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Dynamic Positioning System Using Self-Tuning Control
Copyright © IFAC Intellige nt Tuning and Adaptive Control . Singapore 199 1
DYNAMIC POSITIONING SYSTEM USING SELFTUNING CONTROL E. A. Harros and H. M. Morishita Dept. of Marine Engineering and Naval Architecture. University of Sao Paulo. Caixa Postal 61548. CEP 05508. Sao Paulo. Brazil
Abstract. This paper deals with the application of a generalized minimum variance self-tuning controller to the Dynamic Positioning System (DPS) of a semi-submersible rig . In the control design the sw a y, surge and yaw have been considered as independent motions . There is no explicit fil~ering process to the high frequency motions since the rig itself may be treated as a low pass filter becau s e it is designed to be hydrod ynamically transparent. The control performance is analyzed by computer simulation, verifying influence of direct.ion and magnitude of the wave and current. The results reveal that . the self-~ull1n~ DPS keeps position and heading of rig within the accuracy requirements In sptte of an existence of a small off-set. Keywords Adaptive control; Marine systems; minimum variance control; selftuning regulators ; dynamic positioning system.
INTRODUCTION
of the system that needs be simple to avoid excessive computational load . Balchen et alii(1980) assumed that high frequency motions were purely oscillatory and could be modeled by a second order sinusoidal oscillator with variable center frequency. Kallstron (1983) has taken this model but has kept the frequency constant. Grimble, Patton and Wise(1980) used a fourth order wave model to high frequency motions with fixed low frequency filter gain. Fung and Grimble(1983) propose a self-tuning Kalman Filter decoupling low and high frequency motions .
A dynamic positioning system (DPS) is used to maintain position and heading of an offshore industry floating vessel, mainly in deep waters where, often, it is impractical to use conventional mooring system . A DPS is designed to counteract forces and moments imposed on the vessel by wind, current and waves. It is based on the action of a set of thrusters governed by a control algorithm . It controls only three of the six modes of motion, i.e ., surge, sway and yaw. In the DPS design, the vessel motions are usually divided in two parts, the high and low frequency motions . The high ones are induced by waves, the low ones are determined by characteristics of the DPS together with the wind, wavedrift and current. The high frequency motions are oscillatory in nature.So they must be filtered away to avoid wear and tear of the thruster mechanism and to save energy .
A theoretical development of a semi-submersible rig model is difficult because it involves modelling of wind, wave and current action and of hydrodynamic interaction between the rig and sea water. These considerations suggest the verification of the performance of a DPS with adaptive control assuming the rig as an unknown parameter system with stochastic perturbations . This paper deals with the application of a generalized minimum variance self-tuning controller to the DPS of a semisubmersible rig. The system is described by a stochastic autoregressive moving average model with unknown coefficients . The control law is obtained by the minimization of a cost function in which system output and control effort are weighted.The application of the self-tuning controller requires the estimation of the model parameters. In this paper the implicit way is used to estimate directly the parameters of the controller by means of the recursive least-squares algorithm. A rig is designed to be hydrodynamically transparent and so it may be treated as a low pass filter .
The first generation of DPS is based on conventional PID controller with notch filter. In this case, the wave filter imposes a phase lag on the position error signal that usually restricts the allowable bandwidth of the controller. This means that the tuning of the control gains is a compromise between the effectiveness of a high frequency motion filtering and the stability margins required for satisfactory controller performance. These considerations have led to the development of a second generation of DPS based on linear quadratic theory . This generation of DPS employs Extended Kalman Filter to estimate states and parameters. But performance of EKF depends on mathematical model
307
Taking this characteristic into account the design of a DPS without explicit filtering process to the high frequency motions is attempted .
C(Z·I) = Ej(z·I)A(z·l) + z·jF)z·l) Now , substituing Eq . (3) into Eq. (2) and making al l aU(k) = 0, it follows that the control law is given by:
The development of the controller requires a mathematical model of the rig . The model adopted was developed by using an auto-re~res sive moving average for high frequency motions. For the low frequency motions maneuvering equations were used . Current and seco?d or~e. r wave forces are described through semi empirical relationships . This paper does not emphasize wind action because the usual feedforward control to compensate the wind force has not been included. The performance of the DPS is analyzed by computer simulations with data of GVA 4000 rig, verifying influence of a direction and magnitude of the wave and current.
