Simultaneous Roll Damping and Course Keeping via Sliding Mode Control for a Marine Vessel in Seaway

Transcription

1 Milano (Italy) August 8 - September, 11 Simultaneous Roll Damping and Course Keeping via Sliding Mode Control for a Marine Vessel in Seaway C. Carletti A. Gasparri S. Longhi G. Ulivi Dipartimento di Ingegneria Informatica, Gestionale e dell Automazione, Università Politecnica delle Marche, Ancona, Italy ( carletti@diiga.univpm.it, longhi@univpm.it). Dipartimento di Informatica e Automazione, Università RomaTre, Roma, Italy ( gasparri@dia.uniroma3.it ) Abstract: In this paper the simultaneous roll reduction and course keeping problem for a surface vessel by means of active fins and rudder is addressed. This work extends the results proposed in [Carletti et al. (1)] by considering an alternative nonlinear multivariable ship dynamics where the wave effects are modeled as forces and moments affecting the system dynamics rather than as disturbances affecting the control inputs. A theoretical analysis of the boundedness of the zero dynamics along with the design of a sliding mode control are proposed. Simulations results are provided to corroborate the theoretical analysis. 1. INTRODUCTION Roll damping and course keeping are two important control problems in the marine engineering. Indeed, roll oscillations induced by waves significantly affect the comfort of crew and passengers and increase the possibility of cargo and on-board equipment damage. In addition, tracking a predefined, presumably optimum, path allows to minimize time and cost of the course. These problems have been deeply investigated in literature: active tanks, gyroscopes, active fins represent some of the most popular solutions for the roll damping [Perez (5)]; PID, linear quadratic optimal control, state feedback linearization, SISO sliding mode control, nonlinear back-stepping are some of the control techniques developed for the course autopilot [Fossen ()]. Nonetheless, the roll damping and the course keeping should not be considered separately. In fact, when the rudder is used to turn the ship, at the same time, a roll moment is created. In the same way, a steering motion can arise from the active fins if these are located aft of the Center of Gravity (CG). This coupling between the roll and yaw dynamics is particularly relevant for slender and relatively fast monohull, e.g. motoryachts and patrol vessels, as the one considered in this paper. The Rudder Roll Damping (RRD) technique exploits this rudder double effect to perform the roll motion damping simultaneously with the ship course control. The costeffectiveness of this approach compared with active fins solutions has made it very popular. Unfortunately, roll damping by rudder is not a straightforward control problem due, above all, to the nonlinear coupling in the rollsway-yaw dynamics [Blanke and Christensen (1993)] and the instable zero dynamics in the rudder-roll system [Fossen and Lauvdal (1994)]. Moreover, the effectiveness of RRD control in roll motion reduction has been debated [Stoustrup et al. (1995) and references therein]. The roll damping can be improved by introducing the active fins in the control system. Despite of their cost and resistance, Fins Roll Damping (FRD) is largely utilized in surface vessels due to their effectiveness in roll reduction. In the literature, several works have been proposed for the roll motion reduction by means of both fins and rudder. However, the majority of these approaches, e.g., [Roberts et al. (1997)], [Katebi (4)] and [Koshkouei et al. (7)], are based on a linear model to describe the ship dynamics, and therefore significant nonlinear effects, such as the instability of the zero dynamics, are neglected. Furthermore, the proposed solutions are based on independent FRD and RRD controllers, which do not consider their mutual effect on the roll angle. In [Carletti et al. (1)] an attempt to take into account the nonlinear coupling between yaw and roll dynamics for simultaneous roll reduction and course keeping has been proposed. The twofold control