comparing two different control algorithms applied to dynamic

Proceedings of the 10th Mediterranean Conference on Control and Automation - MED2002 Lisbon, Portugal, July 9-12, 2002.

COMPARING TWO DIFFERENT CONTROL ALGORITHMS APPLIED TO DYNAMIC POSITIONING OF A PIPELINE LAUNCHING BARGE E.A. Tannuri*, C.P. Pesce † Dept. of Mechanical Engineering, Escola Politécnica , University of São Paulo São Paulo, Brazil , Av. Prof. Mello Moraes 2231, SP 05508-900 fax: +55+11+8131886 e-mail: *[email protected] ; †[email protected]

Keywords: sliding mode control, optimal control, 1 Introduction dynamic positioning, launching barge.

Abstract This work was motivated by the necessity of estimating the required power for a Dynamic Positioning System (DPS) to be installed in an already existing pipeline-laying barge, which operates in both intermediate and deep waters (up to 1000 meters). Small-scale experiments were used to obtain current and wind forces acting during operation. Additional environmental effects were estimated using validated models. A numerical simulator was then developed, also including propeller dynamics, thruster allocation logics and the controller. A robust non-linear sliding mode control (SM) was applied, and was compared to the so-common optimal linear LQ control. This paper describes, in details, the application of both controllers and the methodology used for comparison, focusing on: dynamic performance and energy consumption, number of adjusting parameters, implementation simplicity and robustness to modelling errors. The analyses showed that both controllers satisfy operational performance requirements, although the SM controller is more appropriate, due to good robustness properties and fewer number of parameters to be adjusted. The LQ controller is simpler, but extremely dependent on weight matrix adjustment, what is very time-consuming and must be redone whenever the barge heading changes.

BGL1 is a crane and pipe-laying barge (Figure 1), operating in Brazilian waters for more than 20 years. Equipped with 10 mooring lines and operated with the aid of anchor handling tugboats, BGL1 was originally designed for shallow and intermediate depths. Pipe-laying has been successfully accomplished through the so-called SLay operation, shown in Figure 2. BGL1 S-Lay is done from a stern launching ramp, positioned at starboard, with or without a stinger. Pipeline is welded on deck.Traction is sustained by means of a traction-controlling machine, of the caterpillar type.

Figure 1: BGL1 photography With increasing depths, the based on conventional mooring positioning laying operation presents serious technical and economical limitations. S-Lay mode of operation is appropriate for shallow and intermediate depths. For deeper waters the socalled J-Lay launching is a recommended practice (Figure 3), in which the pipe is launched almost vertically.

The operation without DP system requires an extremely time consuming planning of anchor positioning along all the launching path. Furthermore, the complex operation usually comprises 5 anchor handling tugboats and several operators to control the winch pull-in machines.

Figure 2: S-lay launching operation

Figure 3: J-lay launching operation

A robust DP system can then improve stationkeeping ability, for both S- or J-Laying modes, with no loss of safety, enhancing operational time schedule and making the operation economically feasible.

natural extension of the so common uncoupled PID control, taking into account the coupling between horizontal motions. Such controllers are, until today, extensively used since they are structurally simple and do not require the complete modelling of environmental agents. Since the dynamics is non-linear with respect to the yaw heading of the barge, different linear models are obtained for each heading and the controller would have to be redesigned for all situations. Sometimes, new matrix weights must also be selected. A second and totally different approach is to take into account all environmental agents models during the design of the controller, taking profit of all information contained in such models. Furthermore, a non-linear controller can be used, avoiding the linearization problem, enabling the design to be done with the full non-linear model of the barge and environmental agents. However, since this methodology requires estimates of all environmental agents, namely current, wind and waves properties, the controller must also handle errors in these estimates. Environmental agents monitoring still presents practical difficulties evolving high levels of uncertainty. In other words, the controller must be robust with respect to the whole set of parameters variations. Furthermore, since complex hydrodynamic phenomena are involved in the wave and current-body interaction, the controller must also present robustness to modelling errors and guarantee performance and stability for models slightly different than that used in the design.

