Roll Damping and Heading Control of a Marine Vessel by Fins-Rudder VSC

8th IFAC Conference on Control Applications in Marine Systems Rostock-Warnemünde, Germany September 15-17, 2010

Roll Damping and Heading Control of a Marine Vessel by Fins-Rudder VSC C. Carletti ∗ A. Gasparri ∗∗ G. Ippoliti ∗ S. Longhi ∗ G. Orlando ∗ P. Raspa ∗ ∗ Dipartimento di Ingegneria Informatica, Gestionale e dell’Automazione, Universit` a Politecnica delle Marche, Ancona, Italy (e-mail: [email protected], [email protected]). ∗∗ Dipartimento di Informatica e Automazione, Universit` a degli Studi “RomaTre”, Roma, Italy (e-mail: [email protected] )

Abstract: This paper investigates the design of an integrated control system for course keeping and roll damping of a multipurpose naval vessel by means of active fins and rudder. The ship dynamics is described through a nonlinear multivariable model which takes into account the hydrodynamic couplings among sway, roll and yaw due to both the wave effect and the control devices, i.e., rudder and active fins. A stability analysis of the zero dynamics along with the design of a MIMO variable structure control law is provided. Simulation results are given to corroborate the theoretical analysis. Keywords: Ship nonlinear model, fins/rudder roll damping, variable structure control. 1. INTRODUCTION

In this work, an integrated control system for heading and roll damping of a multipurpose naval vessel by means of active fins and rudder is proposed. The ship dynamics is described through a nonlinear multivariable model which takes into account the hydrodynamic couplings among sway, roll and yaw, due to both the wave effect and the control devices. A stability analysis of the zero dynamics of the adopted model along with the design of an integrated MIMO variable structure control (VSC) law is proposed.

The roll motion due to the waves significantly affects the ship security and on-board comfort causing seasickness in the crew and passengers and increasing the possibility of load damage. Moreover, roll oscillations degrade the ship manoeuvrability while increasing the oil consumption. These effects turn out to be particular relevant for vessels, e.g., motoryachts, characterized by a reduced size and a high operational speed. The roll damping is deeply investigated in literature and a wide range of solutions have been proposed [Perez (2005)]. Among these, active fins and rudder are considered very attractive [Tzeng and Wu (2000), Lauvdal and Fossen (1995)]. In particular, active fins have shown the best performances in roll reduction at high speed but they result ineffective at low speed. On the other hand, the rudder roll damping (RRD) provides good performances and is more effective than fins roll damping (FRD) at low speed due to the fact that the rudder is located in the race of the propeller and thus operates in higher speed flows than fins. However the use of rudder for roll and steering control has two critical aspects: the nonlinear coupling between sway-roll-yaw [Blanke and Perez (1994)] and the instable zero dynamics in the rudder-roll system [Lauvdal and Fossen (1995)]. Several works which address the roll motion reduction by exploiting both fins and rudder have been proposed in the literature [Roberts et al. (1997), Katebi (2004)]. However, these works are based on linear models to describe the ship dynamics, and therefore significant nonlinear effects such as the instability of the zero dynamics are neglected [Fossen and Lauvdal (1994)]. Furthermore, the proposed solutions are based on independent FRD and RRD controllers, which do not consider their mutual effect on the roll angle, thus compromising the overall performances.

978-3-902661-88-3/10/$20.00 © 2010 IFAC

The rest of the paper is organized as follows. In Section 2 the nonlinear ship model for the considered multipurpose naval vessel is provided. In Section 3 the properties of the zero dynamics for the considered nonlinear ship model are investigated. In Section 4 a Variable Structure Control (VSC) for simultaneous course keeping and roll damping is proposed. Finally, in Section 5 conclusions are drawn and future work is discussed. 2. NONLINEAR SHIP MODEL The control design deals with a square nonlinear ship model affine in the inputs of the form: x˙ = f (x) + g · (u + wH ) y = h(x)

(1)

where x ∈ R5 is the state vector, y ∈ R2 is the output and u ∈ R2 is the input, while wH represents the external disturbances.Its derivation from the ship manoeuvring model combined with the first order wave disturbance will be shown in this section. 2.1 Reference Frames The six degrees of freedom (6DOF) motion of a ship can be described as the superposition of three translational motions, surge, sway and heave, along three perpendicular reference directions, and three rotational motions, roll, pitch