objective was obtained by means of a MIMO variable structure control law using together fins and rudder, with good performances in different environments. This work represents an extension of the result proposed in [Carletti et al. (1)]. The novelty is the adoption of a more realistic nonlinear multivariable ship dynamics where the wave effects are modeled as forces and moments affecting the system dynamics rather than as disturbances affecting the control inputs.. NONLINEAR SHIP MODEL The control design deals with a nonlinear ship model affine in the inputs and disturbances of the form: ẋ = f(x) + g(x) u + d(x) w y = h(x) where x R 5 is the state vector, y R is the output and u R is the input, while w R 3 represents the external disturbances. This mathematical model is obtained by combining a ship manoeuvring model along with the waves-induced forces/moments obtained from the seakeeping theory. This allows to have a realistic description of (1) Copyright by the International Federation of Automatic Control (IFAC) 13648

2 Milano (Italy) August 8 - September, 11 a steering vessel in seaway [Perez (5)]. The reference vessel is a small relatively fast monohull whose principal characteristic parameters are available in [Perez et al. (6)] and for which a suitable set of manoeuvring coefficients was published in [Blanke and Christensen (1993)] The ship dynamics has been implemented by using the Marine System Simulator (MSS) Matlab toolbox developed by the Norwegian University of Science and Technology (NTNU)[MSS (1)]. Fig. 1. Reference frames and notation for ship motion description.1 Manoeuvring Model 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 and yaw, around the same axes. For manoeuvring analysis the position and orientation of the vessel are traditionally defined with respect to the inertial Earthfixed (n-frame) O n {x n, y n, z n } while the linear and angular velocities and accelerations of the vessel are expressed with respect to the body-fixed reference frame (b-frame) O b {x b, y b, z b } with axes oriented as shown in Figure 1. The 6DOF equation of motion in the b-frame can be written as [Fossen ()]: M RB ν + C RB (ν) ν = τ RB () where ν = [u, v, w, p, q, r] T is the generalized velocity vector according to the SNAME (195) notation, M RB is the generalized mass-inertia matrix, C RB (ν) is the Corioliscentripetal matrix. Finally, τ RB = [τ 1 τ τ 3 τ 4 τ 5 τ 6 ] T is the total vector of forces and moments acting on the ship. 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 (199)]. In particular, as the surge dynamic component is significantly slower compared to the other ones, the variable u can be considered constant and equal to the mean forward speed U. Moreover, let us consider the b-frame such that the reference axes 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 = [x g y g z g ] T can be considered located on the xz-plane of the b-frame, so that y b y g and thus y g =. As a result, a reduced 3DOF ship model in the variables v, p and r is obtained (see Figure 1). 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 cs + τ prop (3) where τ hyd describes the hydrodynamic and hydrostatic forces and moments, τ cs represents forces and moments due to the control surfaces while τ prop is the propulsion forces and moments. The nonlinear hydrodynamic term has been modeled following the approach given in [Blanke and Christensen (1993)] and using the coefficients released therein. The control module τ cs takes into account the effects on the ship motion due to the control surfaces, namely rudder (τ rud ) and active fins (τ fins ). In particular, it can be detailed as follows: τ cs = τ rud + τ fins. (4) In order to have a dynamic model affine in the inputs, the rudder mechanical angle δ rudder and the force N fins have been chosen as control variables. In particular, N fins represents the resulting hydrodynamic force component normal to the fins, defined as a function of the lift L fins and drag force D fins arising on them as L fins cos(α e) + D fins sin(α e) with α e the fins effective angle of attack