A previous work described in details the procedure used to determine the required power for the DPS [13], guaranteeing safe operation in deep-water even with simultaneous failure of 2 thrusters. Environmental forces were evaluated with towing tank tests complemented by validated theoretical models. A numerical simulator was then developed and an “ideal” feedback linearization controller was used in the exhaustive dynamic analysis then worked out. As a natural extension, this work deals with the practical implementation of the controller, taking into account robustness issues, performance under several environmental conditions, design Representing this class of controllers, a robust version of the feedback linearization technique was simplicity and parameter adjusting procedures. developed using the sliding mode methodology Two different controllers were implemented and originally proposed in [14]. Such methodology tested, both presenting very different design presents good robustness properties being adequate philosophies and representing two broad classes of for models containing parametric uncertainties and DPS controllers usually applied. modelling errors. In theory, the controller The LQ controller takes a linearized model of barge guarantees optimal performance with bounded dynamics and all environmental forces are model uncertainties; however, the control action highly oscillatory components considered as disturbances applied to the control presents (chattering), which is unsuitable for most loop. With proper adjustment of matrix weights, the controller can be tuned in order to guarantee applications. The control law can be modified to prescribed performance parameters under a range of solve this problem [8], avoiding chattering and still disturbance amplitudes. This kind of controller is a guaranteeing a prescribed level of performance.

This paper describes the application of both controllers and the comparison between them, focusing on dynamic performance and energy consumption, number of adjusting parameters, implementation simplicity and robustness to modelling errors. The analyses showed that both controllers satisfy operational performance requirements. SM controller is, however, more appropriate, due to good robustness properties and fewer number of parameters to be adjusted. The LQ controller is simpler, but weight matrix adjustment is very time-consuming and must be redone whenever the barge heading changes.

point at a not yet existing moon-pool. The direction of this latter force (launching direction) may change within a sector of ±90o. Fig. 4 shows either applied forces. For simplicity, the forces are ideally considered as constant. y J-Lay Launching Force (20tf) (13.0m;7.0m) (-90.7m;-12.9m)

Launching direction

x

Stinger

S-Lay Launching Force (90tf)

Figure 4: Launching operation. Pipeline applied In the second section, all barge data are described, forces (S-lay and J-lay) including the forces coefficient obtained in towing tank tests. The models used in the numerical simulator are also presented, including Thrusters’ position is shown in Figure 5. The environmental agents, barge and propellers system consists of three fore-body units and three dynamics. In the third section, the application of the stern ones. SM controller is described, with a theoretical y explanation about this control methodology. The LQ controller is also briefly exposed. In the sequence, the comparison between the controllers is presented, with some illustrative simulations x included. The conclusions are then summarized in the last section. (-50 ; 13,15)

#5 #6 (-33,99 ; 19,15)

#4 (-33,99 ; -8,44)

#1 (35,61 ; 13,15)

#2

(59,33 ; -7,64)

(42,81 ; -13,15) #3

2 Barge data and modelling BGL1’s main particulars are presented in Table 1.

Figure 5: Propeller Positions in meters (related to XY axis centered at CG)

Table 1: BGL1 main data Length (L) Beam (B) Draft (T) Position of CG (xG) Mass (M) Surge Added Mass (M11)* Sway Added Mass (M22) * Yaw Added Mass (M66) * Lateral Area (AL) Frontal Area (AF) *

121.9 m 30.48 m 5.18 m -4.18m 17177ton 1717ton 8588ton 1.28.107 ton.m2 1500m2 420m2

at low frequency

For the S-lay operation, a constant force of 900kN applied by the pipe, acting backwards at the end of the stern ramp has been considered. A smaller force of 200kN has been considered for the J-lay mode analysis. Such a force is applied to an anchoring

