Adaptive Control Design for Cargo Ship Steering with Limited Rudder Angle

Proceedings of the 12th IFAC Symposium on Transportation Systems Redondo Beach, CA, USA, September 2-4, 2009

Adaptive Control Design for Cargo Ship Steering with Limited Rudder Angle Nazli E. Kahveci ∗ Petros A. Ioannou ∗∗ ∗

Ford Research and Advanced Engineering, Dearborn, MI 48121 USA (e-mail: [email protected]) ∗∗ University of Southern California, Los Angeles, CA 90089 USA (e-mail: [email protected])

Abstract: Marine transportation offers a cost-effective and viable alternative to cargo aircraft, trains, and trucks in general. Traffic control in congested waterways has recently become a challenging task demanding higher accuracy in ship navigation and advanced methods for collision or grounding avoidance. Modern ship control systems call for new technologies to be applied to ship steering involving station-keeping and course-changing maneuvers. Uncertain cargo ship dynamics require robust steering control design techniques potentially reducing the effects of external conditions and minimizing path deviations in case particularly strong lateral wind or wave forces are experienced. Among the additional issues to be addressed are possible conflicts between rudder angle limitations and controller performance. We present an adaptive steering control design for uncertain cargo ship dynamics subject to input constraints while avoiding performance compromises under changing environmental conditions. Several maneuvering scenarios are simulated to verify the effectiveness of our control design approach. Keywords: Automatic ship steering; parametric uncertainty; robust adaptive control. 1. INTRODUCTION The impact of automatic control on transportation and vehicle systems is well-known. The driving forces behind the increasing use of control in transportation systems are recognized in Kiencke et al. [2006] to be the rising need for transportation services and the demand for higher safety level. Development of unmanned marine vehicles is peculiarly critical in providing cost-effective solutions to several littoral, coastal and offshore problems as described in Sutton and Roberts [2006]. The dynamics of unmanned underwater vehicles, semisubmersibles, and unmanned surface craft are nonlinear in nature and are subject to a variety of disturbances such as varying drag forces, vorticity effects and currents, and are required to be robust in terms of disturbance rejection, varying vehicle speeds and dynamics as discussed in Kiencke et al. [2006]. It is also argued that the navigation of unmanned surface vehicles can be difficult and further complicated if the vehicles operate in areas of non-existent/degraded Global Positioning System (GPS) reception, or in a pack with similar vehicles. On long courses, ships are usually put under automatic steering. However, as explained in Slotine and Li [1991] the dynamic characteristics of a ship strongly depend on several uncertain parameters such as water depth, ship loading, wind and wave conditions. Adaptive controllers ⋆ This work was supported by the National Science Foundation Grant No. 0510921. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

978-3-902661-50-0/09/$20.00 © 2009 IFAC

can be employed to achieve improved control performance under varying operating conditions, as well as to avoid energy losses due to excessive rudder motion. A ship model parameter identification scheme is developed in Casado and Ferreiro [2005] using the experimental data for the temporal variation of the yaw angle in the presence of ship model uncertainties possibly due to winds, waves, currents, and other exogenous effects, and different sailing conditions faced along coastal or open sea sailing routes. A maneuvering design robust to unknown bounded disturbances is presented in Skjetne et al. [2004], and an adaptive version of the maneuvering methodology is presented in Skjetne et al. [2005]. A control law involving feedback from both position and velocity measurements is derived for ships using integrator backstepping in Pettersen and Nijmeijer [2001]. In case only position measurements are available the ship velocity term needs to be generated using an observer. The experimental results demonstrate satisfactory reference tracking performance with errors depending on factors such as parametric uncertainties, measurement noise, waves, currents, and thruster saturation. For small variations in the steering characteristics the feedback controllers can be used to correct the behavior of the ship automatically. For large variations, however, the parameter settings for the autopilot need to be adapted manually, and as pointed in Amerongen and Cate [1975] for large ships with instantly changing dynamics adapting the autopilot settings is critical to handle accurate course changes in narrow coastal waters, whereas a variation in water depth might introduce course instability.