between the foil and the incoming flow. As a result, the control module for the rudder is: [ ] [ ] L rud γ rud τ rud R rud L rud R rud γ rud δ rud (5) LCG L rud LCG γ rud where L rud γ rud δ rud is the lift force acting on the rudder, while R rud and LCG are the distance between the rudder Center of Pressure CP and the ship CG along the z-axis and x-axis of the b-frame respectively. The control module for the fins is: sin(βtilt ) τ fins R fins F CG sin(β tilt ) N fins (6) where R fins 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 β tilt represents the fins tilt angle. Further details can be found in [Perez (5)]. Note that, the propulsion component τ prop will be neglected in the rest of the paper as a consequence of the constant ship cruise speed assumption (u = U). In fact, this implies that the propeller effect is completely compensated by the hull hydrodynamic resistance.. Disturbance Modeling The ship motion in seaway can be described in several ways [Perez and Blanke ()]. A common framework for the design of control algorithms is based on the linear waves theory and assumes the sea elevation at a certain location to be represented by means of a zero mean Gaussian stochastic process. As a consequence, the excitation loads due to an irregular (i.e., random in both time and space) sea are described as summation of several regular waves responses. Moreover, the wave-induced loads can be completely described by their Power Density Spectra (PSD), and thus characterized in the frequency domain. As a result, the wave effect on the ship dynamics is described as an additive disturbance input w in the nonlinear model of eq. (1). Let us assume w = [ w 1 w w 3 ] T to be the 3DOF vector representing the 1st order wave forces and moments acting on the vessel. In the Laplace domain, each component w i can be considered as the output of a white Gaussian noise n i filtered by a linear filter H i (s): w i (s) = H i (s) n i (s), i = 1,, 3. (7) Indeed, the PSD for w i (s) is: P wi (ω) = H i (jω) P ni (ω) = H i (jω), i = 1,, 3. (8) with P ni (ω) the unitary PSD of n i. The filters H i (s) are usually represented as a nd order transfer function: 13649

3 Milano (Italy) August 8 - September, 11 λ wi ω ei σ i s H i = s + λ wi ω ei s + ωei, i = 1,, 3. (9) with λ wi the damping coefficient, ω ei the dominating wave frequency in the encounter frequency domain and σ i a constant describing the wave intensity. For each DOF, the filter parameters have been tuned by comparing the output motion due to the wave disturbance simulated by the shaping filters with the output obtained by the motion RAOs, available in Perez (5). In particular, as the ship moves (U ), the encounter frequency ω ei is considered and computed as: ω ei = ω i ω i U g cos χ with ω i the modal frequency of P wi (ω). Moreover, the constant σ i is computed as: σ i = max(s wi (ω)) and λ wi chosen such that P wi (ω) fits S wi (ω) using leastsquare criterion. Further details can be found in [Fossen ()]..3 State Space Representation In order to obtain the ship dynamics state space representation given in eq. (1), the 3DOF manoeuvring model described in eq. () and the wave disturbance described in eq. (9) have been combined. In addition, the roll angle ϕ and the yaw angle ψ have been considered as part of the state as well. Therefore, the state vector x can be defined as follows: x = [ v p r ϕ ψ ] T R 5, (1) where ψ = ψ ψ d with ψ d the desired yaw angle. At this point, the nonlinear ship dynamics in the augmented state space can be written as follows: M ẋ + C(x) x = T hyd (x) + T cs u + T w w (11) where the input is u = [δ rud N fins ] T, while the matrices M, C, T cs, T w and the vector T hyd are defined as follows: ˆM M = RB + M A, (1) I ] [ĈRB C(x) = E(x) ˆτ hyd [ˆτ rud T hyd (x) =, T cs = 1 E(x) = cos ϕ ] ˆτ fins I3, T w = 3 with opportune completion matrices, I k a k k identity matrix, ˆτ hyd the nonlinear hydrodynamic term in the manoeuvring model given in eq. (3) where the terms proportional to the accelerations are not considered as they have been included in the added mass matrix