cp*(k+d
I k) = Py*(k+d I k)
+ Qu(k) - Rw(k)
=0
(5)
where Q = Q'qo/b o . Considering Eq . (4) with Eq. (5) it follows : Ccp*(k + j
I k) =
Fy(k) + Gu(k) + Hw(k) + cl (6)
where F(z·l) = L pld .j(Z·I) G(Z·I) =L Pjz·jEd./z·l)B(z·l) + C(Z·I)Q(Z·I) H(z · l) = -C(z · I)R(z·l)
SELF-TUNING CONTROLLER
cl =
The generalized minimum variance controller proposed by Clark and Gawthrop(1975) is considered. As this theory is well - known only the essentials of the method are outli~ed hn~. Consider a system described by a Illlear difference equation : A(z·l)y(k) = B(z· l)u(k-d) + C(z·l)v(k) + D
PjEP)D
As the control law is given by setting *(k + d I k) to zero at each stage it can be also obtained by: Fy(k) + Gu(k) + Hw(k) + cl = 0
As the self-tuning control theory is developed to unknown systems it is necessary to adopt some procedure to get the control law parameters . Here the implicit approach is considered for estimating directly the coefficients of F(Z·I), G(Z·I), H(Z·I) and cl • Defining cp (k+d) as:
(1)
where y is the output ; u is control signal; D is a constant · v is an uncorrelated random sequence of zero ~ean disturbing the system; d is the time delay· Z·I is the backward-shift operator, and A(z·I) , B(z·l) and C(z·l) are polynomials given by :
cp(k + d) = Py(k + d) + Qu(k) - Rw(k) and making use of Eq . (3), it follows that
A(z·l)
cp(k+d) = cp*(k+d
B(z·l)
k) + E(k+d)
(8)
E
(k + d)
=
LP;e(k + d - i) i=O
where b .. O· na nb and nc are polynomial order.> and the °root~ of 'C(z · l) lie within the unit circle. The control law is derived by minimization of the following cost function:
Clarke and Gawthrop(1975) show that from Eq. (5) and Eq. (8) a recursive ,least s9..uares ~lgo rithm can be obtained to get F(Z·I) , G(Z·I) , H(Z·I) and 6' that are polynomials with estimated values of the coefficients of F(z·I), G(Z·I), H(z· I) and cl . The control law is then given by:
I = E {[P(z·l)y(k + d)-R(z·1 )w(k)j2 - [Q' (z·l)u(k}F} (2) where w is the reference and P(Z·I), R(Z·I) and Q(Z·I) are weigh i ng polynomials; E is the expectation operator. However, the term y(k +?) makes deduction of control law from Eq. (2) difficult. This problem may be overcome by defining the optimal predictor y*(k + d I k) of y(k + d) as:
I
k) = y(k+j) - e(k+j)
Fy(k) + Gu(k) + HW(k) + t'= 0
I k) = F'jy(k)
+ EjBu(k + j-d) + EP)D
(9)
MODEL OF SEMI- SUBMERSIBLE RIG The four-column semi - submersible drilling rig GVA-4000 was selected to verify the performance of adaptive DPS. In order to develop the mathematical model of the rig it is assumed that total rig motions are the sum of high and low frequency motions . Three different coordinate systems, as is shown in Fig . 1, are used to determine the dynamic equations of the rig. The first one, xyz, is fixed to the rig, and its origin is at the center of gravity of the rig. The coordinate system XOYO is fixed at the earth. This system is used for the integration of the low frequency
(3)
where e(k + j) is the prediction error. The optimal predictor can be obtained recursively as : Cy*(k + j
I
d -I
where
C(Z·I)
y*(k+j
(7)
(4)
for j s d and where F '. (z·l) and E/z · I ) are na-l and j-l order polynomials and their coefficients are determined by :
308
v(k) + cWlv(k-1) .. . . + cWnv(k-n)
equations and it defines wave, wind and current directions. The third coordinate system, XhfY h fZhf is fixed at the posi ti on of the ce n tre of gravity in the low frequency motion . The axes Xhf, Vhf and Zhf are parallel to the axes XO, YO and z respectively .
where v is a zero mean white noise signal. The order of the filter is also four (Kallstrom,1983) . I.ow Frequency Motions An ideal-fluid maneuvering model in the horizontal plane is proposed in order to represent the low frequency motions (Norrbin,1971):
Xc
(l-Xu)U = (l-Yv)vr+ (Yv-Xu)v/ + X/m
CURRENT
WIND
(Xu-1)ur+ (Yv-Xu)u/+Y/m
L(k 2..-N,)i-
=
(10)
(IlL) (Yv-Xu) (u-u c) (v-vc) + N/mL
where Xu ' YvandN; are the added mass and moment in surge, sw a y and yaw, respectively, normalized by the use of the " bis" system (Norrbin,1971); u is the surge velocity , v is the sway velocity and r is the yaw rate ; L is the length of the rig;m is the mass of the rig; kz, is the radius of gyration around z axis; U c and Vc are the components in the x and y direction of a constant homogeneous current and they are related with the earth fixed coordinate by:
x
o L . . - - - - - - - -_ _ _ _ _- - J _
(l-Y')V=
Ye
Fig. 1. The Coordinate System Hi~h
frequency motions
where u co and vco are the components of the current in XO and YO directions respectively and '" is the heading of the rig. The forces X and Y and the moment N consist of three different components:
The high frequency motions are due to oscillatory motions of the waves and in this paper they are obtained by an auto regressive moving average model (ARMA) fed by the wave elevation for each degree of freedom. For instance, the high frequency sway motion, YhC is given by: Yhf(k) + ahlYhf(k-l) + ..... + ahnYhf(k-n) ~
(k) + bh l Hk-l) + ... + bh n Hk-n)
X
X visc + XWdrift + Xthrust
Y
Y visc
N
N visc + Nth rust
+ Y wdrift + Y thrust
where the visc , wdrift and thrust subscripts mean viscous, wave drift and thurst forces respectively. Low frequency motions also depend on wind efforts that are omitted here since their influence is not analyzed in this paper .