34

10.3182/20100915-3-DE-3008.00074

CAMS 2010 Rostock-Warnemünde, Germany, Sept 15-17, 2010

where τ hyd describes the hydrodynamic forces and moments; τ prop is the propulsion forces and moments; τ cs represents forces and moments due to the control surfaces (active fins and rudder); τ env models forces and moments due to waves, wind and currents. Note that, the propulsion component τ prop will be neglected in the rest of the paper, since the propeller effect can be assumed completely compensated by the ship calm water resistance, under the assumption of constant ship cruise speed. Moreover, the environmental term τ env will not appear in the following being modeled as a disturbance affecting the control input channels (Sec. 4).

Fig. 1. Reference frames and notation for ship motion description and yaw, around the same axes. For manoeuvring analysis the position and orientation of the vessel are usually defined with respect to the inertial Earth-fixed n-frame On {xn , yn , zn } while the linear and angular velocities and accelerations of the vessel are expressed with respect to the body-fixed reference frame b-frame Ob {xb , yb , zb } with axis oriented as shown in Fig. 1.

2.3 Hydrodynamic Forces and Moments The hydrodynamic forces and moments in 3DOF are modeled as nonlinear functions of the accelerations v, ˙ p, ˙ r, ˙ the velocities v, p, r and the roll angle φ as shown in the following equations [Blanke and Perez (1994), Perez (2005)]:

2.2 Rigid Body Dynamics

τ2hyd = Yv˙ v˙ + Yr˙ r˙ + Yp˙ p+ ˙ + Y|u|v |u|v + Yur ur + Yv|v| v|v| + Yv|r| v|r|+ + Yr|v| r|v| + Yφ|uv| φ|uv| + Yφ|ur| φ|ur| + Yφuu φu2

The ship dynamic nonlinear model can be obtained considering the vehicle as a rigid body and applying the Newton Mechanics to describe its 6DOF motions [Fossen (2002)]. The equations of motion in matrix form are : M RB · ν˙ + C RB (ν) · ν = τ RB

τ4hyd = + + +

(2)

where ν = [u, v, w, p, q, r]T is the generalized velocity vector in body-fixed frame (b-frame) according to the SNAME (1950) notation, M RB is the generalized mass-inertia matrix, C RB (ν) is the Coriolis-centripetal matrix due to the handing over from an inertial reference system to a moving one. Finally , τ RB = [τ1 τ2 τ3 τ4 τ5 τ6 ]T is the total vector of forces and moments acting on the ship.

τ6hyd = + + +

v˙ p˙ r˙

0 0 m|U | + 0 0 −mzg |U | · 0 0 mxg |U |

v p r

=

τ2 τ4 τ6

MA =

(7)

Yv˙ Yp˙ Yr˙ Kv˙ Kp˙ 0 , Nv˙ 0 Nr˙

(8)

while the residual coefficients representing the hydrodynamic nonlinear coupled terms are collected in the 3DOF ∗ ∗ ∗ vector τˆ ∗hyd = [τ2hyd τ4hyd τ6hyd ]T . The numerical values of the coefficients are taken from [Perez (2005)] 2.4 Control Modules The control module τ cs takes into account the effects on the ship motions due to the control surfaces, i.e. rudder (τ rud ) and active fins (τ fins ). It can be detailed as follows: τ cs = τ rud + τ fins .

(9)

The effectiveness of rudder and fins on roll and yaw motion is different due to their different geometrical location on the hull with respect to the CG. Rudder The 3DOF forces and moments generated by the rudder in the b-frame can be approximated by [Perez (2005)]: " # " #

(3)

where Ix and Iz are the moments of inertia with respect to the origin of the b-frame, and with the obvious definition ˆ RB , C ˆ RB and τˆ RB . of the matrices M The total vector of forces and moments τ RB is generated by different phenomena and it can be expressed by the superposition of terms arising from these: τ RB = τ hyd + τ prop + τ cs + τ env

Nv˙ v˙ + Nr˙ r+ ˙ N|u|v |u|v + N|u|r |u|r + Nr|r| r|r| + Nr|v| r|v|+ Nφ|uv| φ|uv| + Nφu|r| φu|r| + Np p + N|p|p |p|p+ N|u|p |u|p + Nφu|u| φu|u|