Figure 6 shows the simulator block diagram. The full feedforward path is just present in the SM controller. For the LQ controller there is only a wind feedforward path, current and waves effects being treated as disturbances. Detailed descriptions of the controllers are given in sections 4 and 5. Environmental Agents Model s Estimation of Environmental Agents Intensi ties and Di recti ons Waves Set Point

Dynamics Thruster and Al location Controller properties of Al gorithm thurusters Control Control thrust Actual thrust Forces and azimuthal and azimuthal angl e in propellers

Filtered Motions

Wave Filter

Wi nd

Current

Barge Dynamics Model

Measured motions

Figure 6: Block diagram of simulator

The wave filter used is a cascaded notch filter given in [4]. The simulator also models the propeller dynamics. The model also gives an estimate of total power consumed by each thruster. For simulation purposes, a series Ka propeller with a 19A nozzle, with 2m diameter and 1.6m pitch has been used. The torque (KQ), thrust (KT) and nozzle (KTN) coefficients are given in [5]. Drive system efficiency of each thruster is 80%, and the maximum power of each unit is 1650kW (resulting, approximately, a maximum thrust of 300kN). Thruster control parameters were adjusted so that it takes approximately 15s for the thruster to raise from 0kN to 300kN. The thruster allocation algorithm adopted is based on a modified pseudoinverse technique by [10]. An azimuth filtering was implemented (to prevent the thruster from tear and wear), assuming maximum azimuth rotation velocity of 9o/s. The following dynamic model gives horizontal motions of the barge:

1 1 1 F1C = C X ρLTU 2 ; F2C = CY ρLTU 2 ; F6 C = CM ρL2TU 2 2 2 2

(2)

were F1C and F2C are surge and sway current forces, F6C is yaw moment, ρ is water density and U is the barge velocity, relative to the water. The coefficients, obtained in IPT towing tank, are presented in Figure 8. Current forces and moments associated with barge yaw rotation were evaluated using a cross-flow model presented in [7]. 2.5 Cy 10*Cx 10*Cm

2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 -2.5 0

90

180

270

360

Current Related Heading (degrees)

Figure 8: BGL1 Experimental Current Coefficients.

(M + M 11 )x1 − (M + M 22 )x2 x6 − M 26 x6 2 = F1E + F1O + F1T ; (M + M 22 )x2 + M 26 x6 + (M + M 11 )x1 x6 = F2 E + F2O + F2T ; (1) (I Z + M 66 )x6 + M 26 x2 + M 26 x1 x6 = F6 E + F6O + F6T .

Static wind forces were also evaluated using experimental coefficients (CXV, CYV and CMV) obtained in a captive ‘wind’ test conducted, with the barge model superstructure turned upside down where Iz is the moment of inertia about the vertical into the towing tank. Wind coefficients are defined axis; F1E, F2E, F6E are surge, sway and yaw in the standard way by: environmental loads (current, wind and waves), 1 1 2 F1O , F2O , F6O are operation forces and moment due to F1V = C XV ρ a AFV 2 ; F2V = CYV ρ a ALV ; 2 2 the pipe being launched and F1T , F2T , F6T are forces 1 and moment delivered by propulsion system. The F6V = CMV ρ a AL LV 2 (3) 2 variables x1 , x 2 and x 6 are the midship surge, where F1V and F2V are surge and sway components sway and yaw absolute velocities (Figure 7). of the wind force, F6V is the wind yaw moment; ρa (sway) x2 Y is the air density; V, the mean ‘wind’ velocity. The corresponding coefficients are presented in Figure x1 (surge) 9. Gust spectrum simulations consider Harris-DNV x 6 = ψ (yaw) formula:








X

Figure 7: Coordinate systems

  286ω  2  S V (ω ) = 1146.C.V .2 +      V  

−

5 6

(4)

where ω is the frequency of wind oscillation, Sv is Static current forces and moment are evaluated the spectral density and C is a surface drag using the experimental coefficients (CX, CY and CM) coefficient, used as 0.0015 for moderate seas. defined as:

1.5

control action results from the “on-off” controller. However, it can be shown that the use of a smooth transition avoids the occurance of that phenomenum [8], still guaranteeing the stability of the system despite a small performance loss.

