- Email: [email protected]
Model Predictive Rudder Roll Stabilization Control for Ships
Copyright C1l IF AC Manoeuvring and Control of Marine Craft, Aalborg, Denmark, 2000
MODEL PREDICTIVE RUDDER ROLL STABILIZATION CONTROL FOR SHIPS
Tristan Perez' Ching-YawTzeng" Graham C. Goodwin'"
• Department 0} Electricd And Computer Engineering, The University 0} Newcastle, Austrdia •• Institute D) Marine Technology, Nationd Taiwan Ocean University, Taiwan . ••• Department 0} Electricd And Computer Engineering, The University 0} Newcastle, Austrdia
Abstract: In this paper, the problem of rudder roll stabilization for ships is addressed using model predictive control. The problem of rudder roll stabilization arises from using the rudder produced rolling motion of the ship to reject the undesirable wave produced roll. The main limiting factor of this problem is the highly non linear behaviour of the mechanical devices that command the rudder. The rudder mechanisms impose constraints on the maximum slew rate and magnitude attainable for the rudder movement, which directly affect the roll stabilization performance. The application of model predictive control is motivated by its essential ability to handle these kinds of constraints. Copyright© 2000 IFAC Keywords: Ship Control, Model Predictive Control.
1. INTRODUCTION
and excursion attainable for the rudder movement. In order to deal with the saturation and slew rate problems among other things, methods such as automatic gain control, (van Amerongen et d ., 1990), or gain scheduling algorithms, see (Laudval and Fossen, 1998) have been proposed.
Besides controlling the heading of a ship, it is also desirable to reduce the rolling motion produced by the waves so as to prevent cargo damage and improve crew efficiency and passenger comfort. Conventional methods for ship roll stabilization include water tanks, stabilizing fins, and bilge keels (Fossen, 1994).
In this paper, the application of Model Predictive Control (MPC) is proposed for the problem of RRS. The use of the MPC strategy is motivated by its essential ability to directly handle the constraints mentioned above.
Rudder Roll Stabilization (RRS) is attractive because no extra equipment needs to be added to the ship. Conversely, the fundamental drawbacks of RRS are the loss of effectiveness it presents at slow speeds of the ship, and the interference it produces with heading control.
The paper proceeds as follows. In section 2, the models that describe the ship dynamics, the wave disturbances, and the steering machinery are presented. In section 3, the fundamentals of the MPC formulation are described and the design of the Model Predictive Rudder Roll Stabilization Control (MPRRSc) is developed. Section 4, contains simulation results. Finally, in section 5 the paper is concluded.
From the point of view of control design, the main limiting factor of the RRS problem is the highly non linear behaviour of the mechanical devices that command the rudder. These devices impose constraints on the maximum slew rate
45
Roll
The roll motion due to the rudder presents a faster dynamic response than the one of the rudder produced yaw motion due to the different moments of inertia of the hull associated with each direction. This property is exploited in RRS.
(0)
Rudder Angle
Surge
(0)
Pitch Sway
(8)
tU Tu
Yaw
(x)
(
The common method of modelling the ship motion is based on the use of Newton's equations in surge, sway and roll, (Blanke and Christensen, 1993). Using this approach, a non linear model is obtained from which a linearised state space model can be derived by defining the following vector of state space variables:
Heave (z)
(y)
Fig. 1. Magnitudes and Conventions for Ship Motion Description 2. MATHEMATICAL MODELS
X :=[V Tp cfJ1jJJT
2.1 Magnitudes and Conventions Jor Ship Motion Descnption
(1 )
Thus, the linearised model around the equilibrium point x = [0000 O]T, and at a forward ship speed U is of the form:
In order to describe the motion of a ship, six independent coordinates are necessary. The first three coordinates and their time derivatives correspond to the position and translational motion while the other three coordinates and their time derivatives correspond to orientation and rotational motion descriptio~ , (Fossen, 1994). For marine vehicles, the six different motion components are called: su'8e, SUXl1j, heave, roll, pitch, and yaw Accordingly, the most generally used notation for theses quantities are: x, y, Z. q), B, and 'l/J respectively, while their time derivatives are denoted u, v, Uj p, q, and rrespectively. Figure 1 shows the 6 coordinate definitions and the most generally adopted reference frame.
x=Ax+ B 6
(2)
The model depends on the ship speed, but this is not of much concern as the model is quasiconstant over a reasonable speed range. This allows the use of gain scheduling to update the model according to different operating speeds. For more detailed explanation of the model see (Blanke and Christensen, 1993).