(6)

The coefficients proportional to the derivative v, ˙ p˙ and r, ˙ are collected in the 3 × 3 Added Mass matrix (M A ): " #

The 6DOF ship model can be reduced to a 3DOF model by means of some simplifying assumptions. First, being the considered vessel characterized by a slender hull with port-starboard symmetry, it is possible to decouple the sway-roll-yaw dynamics from the surge-heave-pitch one [Faltinsen (2002)]. As a consequence, the surge velocity u can be considered constant and equal to the cruise forward speed U and a 3DOF model in the variables v, p and r is obtained. Moreover, let us consider the b-frame such that the reference axis coincide with the ship principal axes of inertia. This implies the inertia matrix to be diagonal. In addition, by assuming a port-starboard symmetry of the hull, the ship center of gravity CG = [xg yg zg ] can be considered located on the xz-plane of the b-frame, so that yb ≡ yg and thus yg = 0. ˆ RB , C ˆ RB , As a result, the reduced 3DOF ship model (M τˆ RB ) obtained from the 6DOF model (M RB , C RB , τ RB ,) (eq. 2) can be described by: " #" # " #" # " # m −mzg mxg −mzg Ix 0 · mxg 0 Iz

Kv˙ v˙ + Kp˙ p+ ˙ K|u|v |u|v + Kur ur + Kv|v| v|v| + Kv|r| v|r|+ Kr|v| r|v| + Kφ|uv| φ|uv| + Kφ|ur| |ur| + Kφuu φu2 + K|u|p u|p| + Kp|p| p|p| + Kφφφ φ3 − ρg∆GZ(φ)

(5)

τˆ rud ≈

Lrud −Rrud · Lrud −LCG · Lrud

≈

γrud −Rrud · γrud −LCG · γrud

· δrud

(10)

where Lrud ≈ γrud ·δrud is the lift force acting on the rudder, while Rrud and LCG are the distance between the rudder center of pressure CP and the ship CG (Fig.2) along the

(4)

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CAMS 2010 AP Rostock-Warnemünde, Germany, Sept 15-17, 2010FP Lpp

Thus, the ship dynamic model of eq. (13) can be written as in 1 that is repeated here for sake of clarity: x˙ = f (x) + g · (u + wH )

CG

LCG

xb

WL

where x ∈ R5 is the state vector, u = [δrud Nfins ]T ∈ R2 , ˜ T ∈ R2 and: wH = [wH1 wH2 ]T ∈ R2 , h(x) = [φ ψ]

VCG

T

(15)

y = h(x)

Ob

zb

f (x) = −M −1 · C(x) · x + M −1 · T hyd (x)

Fig. 2. Main ship values: body-fixed frame Ob {xb , zb },Center of

g

Gravity (CG), Lateral Center of Gravity (LCG), Vertical Center of Gravity (VCG), Waterline (WL), Draught (T), Aft Perpendicular (AP), Front Perpendicular (FP), Length between Perpendicular (Lpp).

· Nfins

¤

g1 g2 = M −1 · T cs .

3.1 Normal Form

(11)

In order to analyze the multivariable nonlinear ship model, the eq. (13) are transformed to normal form. Let {r1 , . . . , rm } be the vector of relative degree at a point x◦ , i.e., the number of time the i-th output yi must be differentiated to have at least one component of the input vector u explicitly appearing [Isidori (1995)]. Let us now briefly introduce the Lie derivative, i.e., the derivative of λ(·) along f (·), as follows: n X ∂λ

where Rfins is the fins roll arm (the distance of the fin CP from the ship CG) and F CG is the longitudinal distance between the fin CP and the ship CG while βtilt represents the fins tilt angle. Nfins is the resulting hydrodynamic force component normal to the fin, defined as a function of the lift Lfins and drag force Dfins arising on them as Lfins cos(αe ) + Dfins sin(α e) with αe the fins effective angle of attack between the foil and the flow.

Lf λ(x) =

i=1

2.5 State Space Representation

∈R .