Cyv 10*Cmv Cxv

1.0 0.5 0.0 -0.5 -1.0 -1.5 0

90

180 270 Wind related heading (degrees)

360

1.8E+05

1.8E+06

1.6E+05

1.6E+06

1.4E+05

1.4E+06

1.2E+05

1.2E+06

1.0E+05

1.0E+06

8.0E+04

8.0E+05

6.0E+04

6.0E+05

4.0E+04

4.0E+05 Dx(30o) Dy(30o) -Dz(30o)

2.0E+04 0.0E+00 -2.0E+04 0.00

0.50

1.00

1.50

2.00

2.0E+05 0.0E+00 -2.0E+05 2.50

Dz(N/m)

Dx and Dy (N/m2)

Figure 9: BGL1 Experimental Wind Coefficients

The sliding mode control was successfully applied to several non-linear systems, such as robot manipulators [9], ROV´s [15], ship track control [6] and assisted dynamic positioning of moored vessels [11]. The later work describes in detail the application of such a controller to a system similar to the pipe-laying barge here considered. Rewriting (1) in terms of the accelerations and velocities in the OXY fixed coordinate system, being R the reference point that lays in the center line of the barge, xR ahead the midship section, one can easily obtain:  X R   f X , din ( X R )   F1RE + F1RO + F1RT      = + ψ Y f ( ) ( ) X C  R   Y ,din R   F2 RE + F2 RO + F2 RT ψ    F + F + F 6 RO 6 RT  6 RE    fψ ,din ( X R )  







Wave Frequency (rad/s)











    

(5)

Figure 10: BGL1 Wave Drift Coefficients for 30o wave heading angle where fi,din are functions of the inertial properties of A Jonswap sea wave spectrum was adopted and the barge and its velocity, C is a rotation matrix mean drift wave forces were evaluated using the depending on the yaw angle and FiR are the forces drift coefficients obtained from a very well known transferred to the reference point. and validated wave-body interaction computer Assuming that all terms in (5) are accurately code1. Figure 10 presents an illustrative example of known, the “estimated” feedback linearization such coefficients for a 30o wave heading angle. controller that guarantees that the states follows the Second order slow drift forces are evaluated using desired ones (X , Y , Ψ ) is given by: D D D [1]. Current-wave interaction effect on wave drift  X − 2λ X~ − λ X~   fˆ ( X )   Fˆ + Fˆ    forces is estimated following [2]. First order wave  F   ~ ~  fˆ (X )  −  Fˆ + Fˆ  + C  Y − 2λ Y − λ Y  = − C F   motions are included in the model using the well         F  fψ ( X )   Fˆ + Fˆ   ψ − 2λψψ~ − λψ ψ~    known Response Amplitude Operators (RAO’s).   







1RT

X , din

−1

R

1RE

Y ,din

R

2 RE

2 RM

R

6 RE

6 RM





D

Y









6 RT

X



−1



2 RT

D

1RM

, din



2 X 2 Y 2

D

(6)

4 Sliding Mode Controller Design

where (~) denotes the tracking error and (^) the best “estimate” of the corresponding term. The control parameters λ are related to the bandwidth of the closed loop system. It is recommended to keep them smaller either than unmodelled resonant modes or than the inverse of phase lags of the system.