where ~ is the wave elevation . Kallstrom(1983) determined the values of coefficients and order n from scale model tests data. The parameters ah . and bh . were estimated with the maximum lik'elihood 'identification methods and Akaike's information criterion was applied in order to determine a suitable model order, that is four to sway, surge and yaw motions .
vjscous Forces The viscous forces of GVA-4000 with a constant homogeneous current have been determined by scale model tests (Kalls trom,1983):
Wave eleyatjon A stochastic realization of the wave elevation may be obtained through a shaping filter fed by white noise . This filter is chosen to minimize the error between the true sea spectrum and the approximate spectrum being generated. ~hus t.he output of the filter is the wave elevation With the desired significant wave height and peak period. Here the filter determined by Kallstrom(1983) is used that is expressed by an auto-regressive moving average model (ARMA):
NvijmL= -0 . 000946V/sin(2 Yr) - 3 . 75r Irl where Vr is the current speed relative to the rig and y is the current direction relative to the rig. XVISC . and y vlSe . can also be expressed as (Barros, 1989) :
Hk) + aWIHk-1) + .. .. awnHk-n) =
309
The low damping and absence of restoring forces in the horizontal plane are reflected by the transfer function. In the case of u eo = v eo = 0 it follows that:
Wave Drift Forces . The wave drift forces are also obtained from scale model tests: X"d'ir/ m
0.0021(~ - z)2cos(Ywave - Y')
=
Y"d'ir/m = 0 . 0014(~ -z)2 s in( Yw ave - Y') where z is the heave and of wave propagation .
y
H(s ) =
wave is the direction
Resultin~ Model The position X o ' Y 0 and the heading in the low frequency model are obtained by integration of the following equations:
~ S2
0
0
0
bn S2
0
0
0
b)) S2
There is a small difference in the presence of the current. For example, consider u eo = 1 m / s, v eo = 0 . Neglecting the coupling terms, the transfer functions is:
Xo = u coSy:> - v si nY'
b ll
0
S (s + 0 ,021 )
Yo = usinY' + v cOSy:> H(s) =
b 22
0 S
= r
(s + 0,011)
0 0 b))
0
The total horizontal motions are obtained by adding the high frequency motions:
0
S2
DPS DESIGN Y,o, Y'
Yo
'0'
Three independent controllers were designed to sway, surge and yaw motions . The rig dynamics model assumed in the control desaign is described by Eq . (1) with na = 2, nb = 1 and d = 1. The control effort u is, depending on the motion, X'h,ust' Y'h'USI or N'h'uSl' Actually DPS has a set of propellers allocated along the rig. So it is necessary calculate control signals for each of them to get desired forces and moment. Thus, this model makes a suitable simplification that doesn't prevent further improvement. The order of polynomial C(Z·I) was also assumed as 2. The output of the transfer function C(z' 1)/A(z·l) fed by white noise represents the sum of high frequency motion and measurement noise. This assumption is reasonable because the amplitude of high frequency motion of the semi-submersible rig considered in this paper is of the same order as the accuracy of the position measurement system. To define the control structure the following cost function was selected:
+ Y hr
+ Y'hr
Y'
Linear Model Analysis A state space representation for the linearized low frequency model is obtained around the values u = v = r = 0 and u e = u eo and ve = v eo :
x = AX
+ BU
y = CX
where:
X=[uvrXoYoY']T;U
y
=
[Xo Y o
Y'l'
all a l2 a l ) a 21 an a 2 )
A=
a)1 an
1 0 0 1 0 0
0 0 0
0 0 0 0 0 0
bll 0 0 0 0 B = 0 bn 0 0 0 [ o 0 b)) 0 0
0 0 0 0 0 0
0 0 0 0 0 0
The sampling period adopted is 12 s. This period was chosen considering the low frequency dynamics, whose range is 0 - 0.25 rad/s, and the maximum recommended frequency to the operation of the propellers that is 0 . 63 rad/s. The values of Po' qo and ql were selected by preliminary simulation tests imposing ±2° to the heading and ±5 meters to the positions .
~Ol
000100] C= 000010 [o 000 0 1
Simulation eresults Several simulations were made to analyze the performance of the self-tuning DPS. In this paper the influence of coefficient ql and the absence of high frequency motion filter are discussed in particular.
where a ij and b ij are obtained from linearization of Eq. (10) (Barros, 1989)
310
SURGE(m)
08
8 .0
THRUST FORCE (SURGE DIRECTION) (MN)
04
4 .0 TIME (s) 500
0 .0
0 .0 + - - - - - - - - 4 - - - - = - - - + - - 500 1000 T I ME (s)
-4 .0
-0 .4
-8 .0
8 .0
4 .0
TIME (s)
0 .0
-4 .0
-8.0 YAW ( degree)
8 .0
4 .0
TIME (s) 500
1000
0 .0 1.000 -4 .0
- 6 .0
-8 .0
- 12.
