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, 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. 1 Introduction 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 S- Lay 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 so- called J-Lay launching is a recommended practice (Figure 3), in which the pipe is launched almost vertically. Proceedings of the 10th Mediterranean Conference on Control and Automation - MED2002 Lisbon, Portugal, July 9-12, 2002.
Transcript
COMPARING TWO DIFFERENT CONTROL ALGORITHMS APPLIED TO DYNAMIC POSITIONING OF A PIPELINE
LAUNCHING BARGE
E.A. Tannur i* , 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
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 li near 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 modelli ng 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.
1 Introduction
BGL1 is a crane and pipe-laying barge (Figure 1), operating in Brazili an 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 S-Lay 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-controlli ng machine, of the caterpill ar type.
Figure 1: BGL1 photography
With increasing depths, the based on conventional mooring positioning laying operation presents serious technical and economical limit ations. S-Lay mode of operation is appropriate for shallow and intermediate depths. For deeper waters the so-called J-Lay launching is a recommended practice (Figure 3), in which the pipe is launched almost vertically.
Proceedings of the 10th Mediterranean Conferenceon Control and Automation - MED2002Lisbon, Portugal, July 9-12, 2002.
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 abilit y, for both S- or J-Laying modes, with no loss of safety, enhancing operational time schedule and making the operation economically feasible.
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 simplicity and parameter adjusting procedures.
Two different controllers were implemented and tested, both presenting very different design philosophies and representing two broad classes of DPS controllers usually applied.
The LQ controller takes a linearized model of barge dynamics and all environmental forces are considered as disturbances applied to the control loop. With proper adjustment of matrix weights, the controller can be tuned in order to guarantee prescribed performance parameters under a range of disturbance amplitudes. This kind of controller is a
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 modelli ng 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 i nformation 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 diff iculties 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 modelli ng errors and guarantee performance and stabilit y for models slightly different than that used in the design.
Representing this class of controllers, a robust version of the feedback linearization technique was developed using the sliding mode methodology originally proposed in [14]. Such methodology presents good robustness properties being adequate for models containing parametric uncertainties and modelli ng errors. In theory, the controller guarantees optimal performance with bounded model uncertainties; however, the control action presents highly oscill atory components (chattering), which is unsuitable for most applications. The control law can be modified to solve this problem [8], avoiding chattering and still 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 modelli ng 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.
In the second section, all barge data are described, including the forces coeff icient obtained in towing tank tests. The models used in the numerical simulator are also presented, including environmental agents, barge and propellers dynamics. In the third section, the application of the SM controller is described, with a theoretical explanation about this control methodology. The LQ controller is also briefly exposed. In the sequence, the comparison between the controllers is presented, with some ill ustrative simulations included. The conclusions are then summarized in the last section.
2 Barge data and modelli ng
BGL1’s main particulars are presented in Table 1.
Table 1: BGL1 main data Length (L) 121.9 m Beam (B) 30.48 m Draft (T) 5.18 m Position of CG (xG) -4.18m Mass (M) 17177ton Surge Added Mass (M11)
* 1717ton Sway Added Mass (M22)
* 8588ton Yaw Added Mass (M66)
* 1.28.107 ton.m2 Lateral Area (AL) 1500m2 Frontal Area (AF) 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
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.
S-LayLaunching Force (90tf)
x(-90.7m;-12.9m)
yJ-Lay
Launching Force (20tf)
Launchingdirection
(13.0m;7.0m)
Stinger
Figure 4: Launching operation. Pipeline applied forces (S-lay and J-lay)
Thrusters’ position is shown in Figure 5. The system consists of three fore-body units and three stern ones.
x
y
(35,61 ; 13,15)
#1
(42,81 ; -13,15) #3
(59,33 ; -7,64) #2
(-33,99 ; -8,44)
(-33,99 ; 19,15)
(-50 ; 13,15)
#4
#5#6
Figure 5: Propeller Positions in meters (related to XY axis centered at CG)
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.
Thr uster Al l ocation Al gor i thm
Controll er
Curr entWi ndWaves
SetPoint
Wave Fil ter
Dynamics and
pr operties of thur usters
Control For ces Control thr ust
and azimuthal angl e
i n pr opell er s
Actual thr ust and azimuthal
Envi r onmental Agents Model s
Bar ge Dynamics
Model
Measur ed motions
Fil teredMotions
Estimation ofEnvi r onmental
Agents Intensi ti es and Di recti ons
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) coeff icients are given in [5]. Drive system eff iciency 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 pseudo-inverse 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:
( ) ( )( ) ( )( ) .
