified by the PBSID-OPT method of [9]. Pmp . Data-driven PSA, proposed in Sec. III. Fno . Data-driven FICSI, without considering the parameter identification errors. Frb . Robust data-driven FICSI, as proposed in Sec. IV. The test thresholds were all computed with the false alarm rate of 0.003. As indicated by their large magnitudes above the thresholds under the fault-free case, none of the nominal data-driven algorithms can correctly quantify the distribution of their test statistics, since only the innovation signals are considered therein to be stochastic. On the other hand, the robust data-driven FICSI method correctly accounts the composite covariance information contained both in the innovation signals and in the parameter identification errors, as can be seen from the correct theoretical threshold separating the fault-free case from the faulty one. However, the robust data-driven FICSI has a lowered sensitivity to faults, as compared with its nominal counterpart, as can be seen from the reduced gap between the test statistics of respectively the fault-free case and the faulty one. This is of course the conservativeness brought by the robustness. VI. C ONCLUSIONS In this paper, we have analyzed the effect of parameter identification errors in the data-driven FICSI fault detection scheme. Due to the avoidance of SVD and projection onto a parity space, the identification errors propagate linearly into the FICSI residual vector. This linear dependence finally leads to the explicit expression of the residual covariance in terms of both the innovation signals and the parameter errors. A robustified fault detection method is hence been developed, and verified in the simulation study. The focus of the current work is on the detection only. Fault identification problem in the data-driven FICSI framework has been cast into an unknown input estimation prob-

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