Lgj hi = 0 Lgj Lf hi (x) 6= 0

(12)

M · x˙ + C(x) · x = T hyd (x) + T cs · [δrud Nfins ]

A(x) =

(13)

ˆ RB − M A 0 M , 0 I2

·

ˆ RB 02×2 C C(x) = E(x) 02×2

" T hyd (x) =

τˆ ∗hyd 0 0

¸

=

i

0 −1 0 E(x) = 0 0 − cos φ

#

" ,

T cs =

τˆ ∗rud τˆ ∗fins 0 0 0 0

i ∈ {1, 2}, j ∈ {1, 2}

(18)

¸

Lg1 Lf h2 (x) Lg2 Lf h2 (x) a11

a12

¸

(19)

a ˆ21 cos(x4 ) a ˆ22 cos(x4 )

is nonsingular if the following holds:

(14)

h

(17)

Lg1 Lf h1 (x) Lg2 Lf h1 (x)

·

where the matrices M , C , T cs and the vector T hyd are defined as follows: · ¸ M =

fi .

and the matrix A(x) defined as: ·

where ψ˜ = ψ −ψd with ψd the (constant) desired yaw angle. Thus, the augmented state-space representation of the ship nonlinear model given in eq. (3) is: T

∂xi

It can be shown that such a multivariable nonlinear ship model has relative degree {r1 , r2 } = {2, 2} at x0 = [0, 0, 0, 0, 0]. In fact, the Lie derivatives are:

The adopted state space model considers also the roll angle φ and the yaw angle ψ. Therefore, the following augmented state vector is considered: £ ¤T 5 x = v p r φ ψ˜

(16)

The zero dynamics describes the “internal behavior” of a system when input and initial conditions are chosen to force the output to zero [Isidori (1995)]. The analysis of the zero dynamics has an important role in the control design. Indeed, the presence of an unstable zero dynamics, i.e., non-minimum phase system, prevents the application of important control techniques such as feedback linearization. Furthermore, it significantly affects the achievable performances, for instance by imposing limitation on the maximum control gain.

Active Fins The action of the active fins on the ship motions requires a more detailed analysis compared to rudder due to the location of the foils. In this work the following approximate model for (3DOF) forces and moments induced by the fins will be considered (further details can be found in [Perez (2005)]): " # − sin(βtilt ) 2 · Rfins F CG · sin(βtilt )

£

3. ZERO DYNAMICS ANALYSIS

z- and x-axis of the b-frame respectively. To calculate the lift of the rudder, the effective angle of attack δe−rud has been approximated by the mechanical angle of the rudder δrud [Perez (2005)].

τˆ fins ≈

=

#

a11 a ˆ22 6= a12 a ˆ21 ,

(20)

cos(x4 ) 6= 0.

(21)

It should be noticed that while the condition given by eq. (20) is related to the vessel parameters, the second condition given by eq. (21) is always satisfied in practice π as cos(x4 ) = 0 implies that x4 = + k π, k ∈ N, which is 2 an inadmissible operational condition, being x4 the roll angle. Furthermore, the system has degree {2, 2} for any π state x0 = [x1 , x2 , x3 , x4 , x5 ] with x4 6= + k π, k ∈ N.

with 0 opportune completion matrices, I 2 a 2 × 2 identity matrix, τˆ ∗hyd defined in sec. 2.3, τˆ ∗rud and τˆ ∗fins the coefficients vector multiplying δrud in eq. (10) and Nfins in eq. (11) respectively.

2

36

CAMS 2010 Rostock-Warnemünde, Sept 15-17, 2010 Therefore, accordingGermany, to [Isidori (1995)], the nonlinear

being the vector [α β γ 0 0]T an element of the null space base for gT .

system describing the vessel can be transformed to normal form by applying transfor£ the following local coordinates ¤T mation z(x) = z1 (x), z2 (x), z3 (x), z4 (x), z5 (x) at x0 .

As a result, the system equations (13) can be written in normal form as follows:

In particular, since the system has relative degree {2, 2}, the following choice is possible for the first r1 + r2 = 4 variables: µ 1¶ µ ¶ µ ¶ ξ1

ξ1 =

µ ξ2 =

ξ12 ξ22

z1 (x)

=

ξ21

¶

z2 (x)

µ =

z3 (x)

h1 (x)

=

¶ =

z4 (x)

h2 (x)

ξ˙21 = L2f h1 (x−1 (ξ, η)) +

(22)

Lf h1 (x)

µ

ξ˙11 = ξ21

ξ˙22 = L2f h2 (x−1 (ξ, η)) +

(23)

Lf h2 (x)

(36)

(24)

so that the jacobian matrix £ describing the local coordi¤T nates transformation z(x) = z1 (x), z2 (x), z3 (x), z4 (x), z5 (x) , is nonsingular at x0 . Furthermore, since G = span{g} is an involutive set, z5 (x) can be chosen so that: i ∈ {1, 2}.