TIME (s)
Fig . 2.1 Motions of the rig .
Fig. 2 .2 Thruster efforts
The effect of using ql .. 0 is to increase the o~fset of the rig output, mainly if it is perturbed with constant environmental forces . The offset is usually eliminated by the introduction of an integral action, defining q = q However this . I o. became oscilapproac h f al' 1e d since rig motions latory as an undamped system instead of eliminating the offset. The analysis of the closed loop transfer function by the root locus method has shown that the oscillation increases because ql .. qo tends to assign poles to the unit circle. An intermediate solution was obt a ined with ql = 0.3q o· The test was m a de in accordance with the following conditions: I)Wave with Pierso~ Moskowitz spectrum with significant wave height = 5.6 m and peak period = 12 s · 2) y = 2250 ,' 3) Y currenl -- 190 0', current velocity '. Wlve = 1 m/s; 4) The positions and heading are assumed to be measured with additive white noise disturbances with standard deviations of 0.5 m and 0, 05° respectively.
pected there is a small offset mainly in surge directions . This is due to both constant current action and ql ~ q o' Fig . 2.2 shows that the profile of control efforts is suitable if a comparison is made with other results , for instance Kallstron ( 1983) . The effects of the filter abscence were also tested. A simulation without the feedback of the high frequency motion to the controller was performed, other conditions were Kept the same as in previous test. The results shown in Fig. 3 . 1 and Fig . 3.2 . are similar in comparison with Fig . 2.1 and Fig. 2.2 specially the modulation of the control signal. The sampling period selected doesn't reproduce the high frequency motion because its frequency is greater than 0 .3 rad/s. But there is the aliasing effect of this signal that can cause unnecessary motion of the thrusters . This aspect needs to be analyzed further. CONCLUSIONS In this paper the application of self-tuning DPS was shown. The results indicate that there is some difficulty to eliminate the offset only by sear-
The rig motions shown in Fig. 2 . 1 reveal that the ~elf-. tu~ing DPS keeps position and heading of fig Within the accuracy requirements. As ex-
311
SURGE (m) B.O
OB
4 .0
THRUST FORCE (SURGE DIRECTION) ( MN)
0.4 TI M E (s)
500
0 .0
1000 0 .0 500 -0.4
-8.0
-OB SWAY(m)
B.O
0.8
4 .0
TIME (s)
THRUST FORCE (SWAY DI RECTI ON) ( MN)
0.4
0 .0
0.0
-4.0
-0.4
-8.0 8 .0
1000 TI ME (s)
-4 .0
TIME (s)
- 08 YAW (degree)
12 .
4.0
6 .0
TI ME ( s)
500
MOMENT (YAW DI RECTION) ( KNm)
1000
0.0
0 .0 500
1000 TIME (s)
-4 .0
-6.0
-8 . 0
- 1.2
Fig . 3 . 1. Motions of the rig without high frequency components
Fig . 3.2 Thruster efforts without high frequen-
ching for a suitable cost function. The low damping characteristics of the rig at low frequency motions in the horizontal plane impose limitations to the integral gain included in the cost function . The results also suggest that the control effort is not significantly affected by the filter absence for the semi-submersible rig considered. The implementation of this kind of DPS seems easier than linear quadratic control with EKF since it only requires order and time delay knowledge of the system . Thus a self-tuning regulator may be an easy way to cope with the non-linearities and the time variant characteristics of the system.
REFERENCES Balchen, J.G. et alii (1985) . Dynamic positioning system based on Kalman Filter and optimal control. Modelinl: Identification and contro!. Oslo, 1(3), pp. 135-63. Barros, E . A. (1989) Aplica~ao de urn controlador auto-ajusuivel ao sistema de posionamento dinamico de uma platafor-
312
ma semi-submersivel. Ms dissertation, University of Sao Paulo, Sao Paulo Clarke, D.W. and P.J. Gawthrop (1975). Selftuning controller. Proceedinl:s of the Institution of Electrical Enl:ineers , 22(9) London, pp. 929-934 Fung, P.T.K. and M.J. Grimble (1983). I?ynamic ship positioning using a self-tunIng Kalman Filter. IEEE Transactions on Automatic Contro!, AC-28, New York, pp 339-349 . Grimble, M.J., R . J. Patton and D.A. Wise (1980) . Use of Kalman filterinl: techniques in dynamic ship positioninl: systems. lEE Proceedings . Part D:Control theory and Applications, 127(3), London pp. 93-102 . Kallstron , C.G. (1983). Mooring and dynamic positioning of a semi-submersible. A comparative simulations study. Proc 2nd International Symposium on Ocean Enl:ineerinl: and Ship Handlinl:, Gothenburg, pp.417 -440. Norbbin, N.H. (1971) Theory and observations on the use of a mathematical model for ship maneuyerinl: in deep and confined ~. Goteborg, Statens Skeppsprovningsanstalt, (SSPA publication).