M A, and finally ˆτ, rud ˆτ fins the coefficients vectors multiplying δ rud in eq. (5) and N fins in eq. (6), respectively. Finally, the multivariable nonlinear ship model can be written as in (1), that is repeated here for sake of clarity: ẋ = f(x) + g(x) u + d(x) w y = h(x) (13) where x R 5 is the state vector, u = [δ rud N fins ] T R, w = [w 1 w w 3 ] T R 3, h(x) = [ϕ ψ] T R and: f(x) = M 1 C(x) x + M 1 T hyd (x) g(x) = [ g1 g ] = M 1 T cs d(x) = [ d1 d d3 ] = M 1 T w. 3. ZERO DYNAMICS ANALYSIS (14) 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 zero dynamics plays an important role in the control system design. For instance, the presence of an unstable zero dynamics prevents the application of important control techniques, such as feedback linearization, and significantly affects the achievable performances, for example by imposing limitation on the maximum control gain. 3.1 Normal Form Derivation In order to analyze the multivariable nonlinear ship model, the eq. (11) is transformed to normal form. Let {r 1,..., r m} be the vector of relative degree at a point x, i.e., the number of time the i-th output y i must be differentiated to have at least one component of the input vector u explicitly appearing [Isidori (1995)]. Let us now recall the Lie derivative, i.e., the derivative of λ( ) along f( ), as L f λ(x) = n λ f i=1 x i. i It can be shown that such a multivariable nonlinear ship model has relative degree {r 1, r } = {, } at x = [,,,, ] T. In fact, the Lie derivatives with respect to the input u and the disturbance w are respectively: and: L gj h i = L gj L f h i (x) L dj h i = L dj L f h i (x) i {1, }, j {1, } (15) i {1,, 3}, j {1, }. (16) Furthermore the matrix A(x) defined as: T T a 11 â 1 cos(x 4 ) L g1 L f h 1 (x) L g1 L f h (x) L g L f h 1 (x) L g L f h (x) a 1 â cos(x 4 ) A(x) = L d1 L f h 1 (x) L d1 L f h (x) = a 13 â 3 cos(x 4 ) L d L f h 1 (x) L d L f h (x) a 14 â 4 cos(x 4 ) L d3 L f h 1 (x) L d3 L f h (x) a 15 â 5 cos(x 4 ) (17) has full rank if the following holds: a 1i â j a 1j â i, for at least a pair (i, j), (18) with i, j {1,..., 5} cos(x 4 ). (19) It should be noticed that while the condition given by eq. (18) is related to the vessel parameters, the second condition given by eq. (19) is always satisfied in practice as cos(x 4 ) = implies that x 4 = π + k π, k N. Indeed, this is an inadmissible operational condition, being x 4 the roll angle. Thus, the system has relative degree {, } for any state x = [x 1, x, x 3, x 4, x 5 ] T with x 4 π + k π, k N. Therefore, according to [Isidori (1995)], the nonlinear system describing the vessel can be transformed to normal form by applying the following local coordinates transformation [ z(x) = z 1 (x), z (x), z 3 (x), z 4 (x), z 5 (x) ]T at x. In particular, the following choice is possible for the first r 1 + r = 4 variables: 1365

4 Milano (Italy) August 8 - September, 11 ( ξ 1 ) ( ) ( ) ξ 1 1 z1 (x) h1 (x) = = = () ξ 1 z (x) L f h 1 (x) ( ξ ) ( ) ( ) ξ 1 z3 (x) h (x) = = = (1) ξ z 4 (x) L f h (x) while the remaining (n r 1 r = 1) variable transformation can be chosen arbitrarily: η = z 5 (x) () so that the jacobian matrix describing the local coordinates transformation z(x) is nonsingular at x. Furthermore, since G = span{g} is an involutive set, z 5 (x) can be chosen so that [Isidori (1995)]: L gi z 5 (x) = i {1, }. (3) with g = [ g1 g ]. Note that, the constrain given by eq. (3) can be satisfied by assuming: η = z 5 (x) = α v + β p + γ r (4) with: [ α, β, γ,, ] ker ( g T ). (5) In particular, since g T has the following structure: g11 g1 g1 3 g T = g 1 g g 3 a base for the null space is: ( ) ker g T =,, 1 1 Therefore, by choosing: g1 g 3 g1 3 g g1 1 g g1 g 1 g1 3 g 1 g1 1 g 3 g1 1 g g1 g 1 α = g1 g 3 g1 3 g g1 1 g g1 g 1 β = g1 3g 1 g1 1 g 3 g1 1 g g1 g 1 γ = 1 the Lie Derivatives turn out to be: 1 (6) (7) (8) L g1 z 5 (x) = α g1 1 + β g1 + γ g1 3, (9) L g z 5 (x) = α g 1 + β g + γ g 3. (3) which can be re-written in a matrix form as follows: α g11 g1 g1 3 β γ g 1 g g 3 = (31) being the vector [α β γ ] T an element of the null space base for g T. As