J.R. Marx et al.

Icean Engineering 343 (2026) 123388

Fable 2
dentified parameters for vessel DENEB in 10° SI units.
Index A B C D E
„43 1.9482 0.00199 1532.75 -0.83975 0.00000
1,3) 0.00000 0.00000 0.00000 9.00000 D.00000
1,3) 0.00000 0.00000 0.00000 3.00000 0.00000
AN 0.00000 7.00000 0.00000 72.00000 0.00000
7.2) -12.089 0.00132 0.00000 103.55 0.00000
5 -188.60 0.00000 0.00000 0.00000 5.00000
3,1) 0.00000 3.00000 0.00000 0.00000 0.00000
3,2) -0.47304 0.00000 0.00000 0.00000 0.00000
ae 19707 N 00474 0.00000 71369.1 0.00000

1000.00
9.00000
3.00000
3.00000
1000.00
3.00000
3.00000
0.00000
1000.00

Fable 3
dentified parameters for USV MESSIN in SI units.
Index A

Tr

1,1) -0.03095
1,2) 0.00000
1,3) 0.00000
2,1) 0.00000
2,2) -0.00000
2,3) 0.05006
3,1) 0.00000
3,2) 0.135335
9 +0. 07775

0.00358 2.19761 -0.02829
‘.00000 0.00000 0.00000
7.00000 3.00000 7.00000
3.00000 0.00000 3.00000
01143 -0.00000 12.09742
7.00779 3.00000 7.00000
3.00000 0.00000 3.00000
3.01527 0.00000 3.00000
3 01309 -0.02293 10.08327

0.00000
2.00000
-0.20005
0.00000
9.00000
0.08515
0.00000
0.00000
9.00000

1.00000
0.00000
9.00000
0.00000
1.00000
0.58051
0.00000
1.53164
1.090000

Fable 4
dentified parameters for catamaran BELA in SI units.
Index A RP
AD
1,2)
1,3)
'2,1)
2,2)
'2,3)
3,1)
3,2)
‘321

= F

-0.02790 0.00067 2.315813 :0.01912
0.00000 2.00000 0.00000 73.00000
3.00000 3.00000 0.00000 3.00000
0.00000 3.00000 0.00000 3.00000
:0.01645 3.00047 -0.00000 :0.90338
0.30389 :0.00010 0.00000 3.00000
3.00000 3.00000 3.00000 3.00000
0.00528 -0.00003 0.00000 0.00000
0.122302 2.000006 -0.023105 ‚7 30620

0.00000 1.00000
3.00000 0.00000
3.00000 0.00000
3.00000 0.00000
3.00000 1.00000
3.00000 1.31406
3.00000 0.00000
0.00000 -0.21451
0.000000 1.000006

Combining this with the body-fixed dynamics, the full state vector
pecomes x = [n' v'1', resulting in the nonlinear coupled system
Y= (> RG) | + [%°] *
03x23 8, V/v B+
N
A.(x) B.
with output
y=[Isxa Osx3|x-N


Ya = Ya-ı + CoAxx
Ax
Ya
= Ar 1 = B,
Xr+1ı = | Xx + ÄTz.
C.Ax L3 C.Bx
The reference trajectory to be tracked is denoted by
T
T T T
[22 w. 2. WW | .
where w, is the desired output at future time step k within the prediction
1orizon H,. The predicted output trajectory $ is computed using the
nternal model dynamics and is used in the optimization.
The cost function to be minimized at each step is
J(k) = 9 -w)' 00 -w)+Ar'Rar. (13)
Tere, Q is the symmetric positive semi-definite weight matrix penalizng
 the tracking error, and R is the positive definite matrix penalizing
‘he input incerements. Both matrices are typically chosen as diagonal
‘o decouple output and input weightings. Higher values in Q lead to
;tricter adherence to the reference trajectory, while increasing R suppresses
 aggressive control actions. The tuning must strike a balance beween
 tracking precision and actuator effort, often based on simulation,
>hysical intuition, or automated methods such as trial-and-error, LQRderived
 initialization, or optimization-based parameter search.
Egq. (13) leads to the following constrained optimization problem
nin IK 14
Art
st. TAr<b

(9

£

3.2. Controller design

To enable linearized MPC, the nonlinear model (5) is linearized at
each time step around the current state x,. The resulting system matrices
Ax and B„, correspond to the first-order Taylor expansion of the original
system model (without constant offset terms)
dA (x)x
Ar = Dt 2
0X xX4
0B.T
Bı= | =B.
OT x,
Since the input function B, rt is already linear in z7, its Jacobian remains
constant, but it is stated here for consistency as well.
An incremental system formulation is used to ensure offset-free
tracking adapted from Ruscio (2013)
AxX,ı.ı = ArdAx, + Bidrt,