The sliding mode controller is composed by an “estimated” feedback linearization action added to a term responsible to guarantee the prescribed performance and stability in face of limited errors. This term was originally defined in [14] as an onoff controller with amplitude proportional to the maximum modelling error. Due to implementation In order to guarantee a prescribed tracking delays or numerical imprecision, a high oscillatory precision even with errors in the estimates, an extra term proportional to the maximum error is added to (6) given by: 1 WAMIT

 − K X sat ( s X Φ X )      −1  = C  − K Y sat ( sY Φ Y )   − K sat ( s Φ )   ψ ψ ψ   

 ∆F1RT   ∆F2 RT  ∆F  6 RT

The later term is a non-parametric error also added to the controller. The derivatives can be evaluated either by analytical or numerical calculation. Expressions similar to (10) are used for winds and where the variables s are true measures of tracking wave loads, and all together are used in (8). performance added to an integral action defined by ~ ~ ~ s X = X R + 2λ X X R + ∫ X R and analogously for Y 5 LQ controller design (7)



and ψ.

Writing (1) for the reference point R and The gains K = (KX, KY, KΨ) are obtained such that eliminating quadratic terms one obtains: all elements of :  x1R    1 M x 2 R  = FRT + FRE + FRO (11) K − C − max FRE − Fˆ RE (8) x   6R  are positive. It has been assumed that the inertial functions fi,din and the operational forces FRO are being the mass matrix M dependent only on well known with sufficient accuracy compared to the known mass properties. Besides, the velocities in environmental forces. the fixed coordinate system OXY is related to the The parameters Φ are responsible for the surge and sway velocities by: T



(

)

smoothness of the on-off action and must be tuned such that [8]:











XR   cosψ  x1R      ,  YR  = J (ψ ) x 2 R  J (ψ ) =  senψ    0 x    6R  ψ  









Φi =

max( K i ) λi

(9a)



− senψ cosψ 0

0  0 1 

(12)

Defining x R = (x1R x 2 R x6 R X R YR ψ ) as Futhermore, it can be shown that the tracking the state vector, and linearizing the model for precision is limited by: heading angles near the operational one (ψ oper ) one 2Φ i 2 max( K i ) ~ Xi < = λi λ2i



(9b)



T



obtains:  M   0 3×3

0 3×3  0 3×3   0 3×3 I  x R =  3×3 FRT + D x R +   J( ) 0 I 3×3  oper 3×3   0 3×3  −



(13) Since λi is limited by the unmodelled frequencies and time delays, it can be seen that for higher which can be written in the state-space format: uncertainty about the system, higher is the expected x R = Ax R + B.FRT + D tracking error. (14)

The calculation of maximum errors for environmental forces is done based on the uncertainty about the direction and intensity of the corresponding environmental agent. For example, supposing that the estimated current velocity is Uˆ and its direction is αˆ , with a maximum uncertainty of ∆U and ∆α respectively. The maximum error in current forces and moments is obtained using a Taylor expansion ( EiC = FiC − FˆiC ):  ∂E   ∂E  max[EiC (U ,α )] ≤  iC  ∆U +  iC  ∆α + eiC FˆiRC  ∂U Uˆ ,αˆ  ∂α Uˆ ,αˆ

(10)

being A and B directly obtained from (13) and D the disturbance vector containing the environmental and pipe laying forces. In order to include integral action to avoid steady offset errors, it can be defined the extended state space vector

(

x R ,ext = x R

∫(X

R

− X D ).dt

∫ (Y

R

− Y D ).dt

∫ (ψ

R

− ψ D ).dt

)

T

(15) And the new state-space equation is defined by the following extended matrixes:  A 6×6 0 3×3   A ext =  0 I 0 3 3 3 3 3 3 × × ×  

 B6 3  B ext =  ×  (16)  0 3×3 

The control forces are then given by the state ones, as a robustness test. Furthermore, a feedback law FRT = −Kx R such that the gain bandwidth factor (λ) of 0.04 is used for all motions. matrix K is obtained from the associated steadyTable 2: SM design parameters and modelling state Riccati equation: errors imposed in the simulation T T A ext S + SA ext − SB ext R −1 B ext S+Q = 0 T K = R −1 B ext S

Parameter

Real Value U=1.0m/s α=80o

(17) Current Velocity Current Dir.

where Q is the state and R is the control weight Wave Sig. Height matrixes. Wave Dir.