η = z5 (x) = α v + β p + γ r

£

¤

¡

¢

η0 = 0

(27)

g11 g12 g13 0 0

a base for the the null space is:   g12 g23 − g13 g22       0 0    g11 g22 − g12 g21    0   0   g13 g21 − g11 g23       ¡ ¢      g11 g22 − g12 g21     ker g T =   0 , 0 ,  1        0 1     0   0  1 0

                   

V (η) =

g12 g23 − g13 g22 g11 g22 − g12 g21 g13 g21 − g11 g23 β= g11 g22 − g12 g21 γ=1

1 2 η . 2

(39)

In fact, the negative definiteness of the derivative of the Lyapunov candidate: V˙ (η) = η η˙ = (−a|η| − b) η 2

(40)

is guaranteed by the fact that −a|η| − b < 0, ∀ η.

(29)

4. CONTROLLER DESIGN

Therefore, by choosing α=

(38)

At this point, the stability of the system can be simply checked by exploiting the following quadratic Lyapunov candidate:

(28)

g21 g22 g23 0 0

(37)

In order to investigate the stability of the zero dynamics, it should be pointed out that eq. (37) has an unique equilibrium point:

In particular, since gT has the following structure: · ¸ gT =

Let us now investigate the properties of the zero dynamics η˙ = q(0, η), which can be detailed as follows:

with a, b ∈ R and a > 0, b > 0.

(26)

α, β, γ, 0, 0 ∈ ker g T .

3.2 Stability Analysis

η˙ = −a|η|η − bη

(25)

¤ with g = g1 g2 . Note that, the constraint given by eq. (25) can be satisfied by assuming: with:

Lgj Lf h2 (x−1 (ξ, η)) · (uj + wHj )

j=1

η = z5 (x)

Lgi z5 (x) = 0

2 X

η˙ = q(ξ, η)

while the remaining (n−r1 −r2 = 1) variable transformation can be chosen arbitrarily:

£

Lgj Lf h1 (x−1 (ξ, η)) · (uj + wHj )

j=1

ξ˙12 = ξ22

¶

2 X

4.1 Variable Structure Control

(30)

The control problem is to drive the roll angle φ to zero and the yaw angle ψ to a specific value ψd or, equivalently, to get the variable ψ˜ = ψ − ψd to zero. The control inputs are the rudder angle δrud and the normal force Nfins arising on the fins. The choice of Nfins instead of the fins angle αfins is due to the constrain of affine input required by the variable structure control technique. Note that, the fins angle can be always computed from Nfins , e.g., through the approximation Nfins = Lfins given in [Perez (2005)]. According to the ship dynamics given in eq. (1) the input u = [δrud Nfins ]T is assumed to be affected by a 1st-order wave disturbance wH = [wH1 wH2 ]T . Thus, the multivariable nonlinear ship model of eq. (36) can be written in a compact form as follows:

(31) (32)

the Lie Derivative turns out to be: Lg1 z5 (x) = α g11 + β g12 + γ g13 ,

(33)

Lg2 z5 (x) = α g21 + β g22 + γ g23 .

(34)

which can be re-written in a matrix form as follows:   α h i β h i g11 g12 g13 0 0 0 ·γ = (35)   g21 g22 g23 0 0 0 0 0

37

CAMS 2010 angle [deg]15-17, 2010 Rostock-Warnemünde, Germany,RollSept 20

Yaw angle [deg] 15

15

10

10

ψ [deg]

φ [deg]

5

0

5

−5

−10

0

−15

−20

0

100

200

300

400

500 t [s]

600

700

800

900

−5

1000

0

100

200

300

400

500 t [s]

600

700

800

900

1000

Fig. 3. Roll and Yaw angles obtained during a 1000 seconds simulation in sea state 5 - encounter angle 135 deg - encounter frequency 0.86rad/s conditions. The controller action starts at t = 300s. The reference yaw angle is changed from 0 deg to 10 deg at t = 600s. ξ˙11 = ξ21