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Improving the navigational efficiency of the course changing and track keeping performance of ship steering control systems is not an easy task due to the maneuverability difficulties. According to McGookin et al. [2000] the control design challenges are mainly due to the restricted size of the rudder, which has to be deflected by a large amount for significant course changes. Several issues regarding ship steering autopilot systems are discussed in Yang et al. [2003], Kallstrom [2000], Lauvdal and Fossen [1998], Fang and Luo [2005]. The autopilot performance restrictions due to modeling uncertainties are considered in Du et al. [2007] and Peng et al. [2007]. The underactuation related concerns are addressed in Li et al. [2008], Jiang [2002], Do et al. [2003], Do and Pan [2004, 2006a,b]. A control problem for autonomous vehicles with control constraints is described in Naik and Singh [2007]. We introduce simplified cargo ship dynamics with rudder angle constraints and present a nominal control design in Section II. The adaptive control design is developed in order to account for several uncertainties in lateral ship dynamics in Section III. The performance of our yaw angle control strategy is illustrated through simulations in Section IV. Finally, our conclusions appear in Section V.

2. CARGO SHIP DYNAMICS AND NOMINAL CONTROL DESIGN Among several mathematical models describing the ship steering dynamics, the first-order model of Nomoto is recognized to be the simplest one in Amerongen [1984]: τ ψ¨ + ψ˙ = Kδw

(1)

where ψ is the yaw angle of the ship, δw is the rudder angle, K and τ are constants determined by the ship’s speed and length. One can define the ship’s yaw rate using: r = ψ˙

(2)

and derive its relation to the rudder angle through: K r = δw 1 + τs

A=



0 1 0 −1/τ



, B=



0 K/τ



, Cp = [1 0 ]

(7)

where the performance output matrix, Cp is chosen such that the performance output, z is defined as the cargo ship’s yaw angle, ψ . We define symmetric upper and lower magnitude limits on the rudder angle via δ¯w = 30 deg , and use the nominal data: K = −0.05 (sec)−1 and τ = 18 sec . A robust controller is desired to handle the effects of the parametric uncertainties. Following the robust adaptive control design proposed in Kahveci et al. [2007, 2008] we construct our scheme in two steps: The saturation limits are initially ignored, and the controller for the original linear dynamics is developed; the design is then augmented with an anti-windup compensator to recover the corresponding linear system performance. For a given yaw angle reference, ψ a compatible state vector, xr can be derived via the left pseudoinverse of Cp . Our control objective is to regulate the tracking error: e = x − xr

(8)

to zero where xr is a constant state reference. The state reference term, Axr can then be combined with the lateral disturbance term, D representing the effects of the external forces acting on the ship dynamics in Equation (4) in order to form an artificial disturbance, d¯ : d¯ = Axr + D

(9)

Asymptotically rejecting d¯ would reject D , and regulate the state error, e implying in turn a reference tracking response for the original system. In order to reject the artificial disturbance, we augment the state vector as:   e˙ xaug = (10) Cp e and solve the regulator problem for the augmented system using the Algebraic Riccati Equation (ARE): T ATaug P + P Aaug + Qz − P Baug Rz−1 Baug P = 0 (11)

(3)

We impose a saturation nonlinearity on the control input, δw and describe the lateral ship dynamics as:

where the augmented system matrices are defined as:     A 0 B Aaug = , Baug = (12) Cp 0 0

In Equation (11) Qz and Rz are chosen accordingly as: x˙ = Ax + Bsat(δw ) + D z = Cp x

(4) (5)

T

where x = [ ψ r ] is the state vector, D is the effect of the external wind and wave forces acting as disturbances on the lateral dynamics, and sat(δw ) is the effective rudder angle. The scalar input saturation function: sat(δw ) = sign(δw ) min ( |δw | , δ¯w ) , δ¯w > 0

Qz =



 0 0 × Q , Q > 0 , Rz > 0 0I

(13)

and by solving Equation (11) we obtain the controller gain: T K = Rz−1 Baug P , K = [K1 K2 ]

(14)

leading to a PI controller in the form: (6)

has δ¯w and −δ¯w as the upper and the lower saturation limits, respectively. The system matrices are given by:

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u = −K1 e − K2

Z

0

t

Cp e(τ )dτ

(15)

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A possible realization of the above controller would be: x˙ c = Ac xc + Bc (xr − x) (16) u = Cc xc + Dc (xr − x) (17)