;
;
6666126226666
2226111626222
1112
6266222111
TOEZ
TOE
TOE
FFFxxMxMxMI
FFFxxMMxMxMM
FFFxMxxMMxMM
++=+++++=++++++=−+−+
������
������
�����
(1)
where Iz is the moment of inertia about the vertical axis; F1E, F2E, F6E are surge, sway and yaw environmental loads (current, wind and waves),
OOO FFF 621 ,, are operation forces and moment due to
the pipe being launched and TTT FFF 621 ,, are forces
and moment delivered by propulsion system. The variables 1x
�, 2x
� and 6x
� are the midship surge,
sway and yaw absolute velocities (Figure 7).
x1
x2
X
Y
x6= ψ (yaw)
(surge)
(sway)
Figure 7: Coordinate systems
Static current forces and moment are evaluated using the experimental coeff icients (CX, CY and CM) defined as:
21 2
1LTUCF XC ρ= ; 2
2 21
LTUCF YC ρ= ; 226 2
1TULCF MC ρ= (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 coeff icients, 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
-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)
Cy10*Cx10*Cm
Figure 8: BGL1 Experimental Current Coeff icients.
Static wind forces were also evaluated using experimental coeff icients (CXV, CYV and CMV) obtained in a captive ‘wind’ test conducted, with the barge model superstructure turned upside down into the towing tank. Wind coeff icients are defined in the standard way by:
21 2
1VACF FaXVV ρ= ; 2
2 21
VACF LaYVV ρ= ;
26 2
1LVACF LaMVV ρ= (3)
where F1V and F2V are surge and sway components of the wind force, F6V is the wind yaw moment; ρa is the air density; V, the mean ‘wind’ velocity. The corresponding coeff icients are presented in Figure 9. Gust spectrum simulations consider Harris-DNV formula:
6
52
2862...1146)(
−
+=
VVCSV
ωω (4)
where ω is the frequency of wind oscill ation, Sv is the spectral density and C is a surface drag coeff icient, used as 0.0015 for moderate seas.
A Jonswap sea wave spectrum was adopted and mean drift wave forces were evaluated using the drift coeff icients obtained from a very well known and validated wave-body interaction computer code1. Figure 10 presents an ill ustrative example of such coeff icients for a 30o wave heading angle. Second order slow drift forces are evaluated using [1]. Current-wave interaction effect on wave drift forces is estimated following [2]. First order wave motions are included in the model using the well known Response Amplitude Operators (RAO’s).
4 Sliding Mode Controller Design
The sliding mode controller is composed by an “estimated” feedback linearization action added to a term responsible to guarantee the prescribed performance and stabilit y in face of limited errors. This term was originally defined in [14] as an on-off controller with amplitude proportional to the maximum modelli ng error. Due to implementation delays or numerical imprecision, a high oscill atory
1 WAMIT
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 stabilit y of the system despite a small performance loss.
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:
++++++
+
=
RTRORE
RTRORE
RTRORE
din
dinY
dinX
R
R
FFF
FFF
FFF
f
f
f
Y
X
666
222
111
,
,
,
)(
)(
)(
)(
ψψ ψ
C
X
X
X
R
R
R
�
�
�
��
��
��
(5)
where fi,din are functions of the inertial properties of the barge and its velocity, C is a rotation matrix depending on the yaw angle and FiR are the forces transferred to the reference point.
Assuming that all terms in (5) are accurately known, the “estimated” feedback linearization controller that guarantees that the states follows the desired ones (XD, YD, ΨD) is given by:
−−−−−−
+
+++
−
−=
−−
ψλψλψλλλλ
ψψψ~~2
~~2
~~2
ˆˆ
ˆˆ
ˆˆ
)(
)(ˆ)(ˆ
2
2
2
1
66
22
11
,
,
,1
6
2
1
���
���
���
��
�
�
D
YYD
XXD
RMRE
RMRE
RMRE
din
dinY
dinX
RT
RT
RT
YYY
XXX
FF
FF
FF
f
f
f
F
F
F
C
X
X
X
C
R
R
R
(6)
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.