a result, the ship dynamics of eq. (1) can be written in normal form as follows: ξ 1 1 = ξ1 ξ 1 = F 1(ξ, η) + ξ 1 = ξ ξ = F (ξ, η) + G 1j u j + G j u j + η = q(ξ, η) + n(ξ, η) w 3 D 1j w j 3 D j w j (3) where F R 1 with F i = L f h i, G R with G ij = L gj L f h i and D R 3 with D ij = L dj L f h i. 3. Stability Analysis Let us now investigate the properties of the zero dynamics η = q(, η) + n(, η) w, which can be detailed as follows: η = a η η bη + ˆαw 1 + ˆβw + ˆγw 3 (33) where a, b, ˆα, ˆβ, ˆγ R all non-negative real constants. In order to investigate the stability of the zero dynamics, let us consider the following Lyapunov candidate V (η) = 1 η. In addition, let us assume the disturbances to be bounded w i (t) W i with i {1,, 3}. For the sake of the analysis, it should be noticed that the dynamics described in eq. (33) has a forcing term, αŵ 1 + ˆβw + ˆγw 3, which prevents the system to have an equilibrium point at the origin x o. Nevertheless, the Lyapunov analysis can still be used, as explained in [Khalil ()], to prove the boundedness of the solution of the state equation. To this end, let us consider the derivative of the chosen Lyapunov candidate V (η) = 1 η : V (η) = η η = η ( a η η b η + ŵ). (34) where ŵ = ˆαw 1 + ˆβw + ˆγw 3. In particular, it can be shown that: V (η) > i.f.f. η < h(ŵ) (35) V (η) < i.f.f. η > h(ŵ) where: h(ŵ) = b + b + 4 a ŵ. (36) a Therefore, by assuming c > h(ŵ), any state starting in the set {V (x) c} will remain therein for all future time since V is negative definite on the bound V = c. As a result, the solutions of the state system are uniformly bounded. 4.1 Sliding Mode Control 4. CONTROLLER DESIGN In this section, the design of a sliding mode control is discussed. The choice of a robust control technique is motivated by the fact that modeling simplifications, ship parameters variability and their coupling effects between roll and sway-yaw dynamics can significantly affect the overall performance of the control systems. In addition, it should be noticed that the boundedness of the zero dynamics, discussed in Section 3., does not introduce any particular limitation to the application of such a control technique. The control problem is to drive the roll angle ϕ to zero and the yaw angle ψ to a specific value ψ d or, equivalently, to drive the variable ψ = ψ ψ d to zero. The control inputs are the rudder angle δ rud and the normal force N fins arising on the fins. The choice of N fins instead of the fins angle α fins is due to the constrain of affine input required by the sliding mode control technique. Let us consider the multivariable nonlinear ship model given in eq. (3) and let us assume the disturbance w to be bounded by a known bound W = [W 1 W W 3 ], i.e., w i < W i, i {1,, 3}. 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: F = F F G = G Ĝ (37) with F and Ĝ the nominal models. Then, the following two sliding surfaces can be defined: s 1 = ξ 1 + λ 1ξ 1 1 =, λ 1 > s = ξ + λ ξ 1 =, λ > (38) 13651

5 Milano (Italy) August 8 - September, 11 and the Lyapunov candidate V (s) = 1 st s with s = [s 1 s ] T. At this point, the following control law is considered: u1 = Ĝ 1 F1 (ξ, η) λ 1 ξ 1 k 1(ξ, η) sign(s 1 ) u F (ξ, η) λ ξ k (39) (ξ, η) sign(s ) where the invertibility of Ĝ is a consequence of the ship dynamic model, and k = [k 1 k ] T is a gain vector required to guarantee the negative definiteness of V (s). As far as the implementation of the control law is concerned, the function sign(s) has been replaced by the continuos function tanh(ϵ s), ϵ R. This allows to significantly reduce the chattering arising from the sliding mode control without affecting the stability analysis. 