As shown in Brieger et al. [2007] to construct the antiwindup compensator one can define F using the relation:   W −1 F =− R+ BT P ǫ

with Ac = 0 , Bc = I , Cc = K2 , Dc = K1 . The tracking problem is thus solved for a linear plant with no input constraints where it is assumed that the output, u of the controller in Equation (17) can be directly applied to the input of the plant. In applications involving the actuator nonlinearities, however, the corresponding upper and lower saturation limits need to be considered. The tools of stability analysis have been recently used to investigate the control design with anti-windup augmentation in the adaptive context, and upon combining the control structure with an adaptive law, the closedloop stability has been established in Kahveci and Ioannou [2007]. The corresponding adaptive control design has been verified through aircraft control simulations with unknown plant parameters using control inputs which are prone to saturation by Kahveci et al. [2008]. The plant transfer function matrix is defined as G(s) = (sI −A)−1 B where sensors are assumed to be available for the yaw angle and and the yaw rate, implying an identity C matrix.

where P > 0 satisfies the following ARE: AT P + P A − P BRB T P + CpT Cp = 0

When F is defined using Equation (20) and (21), the antiwindup compensator can be augmented into the system as:



F I



, Daw = 0 (19)

(22)

yaw = Caw xaw + Daw (δw − sat(δw ))

(23)

The term yaw2 ∈ R2 modifies the controller realization in Equation (16) and (17) as: x˙ cm = Ac xcm + Bc (xr − x) − Bc yaw2 u = Cc xcm + Dc (xr − x) − Dc yaw2

(24) (25)

with Ac , Bc , Cc , and Dc as defined before. The term yaw1 ∈ R modifies the input u by −yaw1 , and defines:

where F is chosen such that A + BF is a Hurwitz matrix. In Kahveci and Ioannou [2007], and Kahveci et al. [2008], a Linear Matrix Inequality (LMI) based anti-windup compensator design methodology is followed for adaptive control design. Here we use the alternative Riccati based method to construct the plant-order anti-windup compensator for stable plants.

x˙ aw = Aaw xaw + Baw (δw − sat(δw )) T  T . where yaw = yaw1 yaw 2

A possible realization of the anti-windup compensator is (Aaw , Baw , Caw , Daw ) with: Aaw = A + BF , Baw = B , Caw =

(21)

The design parameters R, W and ǫ can be properly chosen for a similar L2 performance as promised by the respective LMI based design methodology. The corresponding antiwindup design ensures the local stability of the nonlinear loop with the region of attraction defined by an ellipsoid as described in Brieger et al. [2007] where a simplified tuning approach is also provided for the choice of parameters to define the region of attraction and an upper bound on the local L2 gain. The reader not familiar with anti-windup synthesis techniques using Riccati equations might also want to consult Sofrony et al. [2007].

−1

If G(s) = N (s) M (s) is a full-order right coprime factorization of the transfer function matrix of the linear plant, the anti-windup compensator can be described using   M (s) − I Kaw (s) = (18) N (s)

(20)

δw = u − yaw1

(26)

where u is the modified control input derived through Equation (24) and (25). The nominal control structure is summarized in Figure 1.

Fig. 1. Nominal control system for cargo ship dynamics subject to input constraints.

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3. AN INDIRECT ADAPTIVE CONTROL DESIGN

term projection can then be implemented as described in Kahveci et al. [2008].

The uncertain model parameters can be estimated online and used to update the controller gains along with the antiwindup compensator. The terms in lateral ship dynamics are first filtered to avoid high frequency sensor noise amplification by the derivative term. We introduce a tuning parameter λ > 0 and define the first order prefilter 1/(s + λ) . For any fixed set of plant parameters we obtain:

The state error dynamics for any constant state reference, xr can be written using the relation:

1 1 s x=A x+B sat(δw ) s+λ s+λ s+λ

(27)

e˙ = Ae + Axr + Bsat(δw )

This state tracking error can be regulated using the proposed adaptive control design methodology. Based on the Certainty Equivalence Principle we construct the rudder angle command as: ˆ 1 (e + yaw ) − yaw (34) ˆ 2 1 (Cp e + yaw ) − K δw = −K 1 2 2 s

The regressor vector would be accordingly defined as: φ=

1 T [ ψ r sat(δw ) ] s+λ

(28)

ˆ 1 and K ˆ 2 are evaluated through Equation (11) where K ˆ replacing the two system with the estimates, Aˆ and B matrices, A and B within the definitions of the augmented system matrices, Aaug and Baug in Equation (12).