2.3 Wave Disturbance lv10del Undesired variations of the roll and yaw angles are mainly generated by waves. These wave disturbances cannot be simply modeled as forces and moments or simply included in (2). The reason is that wave forces act over the entire hull, (Blanke and Christensen, 1993). As a result, the disturbances are considered at the output, and the complete model of the ship dynamics including the output equation is of the form:
A part from the mentioned magnitudes, figure 1 also shows the definition and positive convention for the Rudder Angl~ which is usually denoted 6.
x =Ax +B6 , y = C x + d wa ve
2.2 Rudder Generated Motion
(3)
,
where y := [rJ1 'l/JV is defined as the vector of the output variables of interest in the RRS problem under consideration and dwa ve is the wave induced disturbance on the output variables.
When the rudder moves away from the central position, a lift force, called rudder Jorc~ appears on the rudder. This rudder force is due to the non symmetric pressure distribution over the rudder surface. Since the size of the rudder is relatively small, the rudder force is small. However, the roll moment it produces is large as the rudder is located at the bottom of the hull. This moment produces rolling motion. At the same time, the rudder force also produces a yaw moment. This moment is small but once the ship has a small turning deviation, there exists an angle of attack between the hull and the water flow which produces a lift force acting on the hull. This force, called the hull hydrodynamic Jorc~ supplies the centripetal force necessary for the turning motion, and also produces a roll moment as the hull hydrodynamic force acts halfway along the draft. This moment is opposite and larger than the one produced by the rudder.
The wave disturbances can be characterized in terms of their frequency spectrum, for instance, using the Pierson-Moskowits spectrum, see (Fossen, 1994). This frequency spectrum can be simulated by either a series of added sinusoidal signals, or using filtered white noise. In the latter case, the filter used to approximate the spectrum is a second order one of the form R (s)
=
Kw S
s2
+ 2 ~ w. S + w;
(4)
in which Kw is a constant describing the wave intensity, ~ is a damping coefficient, and We is the encounter frequency, which depends on the wave frequency, forward speed of the ship, and the angle between the heading and the direction of the wave . For more details see (Fossen, 1994).
46
bC
introduces phase lag if the system is rate saturated at the frequency of interest. A simple rule to determine a suitable rudder rate that leads to good roll reduction, arises from considering the problem at just the natural frequency of the ship. Therefore, given the model of the ship and the maximum rudder excursion limit, it is easy to show that to avoid saturation of the steering machinery rate, and at the same time reach the maximum rudder angle limits at the natural frequency of the ship the maximum slew rate should be:
-{f] Excursion Limiter
Rate Limiter
Fig. 2. Schematic Block Diagram of the Rudder Mechanism (van Amerongen) 2.4
Steering Machinery
The ship steering machinery is typically an hydraulic mechanism. A widely used mathematical model of the steering machinery is the one proposed by van Amerongen, see (Fossen, 1994). Figure 2 shows a block diagram representation of the model. The saturation blocks in the block diagram are defined as sat C ) := si gn( ·) min(I · !' f3) , f3
>0
(6)
For example, if the natural frequency of the ship is 0.5 rad/ sec then for a maximum rudder excursion of 20 deg, a maximum ra te of 10 deg/ sec is necessary to avoid saturation whilst reatining full rudder amplitude. Whereas, a rate of 15 deg/ sec would be necessary if the the maximum rudder excursion were 30 deg.
(5)
For merchant ships the rudder rate is within 3 to 4 deg/ sec while for military vessels, it is in the range of 3 to 8 deg/ sec. The maximum rudder excursion is in the range of 20 to 35 deg.
This simple rule can give an indication of the quality of roll reduction achievable or whether it is desirable to upgrade the rudder rate to achieve good roll reduction. If the rudder slew rate is less than the value given in (6), then the steering machinery will saturate in rate. Thus, it will not be possible to obtain the maximum amplitude for the rudder movement at that frequency, and some capability of roll reduction will be lost. Similar considerations for the applicability of RRS are given in (Roberts, 1993).
In order to achieve better roll reduction, new steering machinery systems have been designed, and the rudder rate has been increased to 15-20 deg/sec, (Laudval and Fossen, 1998). 3. CONTROL DESIGN
3.1 Control O~ ectives The design objectives for RRS are:
• Increase the damping and reduce roll amplitude.