In order to guarantee a prescribed tracking precision even with errors in the estimates, an extra term proportional to the maximum error is added to (6) given by:
Φ−Φ−Φ−
=
∆∆∆
−
)(
)(
)(1
6
2
1
ψψψ ssatK
ssatK
ssatK
F
F
F
YYY
XXX
RT
RT
RT
C (7)
where the variables s are true measures of tracking performance added to an integral action defined by
∫++= RRXRX XXXs~~
2~ λ
�
and analogously for Y
and ψ.
The gains K = (KX, KY, KΨ)T are obtained such that all elements of :
( )RERE FFCK ˆmax1 −− − (8)
are positive. It has been assumed that the inertial functions fi,din and the operational forces FRO are known with suff icient accuracy compared to the environmental forces.
The parameters Φ are responsible for the smoothness of the on-off action and must be tuned such that [8]:
i
ii
K
λ)max(
=Φ (9a)
Futhermore, it can be shown that the tracking precision is limited by:
2
)max(22~
i
i
i
ii
KX
λλ=Φ< (9b)
Since λi is limited by the unmodelled frequencies and time delays, it can be seen that for higher uncertainty about the system, higher is the expected tracking error.
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 U and its direction is α , with a maximum uncertainty of U∆ and α∆ respectively. The maximum error in current forces and moments is obtained using a Taylor expansion ( iCiCiC FFE ˆ−= ):
[ ] iRCiCU
iC
U
iCiC Fe
EU
U
EUE ˆ),(max
ˆ,ˆˆ,ˆ
+∆
∂∂+∆
∂∂≤ α
αα
αα
(10)
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 wave loads, and all together are used in (8).
5 LQ controller design
Writing (1) for the reference point R and eliminating quadratic terms one obtains:
RORERT FFFM ++=
R
R
R
x
x
x
6
2
1
��
��
��
(11)
being the mass matrix M dependent only on well known mass properties. Besides, the velocities in the fixed coordinate system OXY is related to the surge and sway velocities by:
=
R
R
R
R
R
x
x
x
Y
X
6
2
1
)(�
�
�
�
�
�
ψψ
J ,
−=
100
0cossen
0sencos
)( ψψψψ
ψJ (12)
Defining ( )TRRRRR YXxxx ψ621
���=Rx as
the state vector, and linearizing the model for heading angles near the operational one ( operψ ) one
obtains:
DF0
Ix
0)J(
00x
I0
0MRT +
=
−
+
×
×
×
××
××
×
33
33
33
3333
3333
33R
operR �
(13)
which can be written in the state-space format:
DFBAxx RT ++= .RR
(14)
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
And the new state-space equation is defined by the following extended matrixes:
=
×××
××
333333
3366ext 0I0
0AA
=
×
×
33
36ext 0
BB (16)
The control forces are then given by the state feedback law RRT KxF −= such that the gain matrix K is obtained from the associated steady-state Riccati equation:
SBRK
0QSBRSBSASAText
1
Text
1extext
Text
−
−
=
=+−+ (17)
where Q is the state and R is the control weight matrixes.
6 Case studies and compar ison between controllers
A common Brazili an 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.
X
Y
0°
90°
180°
270°
Current (80°)U=1.0m/s
Waves(90°)Hs=1.5mTp=8.45s
Wind (100°)V=12m/s
Figure 11: Environmental conditions
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.
R=3
00m
30°
Figure 12: Typical reference path
The SM controller is designed assuming that all environmental agents are colli near, with a direction of 90o. Furthermore, all estimated values used in the design, as well as the modelli ng errors imposed to the simulation, are shown in Table 2. It is important to mention that the controller was designed for conditions very different from the real
ones, as a robustness test. Furthermore, a bandwidth factor (λ) of 0.04 is used for all motions.