4. Case Study In order to prove the effectiveness of the proposed control technique a case study is described. The ship parameters used for the nonlinear ship model described in eq. (1) can be found in [Perez (5)]. Limitations in magnitude and rate for the control variables due to the control surfaces hydraulic machinery are introduced within the simulation model. This has been achieved by using the Simulink blocks implementing the hydroservos model proposed in [Amerongen (198)]. In particular, the rudder angle saturation is set to deg while the rudder speed is limited to deg/s. The saturations values for N fins and Ṅfins are computed by considering a fins maximum angle of 8.8deg (fins static stall angle) and a fins maximum angle rate equal to 3deg/s, that is a maximum fins force of 1 7 N and a maximum fins force rate equal to 1 5 N/s, respectively. Furthermore, the proposed control system has been tested with the ship cruise speed equal to 7m/s, and against different sailing conditions. In particular, a sea state 3 with H s =.3 m and T = 6 s and a sea state 5 with H s =.5 m and T = 7.5 s, where H s is the significant wave height and T is the average wave period, have been simulated. In addition, for each sea state, quartering, beam and bow seas have been considered. As far as the effectiveness of the integrated control system for roll damping is concerned, the percentage ( Reduction ) Statistic of Roll (RSR), defined as RSR = 1 1 Sc S u, is considered, where the subscripts c and u stand for controlled and uncontrolled respectively, and S is the Root Mean Square (RMS) of the roll signal. Figure describes the roll and yaw angles for a simulation with the following operative condition: sea state 5 with an encounter angle χ e = 13deg. In particular, the sliding mode control is not activated the first 3 s and is turned on after 3 s for the rest of the simulation. The yaw reference angle changes from deg to 15deg at t = 65 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 sliding mode control is activated. Figure 3 and Figure 4 describe the control inputs (rudder on the left, fins on the right) efforts in magnitude and velocity respectively. It can be noticed that the rudder angle and the fins force are significantly below the saturation thresholds imposed by their corresponding mechanical systems (deg for the rudder angle, 1 7 N for the fins force). Furthermore, the same holds for the rudder angle rate (saturation value: deg/s) and fins rate (saturation value: 1 5 N/s). Indeed, this is extremely important for the life of the mechanical system. Table 1 and Table give a synoptic overview of the results obtained for the two sea states with respect to different encounter angles χ e, namely 3deg, 9deg and 13deg. 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. The following controller parameters have been used for the simulations: {λ 1, λ } = {.3,.3}, {k 1, k } = {.,.4} and {ϵ 1, ϵ } = {5, 5}. Sea State 3 χ e = 3 χ e = 9 χ e = 13 RSR(ϕ) RSR( ϕ) Table 1. Control system performances for the vessel sailing with U = 7m/s in slight waves. Sea State 5 χ e = 3 χ e = 9 χ e = 13 RSR(ϕ) RSR( ϕ) Table. Control system performances for the vessel sailing with U = 7m/s in rough waves. As far as a comparative analysis is concerned, to the best of the authors knowledge, all the contributions available in the literature rely on independent FRD and RRD controllers under the assumption of decoupled roll and yaw dynamics. In this work, on the one hand the decoupling assumption between roll and yaw dynamics is released, on the other hand, an integrated fins-rudder control law which takes into account the mutual interaction of the two actuators over each one of those dynamics is designed. Nevertheless, the control system proposed in Lauvdal and Fossen (1997) and Koshkouei et al. (7) can be considered for comparison. In particular, compared to these works, our approach turns out to be more effective in terms of roll reduction and control efforts. Finally, Figure 5 describes the evolution of the zero dynamics. According to the theoretical analysis, the zero dynamics η = q(, η) is bounded within ±8. Note that, the peak at time t = 65 s is due to the change of the yaw angle reference from deg to 15deg. 5. CONCLUSIONS In this paper, the problem of simultaneous roll reduction and course keeping for a marine vessel by means of active fins and rudder has been addressed. This work extends the results proposed in [Carletti et al. (1)] by considering a more realistic nonlinear multivariable ship dynamics where the wave-effects are modeled as forces and moments affecting the system dynamics rather than as disturbances affecting the control inputs. A theoretical analysis of the boundedness of the zero dynamics along with the design of a sliding mode control have been proposed. Simulations carried out by exploiting the Marine System Simulator (MSS) have shown the effectiveness of the proposed control system. 1365