3.2 Model Predictive Rudder Roll Stabilization Control
• Control the heading oJ the ship.
Model Predictive Control is an optimization method that selects control inputs by on-line minimization of a performance index. The performance index is defined in terms of both present and predicted system variables and is evaluated using an explicit model to predict fu ture system outputs. The control sequence, which is computed based on an estimate of the current state of the system, is an open loop strategy. However, this is converted into a feedback strategy by continuously re-evaluating the state based on the most recent observations of the output. Typically, the first step of the open-loop control sequence is applied, and the whole procedure is repeated as updated state information becomes available. An essential feature of MPC is the ability to handle constraints directly, including constraints on the control inputs, outputs or internal states. This feature has been one of the keys to its success in industrial applications (Meadows and Rawlings, 1997). In the following, the general MPC design is described, and then applied to the RRS problem.
In order to achieve these control objectives, there must be some frequency separation between the roll and ya w responses to the rudder movement. Basically, high frequency rudder movements are required for roll control while low frequency rudder movements are used to control the yaw or heading. The ship presents a roll natural frequency, and it is within the range of this natural frequency where roll reduction is most desired. On the other hand, the ya w / rudder frequency response is sensitive at low frequency and attenuates quickly at high frequency (Fossen, 1994). In order to achieve high roll reduction, the steering machinery has to be able to generate a rudder movement with enough power spectral content around the ship natural frequency. The power transferred from the rudder to the roll is a function of the maximum rudder excursion and the rudder rate. The maximum rudder angle limits the roll reduction ability directly while the limited rudder speed reduces the amplitude and
The discrete time model of the system is given by 47
x(t + 1) = A x(e) + B ure), y( £) = C x(e),
and,
(7)
where x( €) E IRn , u( £) E IRm, and y( €) E IRP. It is assumed that (A, B, C) is stabilizable and detectable. For this plant, the problem of tracking a constant set-point Ys and rejecting the output disturbance d is considered, i.e., the objective is to regulate the error ere) = (y(e)
+ d(e))
- Ys
Y =
l =k
w
x(k),
(9)
,
.
<1>=
o
AM-IB A M - 2 B .. . AB AMB AM-IB .. . A2B
diag[C.... , C] ,
Q[CAx(k) - Yds]
u
uTWU
+ 2U TF.
(15)
In many control applications, the spectrum of the disturbances can be approximated by the output of a white noise driven filter. Based on this approximation, a Kalman filter can be used to estimate the state of the augmented system comprising the state of the plant and the state of the noise filter.
o o B AB
In the RRS application problem, the noise filter that approximates the disturbance spectrum is given in (4). Consequently, by converting (4) into a state space form and using an equivalent discrete time model, the disturbance model is given by, xd(l + 1) = Ad Xd(e) + Bd 1)(e), dwave(l) = Cd Xd(e),
Q = diag[Q, ... ,Q],
C=
T
In order to implement the MPC algorithm proposed in subsection 3.2, it is necessary to have an estimate of the state and the disturbance; as well as to predict the disturbance over the prediction horizon N described in 13. These aspects of the design are taken into consideration below.
Then, using (10), (11), the following definitions
diag[R, ... , RJ,
c
3.3 Optimal Estimation and Prediction OJ Sate and Output Disturbance
(11)
R. =
(14)
Going back to the RRS application, the state x is the one defined in (1), the output vector is y = [tP ?jJjT, the disturbance is the wave generated one, i.e., dwave , and the control signal, u, is the rudder angle 6.
(10)
o o
R. u
The first control move uopt(k) is the control applied to the plant (7) at time k, and then the whole procedure is repeated at time k + l.
x(k+N)
B
+ d(k) ] +
LU'SK
U=
B AB
Q [y(k) - Ys
cT QC + R.,
=T
U opt = arg min
X(k+l ) ] x(k + 2) [
(13)
Magnitude and rate constraints on the plant input and output can be easily expressed as linear constraints on U of the form LU < K, and the optimal solution can be numerically computed via quadratic programming:
where
x=
+ d(k)?