6 Case studies and comparison between controllers A common Brazilian Campos Basin environmental, condition shown in Figure 11, is considered in the present analysis. Hs is the significant wave height and Tp is the peak period. Y

270°

X 0°

180° Current (80°) U=1.0m/s

Max. Variation(*) ∆U=0.2m/s ∆α=10o

ˆ =90o 

∆HS=1.0m ∆αW=10o



Hs=1.5m αW=90o

S=2.0m

ˆ W =90 

o

Wave Peak Per.

ΤP=8.45s

TˆP = 8.0 s

Mean Wind Dir.

αWi=100

ˆ Wi =90

Current/Wave/ Wind mod. error Wind Velocity Measuring Error

+20%

20%

−10%

10%

o 



P

= 1.0 s

∆αWi=10o

o

(*)considered in the design The resulting path of the barge is presented in Figure 13, and tracking errors in Figure 14. Since the admissible launching error is 7,5m, the controller fully satisfies performance requirement, also keeping the heading error smaller than 1.5o.

Wind (100°) V=12m/s

Waves (90°) Hs=1.5m Tp=8.45s

Estimated Value(*) Û=1.2m/s

100 0

90° Y (m)

-100

Figure 11: Environmental conditions

-200 -300 -400 -500 -600 -200

0

200

400

600 800 X (m)

1000

1200

1400

1600

Figure 13: Resulted path using SM controller Posição Pto Referência

100

0

Y (metros)

6 4 XY error(m )

R=3 00m

The main task of the barge DPS is to follow a prescribed path determined by the pipe being launched. For the S-lay operation, Figure 12 shows a typical path that must be followed with a constant tangential velocity of 0.5m/s.

2

-100 -200 -300 -400

0 -500 -200

30°

0

200

400

-2 0

500

1000

600 X (metros)

1500

800

1000

2000

1200

2500

1400

3000

Ψ error(degrees )

10

Figure 12: Typical reference path

5 beginning of c urve

end o f c urve

0 The SM controller is designed assuming that all environmental agents are collinear, with a direction -5 0 500 1000 1500 2000 2500 3000 of 90o. Furthermore, all estimated values used in Tim e(s ) the design, as well as the modelling errors imposed to the simulation, are shown in Table 2. It is Figure 14: Tracking and heading errors: SM control important to mention that the controller was designed for conditions very different from the real

The LQ controller was also tested under the same environmental conditions and modeling errors shown in Table 2. The weight matrixes were adjusted to reach an acceptable performance. The simulation is shown in Figures 15 and 16. The model was linearized about 0o heading. The performance is worse than the SM controller, though still attending the required maximum 7,5m deviation. The heading error is kept smaller than 3o after the transient. It must be stressed that the weight matrixes determines the performance, and possibly better results could be obtained by other choices not tested here.

can be evaluated by the desired response of the final system. Since the controller “knows” all physical phenomena involved in the process, because all the models of environmental forces and dynamics are included, it does not require any other adjustment. The performance is then guaranteed for all possible maneuvers if the environmental conditions remain inside the predictions. Furthermore, the controller can be continually tuned in order to accommodate slow variations of waves, wind and current. The only requirements are simple estimations of some parameters, namely Hˆ S ,αˆ W , TˆP ,Uˆ ,αˆ ,αˆ Wi .

For both controllers the mean total power consumption was approximately 5300kW with a peak consumption of 7000kW. Some differences between both control philosophies are exemplified by this example. It must be stressed again that both of then satisfied the performance requirements for the pipe-laying operation.

An apparent disadvantage of this class of controller is that it requires estimations of the environmental conditions, what may be expansive and may imply practical difficulties. Due to this problem, the SM controller has been designed to handle variations on those estimations, in such a way that even poor estimations are acceptable. The case study, e.g., showed that the controller is able to handle a 50% variation in the significant wave height.