X 2

ξ˙21 = F1 (ξ, η) +

G1j · (uj + wHj ) j=1

ξ˙12 = ξ22 ξ˙22

(41)

χe = 45◦

χe = 90◦

χe = 135◦

RSR(φ) ˙ RSR(φ)

86.8729

90.8779

93.3571

88.8968

90.5029

91.6357

Table 2. Control system performances for the vessel

2 X

= F2 (ξ, η) +

Sea state 5

sailing with U = 7m/s in rough waves for different encounter angles

G2j · (uj + wHj ) j=1

η˙ = q(ξ, η)

where F ∈ R2×1 with Fi = L2f hi and G ∈ R2×2 with Gij = In addition let us assume the disturbance wH is bounded by a known bound W = [W1 W2 ] i.e., |wH1 | < W1 , |wH2 | < W2 . Furthermore, let us consider ∆F and ∆G to be modeling uncertainties of F and G respectively with known bounds, as described in the following equations: b ∆F = F − F (42) b ∆G = G − G

4.2 Case Study

Lgj Lf hi .

In the following, a case study for the proposed control technique is described. The parameters which can be found in [Perez (2005)] are exploited for the non-linear ship model described by eq. (1). The controller is designed at cruise speed 7 m/s. The rudder saturation is set to 40 deg while the rudder speed is limited to 20 deg /s . The fins saturation, expressed in terms of force, is computed by considering a fins maximum angle of 28.8 deg (fins stall angle) and a fins maximum angle rate equal to 23 deg /s, that is respectively maximum fins force 107 N and maximum fins force rate 1.5 · 105 N/s.

b the nominal models. with Fb and G Then, the following two sliding surfaces are defined: s1 = ξ21 + λ1 ξ11 = 0, s2 =

ξ22

+

λ2 ξ12

= 0,

λ1 > 0

(43)

λ2 > 0

and the Lyapunov candidate V (s) = 12 sT s with s = [s1 s2 ]T is considered. At this point in order to stabilize the system the following control law can be considered: · ¸ · ¸ u1 −Fb1 (ξ, η) − λ1 ξ21 − k1 (ξ, η) sgn(s1 ) b −1 · =G (44) u2 b2 (ξ, η) − λ2 ξ22 − k2 (ξ, η) sgn(s2 ) −F

Furthermore, the input disturbances due to the waves are obtained filtering a zero-mean Gaussian white noise process with unity power spectrum by a linear 2nd-order wave response model [Fossen (2002)]. The filter gain is properly scaled to reproduce the roll and yaw motion similar to those obtained by using the motion frequency response functions (motionRAO) of the vessel in combination with the wave spectrum [Perez (2005)]. In particular, the ITTC (Modified Pierson-Moskowitz) wave spectrum is considered for the following sailing conditions: sea state 3 with Hs = 0.3 m and T = 6 s and sea state 5 with Hs = 2.5 m and T = 7.5 s, where Hs is the significant wave height and T is the average wave period. In addition, for each sea state, quartering, beam and bow seas are considered.

b is a consequence of the ship where the invertibility of G dynamics model, and k = [k1 k2 ]T is a gain vector required to guarantee the negative definiteness of the V˙ (s) 1 . Sea state 3

χe = 45◦

χe = 90◦

χe = 135◦

RSR(φ) ˙ RSR(φ)

89.0501

95.2255

93.9063

90.1423

93.8579

89.7812

Table 1. Control system performances for the vessel sailing with U = 7m/s in slight waves for different encounter angles

As far as the effectiveness of the integrated control system for roll damping is concerned, the percentage Reduction Statistic of Roll (RSR) defined as follows:

1

As far as the implementantion of the control law is concerned, the function sgn(s) is replaced by the continous function tanh(²s). It allows to reduce chattering without affecting the stability analysis

³ RSR = 100

38

1−

Sc Su

´ (45)

CAMS 2010 Rudder Effort [deg] Rostock-Warnemünde, Germany, Sept 15-17, 2010 5

4

6

Fins Effort [N]

x 10

4 4 3

2 2

N [deg]

0

0

f

δ [deg]

1

−1 −2 −2

−3 −4 −4

−5

0

100

200

300

400

500 t [s]