DYNAMIC POSITIONING SYSTEM USING SELFTUNING CONTROL E. A. Harros and H. M. Morishita Dept. of Marine Engineering and Naval Architecture. University of Sao Paulo. Caixa Postal 61548. CEP 05508. Sao Paulo. Brazil
Abstract. This paper deals with the application of a generalized minimum variance self-tuning controller to the Dynamic Positioning System (DPS) of a semi-submersible rig . In the control design the sw a y, surge and yaw have been considered as independent motions . There is no explicit fil~ering process to the high frequency motions since the rig itself may be treated as a low pass filter becau s e it is designed to be hydrod ynamically transparent. The control performance is analyzed by computer simulation, verifying influence of direct.ion and magnitude of the wave and current. The results reveal that . the self-~ull1n~ DPS keeps position and heading of rig within the accuracy requirements In sptte of an existence of a small off-set. Keywords Adaptive control; Marine systems; minimum variance control; selftuning regulators ; dynamic positioning system.
INTRODUCTION
of the system that needs be simple to avoid excessive computational load . Balchen et alii(1980) assumed that high frequency motions were purely oscillatory and could be modeled by a second order sinusoidal oscillator with variable center frequency. Kallstron (1983) has taken this model but has kept the frequency constant. Grimble, Patton and Wise(1980) used a fourth order wave model to high frequency motions with fixed low frequency filter gain. Fung and Grimble(1983) propose a self-tuning Kalman Filter decoupling low and high frequency motions .
A dynamic positioning system (DPS) is used to maintain position and heading of an offshore industry floating vessel, mainly in deep waters where, often, it is impractical to use conventional mooring system . A DPS is designed to counteract forces and moments imposed on the vessel by wind, current and waves. It is based on the action of a set of thrusters governed by a control algorithm . It controls only three of the six modes of motion, i.e ., surge, sway and yaw. In the DPS design, the vessel motions are usually divided in two parts, the high and low frequency motions . The high ones are induced by waves, the low ones are determined by characteristics of the DPS together with the wind, wavedrift and current. The high frequency motions are oscillatory in nature.So they must be filtered away to avoid wear and tear of the thruster mechanism and to save energy .
A theoretical development of a semi-submersible rig model is difficult because it involves modelling of wind, wave and current action and of hydrodynamic interaction between the rig and sea water. These considerations suggest the verification of the performance of a DPS with adaptive control assuming the rig as an unknown parameter system with stochastic perturbations . This paper deals with the application of a generalized minimum variance self-tuning controller to the DPS of a semisubmersible rig. The system is described by a stochastic autoregressive moving average model with unknown coefficients . The control law is obtained by the minimization of a cost function in which system output and control effort are weighted.The application of the self-tuning controller requires the estimation of the model parameters. In this paper the implicit way is used to estimate directly the parameters of the controller by means of the recursive least-squares algorithm. A rig is designed to be hydrodynamically transparent and so it may be treated as a low pass filter .
The first generation of DPS is based on conventional PID controller with notch filter. In this case, the wave filter imposes a phase lag on the position error signal that usually restricts the allowable bandwidth of the controller. This means that the tuning of the control gains is a compromise between the effectiveness of a high frequency motion filtering and the stability margins required for satisfactory controller performance. These considerations have led to the development of a second generation of DPS based on linear quadratic theory . This generation of DPS employs Extended Kalman Filter to estimate states and parameters. But performance of EKF depends on mathematical model
307
Taking this characteristic into account the design of a DPS without explicit filtering process to the high frequency motions is attempted .
C(Z·I) = Ej(z·I)A(z·l) + z·jF)z·l) Now , substituing Eq . (3) into Eq. (2) and making al l aU(k) = 0, it follows that the control law is given by:
The development of the controller requires a mathematical model of the rig . The model adopted was developed by using an auto-re~res sive moving average for high frequency motions. For the low frequency motions maneuvering equations were used . Current and seco?d or~e. r wave forces are described through semi empirical relationships . This paper does not emphasize wind action because the usual feedforward control to compensate the wind force has not been included. The performance of the DPS is analyzed by computer simulations with data of GVA 4000 rig, verifying influence of a direction and magnitude of the wave and current.