100 0

Y (m)

-100 -200 -300 -400 -500 -600 -200

0

200

400

600 800 X (m)

1000

1200

1400

1600

Posição Pto Referência

Figure 15: Resulted path using LQ controller 100 0 Y (metros)

6

XY error(m )

4 2

-100 -200 -300 -400 -500 -200

0

0

200

400

600 X (metros)

800

1000

1200

1400

-2 0

500

1000

1500

2000

2500

3000

Ψ error(degrees )

10

5 beginning of c urve

end o f c urve

0

-5 0

Furthermore, despite the well developed wind sensing technology, some new techniques are being developed to measure current and waves. For example, vertical motions of the ship can be used to estimate the wave incident spectrum, with maximum errors which can be easily handled by a robust controller design [12].

500

1000

1500 Tim e(s )

2000

2500

3000

Figure 16: Tracking and heading errors: LQ control The SM controller requires estimates of all environmental conditions and boundaries in the errors on these estimations. Additionally, it also requires the closed loop system bandwidth, which

Another apparent disadvantage seems to be the complexity of a model-based controller such the SM. However, nowadays, since oceanic systems require sampling time of approximately 1s, this is not a real problem even with the use of common personal computers. Conversely, the LQ controller represents a class of “blind” controllers which consider all environmental forces (except wind) as disturbances in the control loop. As can be seen, the performance requirements can be attended even without any information about environmental conditions. The tuning process is, however, very important, and from different matrixes would result very different performances. Exhaustive simulations must be done to select the “best” weight matrixes guaranteeing that performance

requirements are satisfied environmental conditions.

for

the

possible designed with the model linearized about the actual state of the ship. However, such controller would be extremely complex and the stability of such The second important point that must be analyzed is related to the linearization necessary to the “gain scheduled” system would be questionable. design of LQ controller. For the present, the controller so obtained for an operation heading of 0o guarantees stability for heading angles between – 64o and 64o. The stability boundary can be easily obtained by the calculation of the eigenvalues of the closed loop matrix of system (14) controlled by (17). Figure 17 illustrates the possible headings that the barge may reach safely with the LQ controller. 40

XY error(m )

20

0

-20 -40

0

200

400

600

800

0

200

400

600

800

1000

1200

1400

1600

1800

2000

1000

1200

1400

1600

1800

2000

+64o Stable Headings for the LQ controller

Ψ error(degrees )

20 10

0

-10 -20 -30 Tim e(s )

Figure 18: Rectilinear 50o path: LQ control 40

XY error(m )

20

-64o

Figure 17: Stable headings for the LQ controller

If a linear controller is used, a full stability analysis must be done, defining the allowable headings the system may reach. A possible approach would be the use of several linear controllers, each one

-40 0

200

400

600

800

0

200

400

600

800

100 0

1200

1400

1600

1800

2000

100 0

1200

1400

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20 Ψ error(degrees )

Of course, the controller performance gets worse if the heading angle approximates the boundaries values. The SM controller, conversely, does not require the linearization of the model, what guarantees stability and good performance for all headings. For the sake of illustration, a rectilinear 50o path is considered. The simulations are conducted under the same environmental conditions and modeling errors used in the first case. The initial heading error is –25o and both controllers were applied. As expected, Figure 18 shows that the response of the system under LQ controller is extremely oscillatory, since the heading is close to the stability boundary. The performance of the system gets worse, taking approximately 800s to decrease the oscillation to an acceptable level. The SM controller, for its turn, reaches the desired path and heading after a short transient and without oscillations (Figure 19), keeping the overall good performance presented in the first simulation.

0 -20

10 0 -10 -20 -30 Tim e(s )

Figure 19: Rectilinear 50o path: SM control

7 Conclusions In the present work the control of a pipe laying barge has been analyzed, focusing on the comparison between two different philosophies commonly used in DP systems. A numerical simulator was developed in a previous work and used here. Current and wind forces are evaluated using towing tank tests and additional effects are included using fully validated models. Wave effects are considered using an wave-body interaction software. The simulator includes the 6 azimuthal propeller dynamics and a thrust allocation algorithm. The first class of controllers tested is a linear and non-model based control, commonly used since early times in DPS applications, designed by LQ theory. Second category is composed by non-linear model-based control, here represented by a Sliding Mode controller.