(29)

The anti-windup modification terms are generated using:

The estimation model would be defined through: zˆ1 = θ1T φ , zˆ2 = θ2T φ

(33)

using the estimates θ1 and θ2 at each time t : 

θ1 (t) = a ˆ11 a ˆ12 ˆb1

T



, θ2 (t) = a ˆ21 a ˆ22 ˆb2

T

(30)

where a ˆ11 , a ˆ12 , a ˆ21 , a ˆ22 , ˆb1 , ˆb2 are the estimates of the plant parameters. As a next step we construct the normalized estimation errors, ǫ1 and ǫ2 : ǫ1 = (z1 − θ1T φ)/m2 , ǫ2 = (z2 − θ2T φ)/m2

ˆ Fˆ )−1 B ˆ [ δw − sat(δw ) ] yaw1 = Fˆ (sI − Aˆ − B ˆ Fˆ )−1 B ˆ [ δw − sat(δw ) ] yaw = (sI − Aˆ − B 2

(36)

Above Fˆ is generated through Equation (20) and (21) ˆ. using also the estimates Aˆ and B A block diagram for the proposed adaptive control scheme is demonstrated in Figure 2.

(31) 4. SIMULATIONS

using the following proper normalization signal: m2 = 1 + φT φ

(35)

(32)

For the adaptive law we employ the discrete version of the Least-Squares Algorithm (LSA) given in Ioannou and Sun [1996] and apply a modified LSA with robust weighting as explained in detail in Kahveci et al. [2008]. Assuming that the lower and upper limits for each entry of A and B matrices are known, an orthogonal term-by-

4.1 Uncertain Cargo Ship Dynamics with No Control Input Constraints In all our simulations same unknown conditions are applied for cargo ship dynamics. Here the target yaw angle is a sinusoidal signal with amplitude 0.6 deg and frequency 0.025 Hz Since the rudder angle constraints are not included, the input saturation phenomenon is not

Fig. 2. Adaptive control system for cargo ship dynamics subject to control input constraints.

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0.6

6

0.4

4 Rudder angle (deg)

Yaw angle (deg)

5

0.8

0.2 0 −0.2 −0.4

2 0 −2 −4

−0.6 −0.8 0

x 10

−6

Reference signal System response 10

20

30

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50 Time (sec)

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−8 0

100

Fig. 3. System response: Yaw angle (Section 4.1)

Augmented control input Effective control signal 10

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50 Time (sec)

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Fig. 6. Commanded control signal and effective rudder angle (Section 4.2)

80

The reference yaw angle and the unstable system response are shown in Figure 5. The commanded input goes beyond the rudder angle magnitude constraints whereas the effective input is bound to remain between the lower and upper saturation limits as presented in Figure 6.

Rudder angle (deg)

60

40

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0

4.3 Input Constraints and Modified Adaptive Control −20

−40 0

10

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50 Time (sec)

60

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The same sinusoidal signal is applied as the yaw angle reference along with the same rudder angle magnitude constraints. The adaptive controller with anti-windup modi-

Fig. 4. Control signal: Rudder angle (Section 4.1) 0.8 0.6

observed. Despite the parametric uncertainties, the cargo ship yaw angle efficiently follows the reference signal. The system response and the commanded control input are demonstrated in Figure 3 and Figure 4, respectively.

Yaw angle (deg)

0.4 0.2 0 −0.2 −0.4

4.2 Input Constraints and Unmodified Adaptive Control

−0.6

The adaptive control design with no anti-windup modification is applied to the uncertain cargo ship dynamics with a magnitude limitation of 30 deg imposed on the control input. There is a significant deviation in the system response which cannot be avoided.

−0.8 0

10

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50 Time (sec)

60

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Fig. 7. System response with anti-windup (Section 4.3) 200

40 35

Reference signal System response

Augmented control input Effective control signal

Reference signal System response 150

30

Rudder angle (deg)

Yaw angle (deg)

25 20 15 10 5

100

50

0 0 −5 −10 −15 0

10

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50 Time (sec)

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−50 0

100

Fig. 5. System response in the presence of input saturation limits (Section 4.2)

10

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50 Time (sec)

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Fig. 8. Control signal including anti-windup modification (Section 4.3)

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−40

Rudder angle (deg)

Anti−windup modification

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−60 −80 −100

10 0 −10 −20

−120 −140

−30

−160 0

−40 0

10

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50

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Augmented control input Effective control signal 10

20

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Time (sec)

Fig. 9. Anti-windup modification, yaw 1 (Section 4.3) y y

70

80

90

100

2 aw21 aw22

1.5

0.06

Anti−windup modification

Anti−windup modification

60

Fig. 12. Control signal with anti-windup (Section 4.4)

0.08 0.07

50 Time (sec)

0.05 0.04 0.03 0.02 0.01 0 −0.01 0

1 0.5 0 −0.5 −1 −1.5

10

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−2 0

100

Time (sec)

10

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Time (sec)

Fig. 10. Anti-windup modification, yaw 2 (Section 4.3)

Fig. 13. Anti-windup modification, yaw 1 (Section 4.4)

fication is implemented, and hence the system saturation is handled with minimal deviation from the guideline.