cp*(k+d
I k) = Py*(k+d I k)
+ Qu(k) - Rw(k)
=0
(5)
where Q = Q'qo/b o . Considering Eq . (4) with Eq. (5) it follows : Ccp*(k + j
I k) =
Fy(k) + Gu(k) + Hw(k) + cl (6)
where F(z·l) = L pld .j(Z·I) G(Z·I) =L Pjz·jEd./z·l)B(z·l) + C(Z·I)Q(Z·I) H(z · l) = -C(z · I)R(z·l)
SELF-TUNING CONTROLLER
cl =
The generalized minimum variance controller proposed by Clark and Gawthrop(1975) is considered. As this theory is well - known only the essentials of the method are outli~ed hn~. Consider a system described by a Illlear difference equation : A(z·l)y(k) = B(z· l)u(k-d) + C(z·l)v(k) + D
PjEP)D
As the control law is given by setting *(k + d I k) to zero at each stage it can be also obtained by: Fy(k) + Gu(k) + Hw(k) + cl = 0
As the self-tuning control theory is developed to unknown systems it is necessary to adopt some procedure to get the control law parameters . Here the implicit approach is considered for estimating directly the coefficients of F(Z·I), G(Z·I), H(Z·I) and cl • Defining cp (k+d) as:
(1)
where y is the output ; u is control signal; D is a constant · v is an uncorrelated random sequence of zero ~ean disturbing the system; d is the time delay· Z·I is the backward-shift operator, and A(z·I) , B(z·l) and C(z·l) are polynomials given by :
cp(k + d) = Py(k + d) + Qu(k) - Rw(k) and making use of Eq . (3), it follows that
A(z·l)
cp(k+d) = cp*(k+d
B(z·l)
k) + E(k+d)
(8)
E
(k + d)
=
LP;e(k + d - i) i=O
where b .. O· na nb and nc are polynomial order.> and the °root~ of 'C(z · l) lie within the unit circle. The control law is derived by minimization of the following cost function:
Clarke and Gawthrop(1975) show that from Eq. (5) and Eq. (8) a recursive ,least s9..uares ~lgo rithm can be obtained to get F(Z·I) , G(Z·I) , H(Z·I) and 6' that are polynomials with estimated values of the coefficients of F(z·I), G(Z·I), H(z· I) and cl . The control law is then given by:
I = E {[P(z·l)y(k + d)-R(z·1 )w(k)j2 - [Q' (z·l)u(k}F} (2) where w is the reference and P(Z·I), R(Z·I) and Q(Z·I) are weigh i ng polynomials; E is the expectation operator. However, the term y(k +?) makes deduction of control law from Eq. (2) difficult. This problem may be overcome by defining the optimal predictor y*(k + d I k) of y(k + d) as:
I
k) = y(k+j) - e(k+j)
Fy(k) + Gu(k) + HW(k) + t'= 0
I k) = F'jy(k)
+ EjBu(k + j-d) + EP)D
(9)
MODEL OF SEMI- SUBMERSIBLE RIG The four-column semi - submersible drilling rig GVA-4000 was selected to verify the performance of adaptive DPS. In order to develop the mathematical model of the rig it is assumed that total rig motions are the sum of high and low frequency motions . Three different coordinate systems, as is shown in Fig . 1, are used to determine the dynamic equations of the rig. The first one, xyz, is fixed to the rig, and its origin is at the center of gravity of the rig. The coordinate system XOYO is fixed at the earth. This system is used for the integration of the low frequency
(3)
where e(k + j) is the prediction error. The optimal predictor can be obtained recursively as : Cy*(k + j
I
d -I
where
C(Z·I)
y*(k+j
(7)
(4)
for j s d and where F '. (z·l) and E/z · I ) are na-l and j-l order polynomials and their coefficients are determined by :
308
v(k) + cWlv(k-1) .. . . + cWnv(k-n)
equations and it defines wave, wind and current directions. The third coordinate system, XhfY h fZhf is fixed at the posi ti on of the ce n tre of gravity in the low frequency motion . The axes Xhf, Vhf and Zhf are parallel to the axes XO, YO and z respectively .
where v is a zero mean white noise signal. The order of the filter is also four (Kallstrom,1983) . I.ow Frequency Motions An ideal-fluid maneuvering model in the horizontal plane is proposed in order to represent the low frequency motions (Norrbin,1971):
Xc
(l-Xu)U = (l-Yv)vr+ (Yv-Xu)v/ + X/m
CURRENT
WIND
(Xu-1)ur+ (Yv-Xu)u/+Y/m
L(k 2..-N,)i-
=
(10)
(IlL) (Yv-Xu) (u-u c) (v-vc) + N/mL
where Xu ' YvandN; are the added mass and moment in surge, sw a y and yaw, respectively, normalized by the use of the " bis" system (Norrbin,1971); u is the surge velocity , v is the sway velocity and r is the yaw rate ; L is the length of the rig;m is the mass of the rig; kz, is the radius of gyration around z axis; U c and Vc are the components in the x and y direction of a constant homogeneous current and they are related with the earth fixed coordinate by:
x
o L . . - - - - - - - -_ _ _ _ _- - J _
(l-Y')V=
Ye
Fig. 1. The Coordinate System Hi~h
frequency motions
where u co and vco are the components of the current in XO and YO directions respectively and '" is the heading of the rig. The forces X and Y and the moment N consist of three different components:
The high frequency motions are due to oscillatory motions of the waves and in this paper they are obtained by an auto regressive moving average model (ARMA) fed by the wave elevation for each degree of freedom. For instance, the high frequency sway motion, YhC is given by: Yhf(k) + ahlYhf(k-l) + ..... + ahnYhf(k-n) ~
(k) + bh l Hk-l) + ... + bh n Hk-n)
X
X visc + XWdrift + Xthrust
Y
Y visc
N
N visc + Nth rust
+ Y wdrift + Y thrust
where the visc , wdrift and thrust subscripts mean viscous, wave drift and thurst forces respectively. Low frequency motions also depend on wind efforts that are omitted here since their influence is not analyzed in this paper .