Despite the simplicity of the LQ control and the fact that it does not require any information about the environmental conditions, the weight matrixes selection is a very time-consuming task, and determines the overall performance of the controller. Exhaustive simulations are required to verify if the performance requirements are satisfied for all environmental conditions and desired reference paths since the controller design cannot guarantee itself the performance for all those conditions. Furthermore, a stability analysis must be carried out, and several controllers must be used for different headings since nonlinear effects are disregarded in the design.

[3] D. Bray, “Dynamic Positioning”, The Oilfield Seamanship Series, Vol.9, Oilfield Publications Ltd. (OPL), (1998). [4] T.I., Fossen, “Guidance and Control of Ocean Vehicles”, John Wiley and Sons, Ltd., (1994). [5] E.V. Lewis, “Principles of Naval Architecture”, SNAME, Jersey City, (1988). [6] F.A., Papoulias,A.J., Healey, “Path control of surface ships using sliding modes”, J. Ship Res., Vol.36, No.2, pp141-153, (1992).

[7] A.N. Simos, et al., “A quasi-explicit hydrodynamic model for the dynamic analysis of a moored FPSO under current action”, J. of Ship The SM control is, as far implementation is Res., Vol.45, No.4, pp289-301, (2001). concerned, more complex, because it encloses a model of the system and environmental agents and [8] J.J.E., Slotine, “Sliding controller design for requires estimates of environmental conditions. non-linear systems”, Int. J. Control, Vol.40, No.2, However, due to robustness properties, the pp421-434, (1984). controller guarantees stability and performance in [9] J.J.E., Slotine, “The robust control of robot the presence of estimation and modeling errors, manipulators”, Int. J. Robotics Research, Vol. 4, bounded by prescribed maximum values. No. 2, pp49-64,(1985). Therefore, simple systems for environmental conditions monitoring (or even real time estimators [10] O.J. Sørdalen, “Optimal Thrust Allocation for based on ship motion) can be used, since the Marine Vessels”, Control Engineering Practice, controller can handle large errors on these Vol.5, No.9, pp1223-1231, (1997). estimations. The complexity of the controller is not [11] E.A. Tannuri, D.C. Donha, C.P. Pesce, a problem for the computational capacity of modern “Dynamic positioning of a turret moored FPSO computers added to the fact that the ocean systems using sliding mode control”, Int. J. Robust and dynamics are extremely slow. Nonlinear Cont., Vol.11, 13, pp1239-1256, (2001). [12] E.A. Tannuri, C.P. Pesce, D.C. Donha, Acknowledgements “Assisted dynamic positioning system for a FPSO based on minimization of a cost function”, This work has been supported by Petrobras, State CAMS2001 Proceedings, Scotland, (2001). of São Paulo Research Foundation (FAPESP – Proc.No 98/13298-6) and CNPq/CTPETRO [13] E.A. Tannuri et al., “Dynamic Positioning of a (process no. 469095/00). The authors thank Mr. R. Pipeline Launching Barge”, ISOPE 2002 Conference, Kyushu, Japan, (2002). Pesce for helping with simulations and diagrams.

References

[14] V.I., Utkin, “Sliding Modes and their application to variable structure systems”, MIR Publishers, Moscow, (1978).

[1] J.A.P.Aranha, A.C. Fernandes, “On the secondorder slow drift force spectrum”, Applied Ocean [15] D.R., Yoerger, et al., “Supervisory Control System for the JASON ROV”, IEEE J. on Oceanic Res., Vol.17, pp.311-313, (1995). Eng., Vol. OE-11, No. 3, pp392-400, (1986). [2] J.A.P. Aranha, “Second order horizontal steady forces and moment on a floating body with small forward speed”, J. Fluid Mec., Vol 313, (1996).