0 50 100 150

2。1。

Detectability and isolability

If a fault affects the residual vector, the fault is structurally detectable。 If a  particular fault has a unique pattern in the residual vector’s ele- ments, it is structurally isolable。 In the pres- ence of only one fault, isolable constraints are (m1, m2, m3, m7, m8, m9, m10, m11, m13+j )。 The rest are detectable。 Considering the faults on the

mooring line, violations of constraints c2i+5 and c2i+6 are detectable and their residual vectors are unique from those of the other cases。 On breakage of a line, the measured tension on m13+j becomes zero。 In contrast, with loss of a buoyancy element, the tension will increase。 One can hence distinguish between these two faults by detecting the tension

Figure 3: Time history, autocorrelation, PDF and CDF for raw residual r1 with loss of buoyancy element at t = 580s。 The PDF is shown before and after loss of the MBLE。 A Gaussian probability plot is used for the  CDF。

change of the measurement。 The constraints m13+j are isolable and thus the fault on tension measure- ment equipment can be distinguished from the fault on the mooring line。

If two or more faults are present simultaneously, fault isolation, and therefore also fault accommod- ation, becomes more complex and active fault isol- ation [27] may be required。 This was demonstrated on the PM system in [17]。

age (ARMA) model by a scalar process e(kTs) with independent increments, where k is increments in time and Ts  the sampling time,

where F (q) = a1q1 + a2q2 + · · · + anqn, and C(q) = c1q1 + a2q2 + · · · + anqm are polynomials in the delay operator q, then, provided C(q) is   Hur-

witz, the original residual, ri(k) for brevity, would be whitened to rw (k) by

Standard identification methods can be used to

determine the polynomials in the ARMA model Eq。 (7)。 However, C(q) need be restricted to have eigenvalues within the unit circle。  The orders  of F (q) and B(q) are determined as part of the iden- tification。 The main issue in this procedure is that a model as Eq。 (7) requires e(k) to be stationary,

Figure 4:  Time history, autocorrelation, PDF and CDF   for

raw residual r5 with sudden loss of a buoyancy element (MBLE) at t = 580s。 The PDF plot shows histograms of probability density before and after loss of the MBLE。 The CDF is the one before the fault (H0) plotted in a Gaussian probability plot。

2。2。Properties of residuals

The time-histories, autocorrelation and distribu- tions of the residuals r1  and r5+i  before and  after a fault are shown in Figures 3 and 4。 A similar analysis of residual r2 was also made。 In the ex- periment, the wave-making starts at t = 150s  and a buoyancy element loss at line no。4 is injected at time t = 580s。 Vessel data and experiment details are described in Section 5。1。 The auto-correlation functions for r1  and r5  for the no-fault   condition

(H0) and the PDFs p(r, H0) are calculated for    the

but waves of a short-crested sea are not station- ary。 Furthermore, there are non-linear phenomena involved in forming the mooring line tensions, so a linear whitening process will not suffice。 Therefore, the whitening can at best be approximate。

When applying the whitening procedure, the dis- tribution of the raw whitened residual, shown in Fig。 (6) appears to have a mixed distribution where the bulk of samples are Gaussian but the tails are differently distributed; the tails have some resemb- lance to outliers。   One approach would be to     at-