where ~ is the wave elevation . Kallstrom(1983) determined the values of coefficients and order n from scale model tests data. The parameters ah . and bh . were estimated with the maximum lik'elihood 'identification methods and Akaike's information criterion was applied in order to determine a suitable model order, that is four to sway, surge and yaw motions .
vjscous Forces The viscous forces of GVA-4000 with a constant homogeneous current have been determined by scale model tests (Kalls trom,1983):
Wave eleyatjon A stochastic realization of the wave elevation may be obtained through a shaping filter fed by white noise . This filter is chosen to minimize the error between the true sea spectrum and the approximate spectrum being generated. ~hus t.he output of the filter is the wave elevation With the desired significant wave height and peak period. Here the filter determined by Kallstrom(1983) is used that is expressed by an auto-regressive moving average model (ARMA):
NvijmL= -0 . 000946V/sin(2 Yr) - 3 . 75r Irl where Vr is the current speed relative to the rig and y is the current direction relative to the rig. XVISC . and y vlSe . can also be expressed as (Barros, 1989) :
Hk) + aWIHk-1) + .. .. awnHk-n) =
309
The low damping and absence of restoring forces in the horizontal plane are reflected by the transfer function. In the case of u eo = v eo = 0 it follows that:
Wave Drift Forces . The wave drift forces are also obtained from scale model tests: X"d'ir/ m
0.0021(~ - z)2cos(Ywave - Y')
=
Y"d'ir/m = 0 . 0014(~ -z)2 s in( Yw ave - Y') where z is the heave and of wave propagation .
y
H(s ) =
wave is the direction
Resultin~ Model The position X o ' Y 0 and the heading in the low frequency model are obtained by integration of the following equations:
~ S2
0
0
0
bn S2
0
0
0
b)) S2
There is a small difference in the presence of the current. For example, consider u eo = 1 m / s, v eo = 0 . Neglecting the coupling terms, the transfer functions is:
Xo = u coSy:> - v si nY'
b ll
0
S (s + 0 ,021 )
Yo = usinY' + v cOSy:> H(s) =
b 22
0 S
(s + 0,011)
0 0 b))
0
The total horizontal motions are obtained by adding the high frequency motions:
0
S2
DPS DESIGN Y,o, Y'
Yo
'0'
Three independent controllers were designed to sway, surge and yaw motions . The rig dynamics model assumed in the control desaign is described by Eq . (1) with na = 2, nb = 1 and d = 1. The control effort u is, depending on the motion, X'h,ust' Y'h'USI or N'h'uSl' Actually DPS has a set of propellers allocated along the rig. So it is necessary calculate control signals for each of them to get desired forces and moment. Thus, this model makes a suitable simplification that doesn't prevent further improvement. The order of polynomial C(Z·I) was also assumed as 2. The output of the transfer function C(z' 1)/A(z·l) fed by white noise represents the sum of high frequency motion and measurement noise. This assumption is reasonable because the amplitude of high frequency motion of the semi-submersible rig considered in this paper is of the same order as the accuracy of the position measurement system. To define the control structure the following cost function was selected:
+ Y hr
+ Y'hr
Y'
Linear Model Analysis A state space representation for the linearized low frequency model is obtained around the values u = v = r = 0 and u e = u eo and ve = v eo :
x = AX
+ BU
y = CX
where:
X=[uvrXoYoY']T;U
y
=
[Xo Y o
Y'l'
all a l2 a l ) a 21 an a 2 )
A=
a)1 an
1 0 0 1 0 0
0 0 0
0 0 0 0 0 0
bll 0 0 0 0 B = 0 bn 0 0 0 [ o 0 b)) 0 0
0 0 0 0 0 0
0 0 0 0 0 0
The sampling period adopted is 12 s. This period was chosen considering the low frequency dynamics, whose range is 0 - 0.25 rad/s, and the maximum recommended frequency to the operation of the propellers that is 0 . 63 rad/s. The values of Po' qo and ql were selected by preliminary simulation tests imposing ±2° to the heading and ±5 meters to the positions .
~Ol
000100] C= 000010 [o 000 0 1
Simulation eresults Several simulations were made to analyze the performance of the self-tuning DPS. In this paper the influence of coefficient ql and the absence of high frequency motion filter are discussed in particular.
where a ij and b ij are obtained from linearization of Eq. (10) (Barros, 1989)
310
SURGE(m)
08
8 .0
THRUST FORCE (SURGE DIRECTION) (MN)
04
4 .0 TIME (s) 500
0 .0
0 .0 + - - - - - - - - 4 - - - - = - - - + - - 500 1000 T I ME (s)
-4 .0
-0 .4
-8 .0
8 .0
4 .0
TIME (s)
0 .0
-4 .0
-8.0 YAW ( degree)
8 .0
4 .0
TIME (s) 500
1000
0 .0 1.000 -4 .0
- 6 .0
-8 .0
- 12.