600

700

800

900

−6

1000

0

100

200

300

400

500 t [s]

600

700

800

900

1000

Fig. 4. Rudder and fins efforts during the simulations in sea state 5 - χe = 45 deg conditions. The controller action starts at t = 1000 s. The reference yaw angle is changed from 0 deg to 10 deg at t = 1500 s. The values for the rudder angle δ and the fins force Nf do not ever achieve their saturation value , i.e. 40 deg and 107 N respectively, and a good gap from those is kept.

where the subscripts c and u stand for controlled and uncontrolled respectively, and S is the Root Mean Square (RMS) of the roll signal.

the VSC is activated at time t = 300 s. Furthermore, the peak at time t = 600 s is due to the switch of the yaw angle reference from 0 deg to 10 deg.

Fig. 3 describes the roll and yaw angles for a simulation with the following operative condition: sea state 5 with an encounter angle of 135 deg. In particular, the VSC is not activated the first 300 s and is turned on after 300 s for the rest of the simulation. The reference angle in yaw changes from 0 deg to 10 deg at t = 600 s. It should be noticed that the roll motion is significantly reduced while the yaw angle is promptly regulated to the desired reference after the VSC is activated.

5. CONCLUSION In this work, the design of an integrated control system for heading and roll damping of a multipurpose naval vessel by means of active fins and rudder has been investigated. A multivariable nonlinear model which takes into account the hydrodynamic couplings among sway, roll and yaw due to both the wave effect and the control devices has been considered. A stability analysis of the zero dynamics for the proposed nonlinear model along with the design of a MIMO variable structure control law has been performed. Simulation results have been carried out to corroborate the theoretical analysis. Moreover, a validation of the proposed mathematical analysis for several parameters data-set describing different operative conditions for different vessels will be performed.

Fig. 4 describes the rudder and fins efforts. In particular, it can be noticed that the rudder angle is bounded within ±5 deg and the fins force is bounded within ±6 · 104 N . Indeed, these results are interesting as they are always significantly below the saturation thresholds imposed by their corresponding mechanical systems. Table 1 and Table 2 give a synoptic overview of the results obtained for the two sea states with respect to different encounter angles, namely 45 deg, 90 deg and 135 deg. In particular, it can be noticed that a substantial reduction of the roll damping is achieved for both slight and rough sea conditions. Again, it is worthy to notice that the control inputs required to obtain these performances never reach the saturation limits, both in amplitude and rate.

REFERENCES Blanke, M. and Perez, T. (1994). Rudder roll damping autopilot robustness to sway-yaw-roll couplings. Technical report, Technical University of Denmark. ISSN 0908-1208. Faltinsen, O. (2002). Sea Loads on Ships and Offshore Structures. Marine Cybernetics, Trondheim, Norway. Fossen, T. (2002). Marine Control Systems. Guidance, Navigation, and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics, Trondheim, Norway. Fossen, T. and Lauvdal, T. (1994). Nonlinear stability analysis of ship autopilots in sway, roll and yaw. In Proc. of the 3rd Conf. on Manoeuvring and Control of Marine Craft (MCMC’94), 113–124. Isidori, A. (1995). Nonlinear Control Systems. Springer-Verlag New York, Inc., Secaucus, NJ, USA. Katebi, R. (2004). A two layer controller for integrated fin and rudder roll stabilization. In Proc. of the Conference on Control Applications in Marine Systems (CAMS 2004), 101–106. Lauvdal, T. and Fossen, T. (1995). Nonlinear non-minimum phase rudder-roll damping system for ships using sliding mode control. Perez, T. (2005). Ship Motion Control. Springer-Verlag, London. Roberts, G., Sharif, M., Sutton, R., and Agarwal, A. (1997). Robust control methodology applied to the design of a combined steering/stabiliser system for warship. IEEE Proc. Control Theory Appl., 144(2). Tzeng, V. and Wu, C. (2000). On the design and analysis of ship stabilizing fin controller. Marine Science and Technology, 8(2), 117–124.

Zero Dynamics 0.06

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Fig. 5. Zero Dynamics behavior. Finally, Fig. 5 describes the behavior of the zero dynamics. According to the theoretical analysis, the zero dynamics η˙ = q(0, η) settles around the equilibrium point η0 = 0 after

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