−3

1

x 10

0.8

The yaw angle is shown in Figure 7, and the rudder angle is presented in Figure 8. The first anti-windup modification term, yaw 1 is demonstrated in Figure 9, and the other two anti-windup modification terms, yaw 21 and yaw 22 are displayed in Figure 10.

Anti−windup modification

0.6

4.4 A Sequence of Square Waves as a Yaw Angle Reference

0.4 0.2 0 −0.2 −0.4 −0.6 y −0.8

A sequence of square waves is applied as a reference signal for the yaw angle, and the system is observed to experience saturation. The adaptive anti-windup compensator is implemented for remedy.

−1 0

y

aw21 aw22

10

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80

90

100

Time (sec)

Fig. 14. Anti-windup modification, yaw 2 (Section 4.4)

0.08 Reference signal System response

0.06

The yaw angle is shown in Figure 11, and the rudder angle is demonstrated in Figure 12. The first anti-windup modification term, yaw 1 is presented in Figure 13, and the other two anti-windup modification terms, yaw 21 and yaw 22 are presented in Figure 14.

Yaw angle (deg)

0.04 0.02 0 −0.02 −0.04

4.5 Wind and Wave Effects: Unstable System Response

−0.06 −0.08 0

10

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30

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50 Time (sec)

60

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80

Fig. 11. System response: Yaw angle (Section 4.4)

90

100

The lateral cargo ship dynamics can be severely affected by the wind and wave forces, possibly resulting in unstable oscillations at the control actuators in case an anti-windup compensator is not properly augmented.

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12th IFAC CTS (CTS 2009) Redondo Beach, CA, USA, September 2-4, 2009

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250

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150

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Yaw angle (deg)

Augmented control input Effective control signal

−10

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Disturbance Reference signal System response

−25 0

100

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Fig. 15. System response with no anti-windup (Section 4.5)

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Fig. 18. Control signal with anti-windup (Section 4.6)

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x 10

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Augmented control input Effective control signal

0

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Anti−windup modification

Rudder angle (deg)

2

1.5

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−40 −60 −80 −100 −120 −140

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−160 −0.5 0

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−180 0

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Fig. 16. Control signal with no anti-windup (Section 4.5)

Fig. 19. Anti-windup modification, yaw 1 (Section 4.6) 0.1

We simulate a scenario with no anti-windup in the presence of external disturbances, and present the results in Figure 15 and Figure 16.

y y

Anti−windup modification

4.6 Wind and Wave Effects: Disturbance Rejection The yaw angle of the cargo ship is assigned to track the sinusoidal reference signal in the presence of the same external disturbance as in Section 4.5. The adaptive control design has the potential to accordingly update the system parameters online, and adapt to the changing environmental conditions. 0.6

aw22

0.06

0.04

0.02

0

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Time (sec)

Fig. 20. Anti-windup modification, yaw 2 (Section 4.6) The yaw angle reference, the disturbance, and the system response are presented in Figure 17. The rudder angle is demonstrated in Figure 18. The anti-windup modification terms are shown in Figure 19 and Figure 20.

0.4

0.2 Yaw angle (deg)

aw21

0.08

0

5. CONCLUSION

−0.2

−0.4 Disturbance Reference signal System response

−0.6

−0.8 0

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50 Time (sec)

60

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Fig. 17. System response: Yaw angle (Section 4.6)

90

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Automatic ship steering is an active research field demanding parametric uncertainty to be taken into account for robust cargo ship performance. Relevant challenges for steering control design include uncertain ship dynamics, lateral winds and wave effects acting as external disturbances, and control input constraints limiting the achievable performance of marine vehicles.

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Based on a constrained cargo ship model for steering dynamics we consider changing environmental conditions and implement a robust adaptive control design using the rudder angle as the control input. Our ship steering strategy is verified through simulation scenarios where lateral wind and wave forces are experienced and efficiently rejected. As a result, cargo ship maneuverability issues are addressed under various loadings and sea conditions despite the significant actuator saturation constraints.

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