6 . Preprints of the 18th IFAC World Congress Milano (Italy) August 8 - September, 11 φ [deg] Roll Angle [deg] ψ [deg] Yaw Angle [deg] Fig.. Roll and Yaw angles obtained during a 1 seconds simulation in sea state 5 - χ e = 13deg conditions. The controller action starts at t = 3s. The reference yaw angle is changed from deg to 15deg at t = 65s. 1 Rudder Angle Effort [deg] 1 x 15 Fins Force Effort [N] δ [deg] 5 N fins [N] Fig. 3. Rudder and fins efforts during the simulations in sea state 5 - χ e = 13deg conditions. The controller action starts at t = 3 s. The reference yaw angle is changed from deg to 15deg at t = 65 s. The values for the rudder angle δ and the fins force N fins do not ever achieve their saturation values, i.e. deg and 1 7 N respectively, and a good gap from those is kept. Rudder Angle Rate [deg/s] x 15 Fins Rate [N/s]. [deg/s] δ 1 [N/s] N fins Fig. 4. Rudder and fins rate during the simulations in sea state 5 - χ e = 13deg conditions. The values for the rudder angle rate δ and the fins force rate Ṅfins do not ever achieve their saturation values, i.e. deg/s and 1 5 N/s respectively. Future work will be focused on an experimental validation of the proposed control system to validate its effectiveness in a real environment Zero Dynamics Evolution Fig. 5. Zero Dynamics evolution. Note that, the zero dynamics is bounded within ±8. The peak at time t = 65 is a consequence of the yaw reference angle change. REFERENCES Amerongen, J.V. (198). Adaptive steering of ships - A model reference approach to improved manoeuvering and economic course keeping. PhD thesis, Delft University of Technology, Netherlands. Blanke, M. and Christensen, A. (1993). Rudder roll damping autopilot robustness to sway-yaw-roll couplings. In Proc. of the 1th Int. SCSS 93. Carletti, C., Gasparri, A., Ippoliti, G., Longhi, S., Orlando, G., and Raspa., P. (1). Roll damping and heading control of a multipurpose vessel by fins-rudder vsc. In 8th IFAC Conference on Control Applications in Marine Systems, (CAMS 1). Faltinsen, O. (199). Sea Loads on Ships and Offshore Structures. Marine Cybernetics, Trondheim, Norway. Fossen, T. (). 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), Isidori, A. (1995). Nonlinear Control Systems. Springer-Verlag New York, Inc., Secaucus, NJ, USA. Katebi, R. (4). A two layer controller for integrated fin and rudder roll stabilization. In Proc. of the Conference on Control Applications in Marine Systems (CAMS 4), Khalil, H.K. (). Nonlinear Systems (3rd Edition). Prentice Hall, Upper Saddle River, New Jersey. Koshkouei, A., Burnham, K., and Law, Y. (7). A comparative study between sliding mode and proportional integrative derivative controllers for ship roll stabilisation. Control Theory Applications, IET, 1(5), Lauvdal, T. and Fossen, T. (1997). Nonlinear rudder-roll damping of non-minimum phase ships using sliding mode control. In Proceedings of the European Control Conference (ECC 97). MSS (1). Mss, marine systems simulator. Available at Perez, T. (5). Ship Motion Control. Springer-Verlag, London. Perez, T. and Blanke, M. (). Simulation of ship motion in seaway. Technical Report EE37, Technical University of Denmark. Perez, T., Ross, A., and Fossen, T. (6). A 4-dof simulink model of a coastal patrol vessel for manoeuvring in waves. In Proc. of the 15th Conference on Marine Craft Maneuvering and Control (MCMC 6). Lisboa, Portugal. 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(). Stoustrup, J., Niemann, H., and Blanke, M. (1995). A multi objective h solution to the rudder roll damping problem. In Proceedings of IFAC Workshop on Control Applications in Marine Systems