2。Structural analysis

Modelling of the PM system is defined as the set of the constraints, shown in Appendix A。 A typ- ical PM system is shown in Fig。 (2)。 It should be noticed that the numbers of thrusters, sensors and

Figure 2: A typical PM system。 By courtesy of Berntsen [23]

The analytical redundancy relation arr is a vector expressing x, y linear acceleration balance, arr is a scalar with angular acceleration balance and arr where i = 1, 2, 。。, 6 is a vector expressing the balance between estimated and measured tensions in the six mooring lines。 For brevity, other residuals, which are not related to mooring lines, are not included here。

The parity relations 4 to 6 below, are found by inserting the constraints listed in Appendix A in the symbolic expressions 1 to 3 and    introducing,

measurement units depend on the class of DP sys-

for brevity,  prel  ¾ pm

− R(q1)lH1, the analytical

tem。  Detailed class regulations are found in   [24]

and [25]。 The modelling here presents the normal behaviour of the PM system, and structural ana- lysis is used to find the over-determined subsystem。 It finds a complete matching with respect to un- known variables and uses non-matched constraints, if these exist, as the analytical redundancy rela- tions, also known as the parity relations or residual

form of residuals become Eqs。 (4)-(6)。

In Eqs。 (4)-(6), r1 and r2 are the x and y com- ponents of arr。 The impact  on  yaw acceleration arr is minor for a loss of a buoyancy element or breakage of line j, would affect the residual vector components [r1, r2, 。。。, r5+j ]。

In Eqs。 (4)-(6), ψ1, ψ2, ψ3 are the yaw angle meas-

vectors。

urements, ψ, ψ˙

are  the  yaw  angle  and  yaw rate,

SaTool is a software developed for the structural analysis technique。 With SaTool, a set of parity relations is generated as the result of structural analysis in symbolic form。  Occurrence of a  fault is considered as a deviation from the constraints。 This deviation will affect a parity relation if this parity relation is built from the constraints。 The parity relations can then be used as residual gener- ators for fault detection in the system。 Among the analytical redundancy relations that exist for the system, Eqs。 (1) - (3) are sensitive to faults on the ith mooring line:

pG1, pG2, pH1 are the position measurements, p, p˙ are vessel position and velocity, q1, q2, q3 are the vertical reference measurements, z, φ, θ are vessel heave, roll and pitch, wm1, wm2, cm are wind and current measurements, vw, vc are wind and current velocity, Twave is the wave force, Tmoj is the moor- ing line tension, Tmbi is the MLBE force, Tmj is the mooring line tension measurement in line i, v is the vessel velocity, vm is velocity measurement, u1, u2, 。 。 。 uk are thruster input, T1, T2, T3 are the thruster forces。 The detailed modelling for a PM system is described, e。g。 in [26]。

1    =  c1(a1(u1), a2(u2), c6(m3(ψ3), m9(m3), m6(pmH1)), m3(ψ3),

c2i+5(c6(m3(ψ3), m9(m3), m6(pmH1)), m3(ψ3),

c2i+6(c6(m3(ψ3), m9(m3), m6(pmH1)), m3(ψ3))), m12(wm2), d3(m3(ψ3))),

(1)

2    =  c2(a3(u3), c6(m3(ψ3), m9(m3), m6(pmH1)), m3(ψ3),

c2i+5(c6(m3(ψ3), m9(m3), m6(pmH1)), m3(ψ3),

c2i+6(c6(m3(ψ3))), m12(wm2), d3(m3(ψ3)), m3(ψ3), m9(m3), m6(pmH1)), d4(d3(m3(ψ3))))

+ M−1 [gx (wm1)  gy (wm1)]T + 。 Axy (prel, ψ1), Txy

(gmo(prel, ψ1, gmb(prel, ψ1)))

。 r3 。 = I−1 。Hψ T[g1(u1, u2, 。 。 。 , uk), g2(u1, u2, 。 。 。 , uk), g3(u1, u2, 。 。 。 , uk)]T + gψ (wm1)。