F =T
In the presence of constraints on the input and output, the above dynamic optimization problem amounts to finding the solution of a nondynamic constrained quadratic program (QP) (Muske and Rawlings, 1993). Indeed, letting (7) evolve N steps from the initial condition x(k), the following equality holds:
+A
+ N)
,
Where in (14), ] is independent of U and
In (9), N is the prediction horizon; M :S N is the control horizon; and Q 2: 0 and R > 0 are the state and control weighting matrices, respectively. The minimization of (9) is performed on the assumption that the control reaches its steady state value after M steps. Note that, if the model contains a pure integrator, then the desired steady state input can be zero, i.e., u(€) = 0, ve 2: k + M.
= U
Ys - d(k
[Y - Yds]T Q [Y - Yds] + UT = J + UT.\f U + 2UTF .
i=k
x
+ + 2)
[
J = [y( k ) - Ys
(8)
u(ef Ru(l).
Il]
YS - d(k Ys - d(k .
the functional (9) can be expressed as:
k+M-J
I: eT(f)Qe(l) + I:
l
y(k+ N)
In order to do this, the following standard MPC algorithm is used: given the external signal d(k) and the current state measurement x(k) (these signals will be later replaced by on-line estimates), find the M -move control sequence {u (k) , u(k + l), ... ,u(k + M -I)} that minimizes the finite horizon performance index: k+N
_ Yds -
[
to zero.
J =
Y(k+l ) ] y(k + 2) .
(12)
(16)
Where T/( €) is a discrete time white noise vector.
48
.d"...,es ,
I
Ys--; Diswrbance Predictor . QP "
,
i
0
i Stecrin\!- Machinery :~ i
.:. +
,
I
'p
;
~
.
I
.
I"
V. V i
I
:
~
I, :
"
I
"
I
"
'J
· 1
f}~~~~~~Nv~~
'd
Fig. 3. Schematic Block Diagram of the Control Structure
i: .
02040
The plant model is given in (7), but for the sake of filtering purposes, state and measurement noise are considered, i.e.,
+ 1)
" ~ ~ ~ :'
"
,
Xp(£
"
,) y, ~" '"
"
I
MPRRSCoff MPRRSCon
I
.~~ ~' "
,"
!
I K.. lrnanFiller
i
Ship DvnamiCS~Or)
I
I, I
:
= Ap Xp(£)
y(£) = C p xp(£)
+ Bp u(£) + v(e) , + d wa ve (£) + v(£) ,
I
n
60
a>
I
I
60
80
120
I~
160
I
.
1
.
.
120
14()
160
100
1~
r""'lsecl
~
~
f~~~~!v ifvVVVvVVVV\AJ
(I 7)
er: -'0
o
"
20
40
100
180
'
200
rtmelsocl
Now, by defining the augmented state vector
1,
Fig. 4. MPRRSC Performance. a- Roll Angle; bYaw Angle; and c- Rudder Angle.
(18)
4. SIMULATION RESULTS the total evolution and output generation equations are
In this section, simulation results will be presented to verify the performance of the proposed MPRRSC scheme. The ship model data was taken from (Blanke and Christensen, 1993) at a ship speed of 9 mj sec. The continuous time model matrices are given in the appendix. The discrete time model is obtained using the zero order hold eqUivalent, See (Fralnklin et d., 1998), for a 0.5 sec sampling period. The prediction and control horizons are 20 and 18 samples respectively. During the first simulation, both output references are kept to zero. Figure 4 shows the results obtained. Figure 4a shows the roll angle when MPRRSC is both on and oJl respectively. When MPRRSC is off, the rudder is kept at the central position. Therefore, the roll depicted is that produced by wave action. Figure 4b shows the yaw angle for the same situation, while 4c shows the rudder angle when the MPRRSC is on. For this case, the maximum rudder excursion is 30 deg. while the slew ra te is 15 deg j sec. In the second simulation, a step-wise refrence for the yaw of ±30 deg. is generated , while the MPRRSC is on. The results are shown in figure 5.