Table 2: SM design parameters and modelli ng errors imposed in the simulation
Parameter Real Value
Estimated Value(*)
Max. Var iation(*)
Current Velocity U=1.0m/s Û=1.2m/s ∆U=0.2m/s Current Dir. α=80o �ˆ =90o ∆α=10o
Wave Sig. Height Hs=1.5m �
S=2.0m ∆HS=1.0m Wave Dir. αW=90o
W�ˆ =90o ∆αW=10o
Wave Peak Per. ΤP=8.45s s0.8TP = s0.1 �
P =
Mean Wind Dir. αWi=100o Wi
�ˆ =90o ∆αWi=10o
Current/Wave/ Wind mod. error
+20% 20%
Wind Velocity Measuring Error
−10% 10%
(*)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.
-200 0 200 400 600 800 1000 1200 1400 1600-600
-500
-400
-300
-200
-100
0
100
X (m)
Y (
m)
-200 0 200 400 600 800 1000 1200 1400-500
-400
-300
-200
-100
0
100
X (metros)
Y (
met
ros)
Posição Pto Referência
Figure 13: Resulted path using SM controller
0 500 1000 1500 2000 2500 3000-2
0
2
4
6
XY
err
or(
m)
0 500 1000 1500 2000 2500 3000-5
0
5
10
Ψ e
rro
r(d
eg
ree
s)
Tim e(s)
beginning of curve end o f curve
Figure 14: Tracking and heading errors: SM control
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.
For both controllers the mean total power consumption was approximately 5300kW with a peak consumption of 7000kW. Some differences between both control philosophies are exempli fied by this example. It must be stressed again that both of then satisfied the performance requirements for the pipe-laying operation.
-200 0 200 400 600 800 1000 1200 1400 1600-600
-500
-400
-300
-200
-100
0
100
X (m)
Y (
m)
-200 0 200 400 600 800 1000 1200 1400-500
-400
-300
-200
-100
0
100
X (metros)
Y (
met
ros)
Posição Pto Referência
Figure 15: Resulted path using LQ controller
0 500 1000 1500 2000 2500 3000-2
0
2
4
6
XY
err
or(
m)
0 500 1000 1500 2000 2500 3000-5
0
5
10
Ψ e
rro
r(d
eg
ree
s)
Tim e(s)
beginning of curve end o f curve
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
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
WiPWS UTH ααα ˆ,ˆ,ˆ,ˆ,ˆ,ˆ .
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 diff iculties. 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.
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].
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 for the possible environmental conditions.
The second important point that must be analyzed is related to the linearization necessary to the design of LQ controller. For the present, the controller so obtained for an operation heading of 0o guarantees stabilit y for heading angles between –64o and 64o. The stabilit y boundary can be easily obtained by the calculation of the eigenvalues of the closed loop matrix of system (14) controlled by (17). Figure 17 ill ustrates the possible headings that the barge may reach safely with the LQ controller.
+64o
-64o
Stable Headingsfor the
LQ controller
Figure 17: Stable headings for the LQ controller
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 stabilit y and good performance for all headings. For the sake of ill ustration, a rectili near 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 oscill atory, since the heading is close to the stabilit y boundary. The performance of the system gets worse, taking approximately 800s to decrease the oscill ation to an acceptable level. The SM controller, for its turn, reaches the desired path and heading after a short transient and without oscill ations (Figure 19), keeping the overall good performance presented in the first simulation.
If a linear controller is used, a full stabilit y analysis must be done, defining the allowable headings the system may reach. A possible approach would be the use of several li near controllers, each one
designed with the model li nearized about the actual state of the ship. However, such controller would be extremely complex and the stabilit y of such “gain scheduled” system would be questionable.
Tim e(s ) Figure 19: Rectili near 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 stabilit y analysis must be carried out, and several controllers must be used for different headings since nonlinear effects are disregarded in the design.
The SM control is, as far implementation is concerned, more complex, because it encloses a model of the system and environmental agents and requires estimates of environmental conditions. However, due to robustness properties, the controller guarantees stabilit y and performance in the presence of estimation and modeling errors, bounded by prescribed maximum values. Therefore, simple systems for environmental conditions monitoring (or even real time estimators based on ship motion) can be used, since the controller can handle large errors on these estimations. The complexity of the controller is not a problem for the computational capacity of modern computers added to the fact that the ocean systems dynamics are extremely slow.
Acknowledgements
This work has been supported by Petrobras, State of São Paulo Research Foundation (FAPESP – Proc.No 98/13298-6) and CNPq/CTPETRO (process no. 469095/00). The authors thank Mr. R. Pesce for helping with simulations and diagrams.
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