TIME (s)
Fig . 2.1 Motions of the rig .
Fig. 2 .2 Thruster efforts
The effect of using ql .. 0 is to increase the o~fset of the rig output, mainly if it is perturbed with constant environmental forces . The offset is usually eliminated by the introduction of an integral action, defining q = q However this . I o. became oscilapproac h f al' 1e d since rig motions latory as an undamped system instead of eliminating the offset. The analysis of the closed loop transfer function by the root locus method has shown that the oscillation increases because ql .. qo tends to assign poles to the unit circle. An intermediate solution was obt a ined with ql = 0.3q o· The test was m a de in accordance with the following conditions: I)Wave with Pierso~ Moskowitz spectrum with significant wave height = 5.6 m and peak period = 12 s · 2) y = 2250 ,' 3) Y currenl -- 190 0', current velocity '. Wlve = 1 m/s; 4) The positions and heading are assumed to be measured with additive white noise disturbances with standard deviations of 0.5 m and 0, 05° respectively.
pected there is a small offset mainly in surge directions . This is due to both constant current action and ql ~ q o' Fig . 2.2 shows that the profile of control efforts is suitable if a comparison is made with other results , for instance Kallstron ( 1983) . The effects of the filter abscence were also tested. A simulation without the feedback of the high frequency motion to the controller was performed, other conditions were Kept the same as in previous test. The results shown in Fig. 3 . 1 and Fig . 3.2 . are similar in comparison with Fig . 2.1 and Fig. 2.2 specially the modulation of the control signal. The sampling period selected doesn't reproduce the high frequency motion because its frequency is greater than 0 .3 rad/s. But there is the aliasing effect of this signal that can cause unnecessary motion of the thrusters . This aspect needs to be analyzed further. CONCLUSIONS In this paper the application of self-tuning DPS was shown. The results indicate that there is some difficulty to eliminate the offset only by sear-
The rig motions shown in Fig. 2 . 1 reveal that the ~elf-. tu~ing DPS keeps position and heading of fig Within the accuracy requirements. As ex-
311
SURGE (m) B.O
OB
4 .0
THRUST FORCE (SURGE DIRECTION) ( MN)
0.4 TI M E (s)
500
0 .0
1000 0 .0 500 -0.4
-8.0
-OB SWAY(m)
B.O
0.8
4 .0
TIME (s)
THRUST FORCE (SWAY DI RECTI ON) ( MN)
0.4
0 .0
0.0
-4.0
-0.4
-8.0 8 .0
1000 TI ME (s)
-4 .0
TIME (s)
- 08 YAW (degree)
12 .
4.0
6 .0
TI ME ( s)
500
MOMENT (YAW DI RECTION) ( KNm)
1000
0.0
0 .0 500
1000 TIME (s)
-4 .0
-6.0
-8 . 0
- 1.2
Fig . 3 . 1. Motions of the rig without high frequency components
Fig . 3.2 Thruster efforts without high frequen-
ching for a suitable cost function. The low damping characteristics of the rig at low frequency motions in the horizontal plane impose limitations to the integral gain included in the cost function . The results also suggest that the control effort is not significantly affected by the filter absence for the semi-submersible rig considered. The implementation of this kind of DPS seems easier than linear quadratic control with EKF since it only requires order and time delay knowledge of the system . Thus a self-tuning regulator may be an easy way to cope with the non-linearities and the time variant characteristics of the system.
REFERENCES Balchen, J.G. et alii (1985) . Dynamic positioning system based on Kalman Filter and optimal control. Modelinl: Identification and contro!. Oslo, 1(3), pp. 135-63. Barros, E . A. (1989) Aplica~ao de urn controlador auto-ajusuivel ao sistema de posionamento dinamico de uma platafor-
312
ma semi-submersivel. Ms dissertation, University of Sao Paulo, Sao Paulo Clarke, D.W. and P.J. Gawthrop (1975). Selftuning controller. Proceedinl:s of the Institution of Electrical Enl:ineers , 22(9) London, pp. 929-934 Fung, P.T.K. and M.J. Grimble (1983). I?ynamic ship positioning using a self-tunIng Kalman Filter. IEEE Transactions on Automatic Contro!, AC-28, New York, pp 339-349 . Grimble, M.J., R . J. Patton and D.A. Wise (1980) . Use of Kalman filterinl: techniques in dynamic ship positioninl: systems. lEE Proceedings . Part D:Control theory and Applications, 127(3), London pp. 93-102 . Kallstron , C.G. (1983). Mooring and dynamic positioning of a semi-submersible. A comparative simulations study. Proc 2nd International Symposium on Ocean Enl:ineerinl: and Ship Handlinl:, Gothenburg, pp.417 -440. Norbbin, N.H. (1971) Theory and observations on the use of a mathematical model for ship maneuyerinl: in deep and confined ~. Goteborg, Statens Skeppsprovningsanstalt, (SSPA publication).