(19)
These last equations, are in the form where one can immediately apply standard state estimation procedures, i.e., Xa(£
+ 1)
= Aa xa(£) + Ba u(£) + r w w(£),
y(e ) = Ca Xa(£)
+ v(£) ,
(20)
Now using the stationary Kalman filter, an estimation of the plant state and wave disturbances, say xp(e) and dwave(e), at the instant e = k are obtained given u(k) and y(k) . However, the problem of predicting d wave over the prediction horizon N still remains. The optimal n-step-ahead predictor given xd(k) for the system (16) is of the form (Goodwin and Sin, 1984): (21)
Thus, the prediction of the disturbance is given by d(k d(k
In order to measure the roll reduction produced by MPRRSC, the following percent quantity is used, (Fossen, 1994):
1)]
+ + 2)
RR%
(22)
rd(k~N) pl~t
and
x 100,
(23)
in which AP is the standard deviation of the measured roll when the Roll stabilization controller is oJt while ReS is the standard deviation of the measured roll when the Roll stabilization controller is on The results obtained for the simulation shown in figure 4, and as well as for other simulations with different maximum rudder excursions and slew rates are summarized in the following table:
Finally, a schematic block diagram of the control structure is shown in figure 3.
I The subindices p and d refer to state variables respectively.
- RCS = AP AP
disturb~ce -}ilter
49
Finally, even though the results presented are based on digital simulation, the algorithm could be easily implemented in real-time. Indeed, the simulation required less than real time using MATLAB on a 300Mhz Pentium II standard Pc.
6. REFERENCES
Bitmed, R, M. Gevers and V. Wertz (1990). Adap-
tive Optimal Contra: the thinking's man GPC _ 401 ' , ~; 201I "l (I 1\~ i" \ ;C~ 1 i ?
Prentice-Hall, Inc. Blanke, M. and A. Christensen (1993). Rudder roll dumping autopilot robustness to swayyaw-roll couplings. In: Proceedingj DJ 10th
1~' '".nr" !'. ; 1 \ I· 1\1\ \'1 11 \I i rI ) \I ! 1 I '\ \' 1 ' I.; \ ) \: I I i UJ V LI V U
l~,!\I :!\!\I; "I\ fW,
;; a., \' \!
~ -20~
I 1 i.1 ! ' !\ I \ i \! " ! I , ! \1
l! ~ V U V
-40 '
o
/ \
V
\
, I
j
\
SCSS, Otauxl, Canadapp. 93--119.
,
20
4C
60
80
100
12C
140
160
180
Fossen, Thor 1. (1994). Guidance and Control DJ Ocean Marine Vehicles John Wiley and Sons Ltd. New York. Fralnklin, G. E, J. D. Powell and M. Workman (1998) . Digital Control DJ Dynamic Systems 3rd ed .. Addison Wesley. Good win, G. C. and K. S. Sin (1984). Adaptive filtering, Presiction, and Control Prentice-Hall. Laudval, Trygve and Thor 1. Fossen (1998). Rudder roll stabilization for ships subject to input rate saturation using a gain scheduling control law. In: Proceeding DJ IFAC CAMS'98 pp. 121--126. Meadows, E. S. and J. B. Rawlings (1997). Non Linear Process Control Prentice-Hall. Muske, K. Rand J. B. Rawlings (19 93). Model predictive control with linear models. AIChe
200
TIme jsecj
Fig. 5. MPRRSC Performance. a- Roll Angle; bYaw Angle; and c- Rudder Angle. Max. ExcurSIon I Max. Slew Rote 30deg/15deg/sec 30 deg /12 deg /sec 30 deg /8 deg / sec 20 deg /15 deg / sec 20 deg /12 deg /sec 20 deg /8 deg /sec
RR"Io 59 53 41 49 47 42
Table 1. Roll Reduction Obtained for Different Maximum Excursions and Slew Rates.
5. CONCLUSIONS
lournal3~2),
In this paper, a solution to the problem of RRS using MPC was proposed. The results obtained are very encouraging. The rudder constraints are very well managed by the MPC controller. However, some other aspects of the problem still remain the subject of on-going research. For instance, while tuning the controller, a tradeoff was faced when choosing values for the matrices Q and R (in the functional given in (9» since more roll reduction results in more interference in the heading angle. This tradeoff is essentially produced by the single input two outputs (SITO) characteristic of the plant. Therefore, an interesting research question is to obtain a better understanding of the fundamental limitations that the SITO nature of the problem imposes regardless of the control strategy being used . It would also be interesting to link these limitations to the specific solution proposed in this paper.
262--2872.
Roberts, G. N. (1993). A note on the applicability of rudder roll stabilization for ships. In Proceedings DJ ACe, San Francisco, CaliJornia, USA. pp. 2403--2407. van Amerongen, J., P. van der Klugt and H. van Nauta Lemke (1990). Rudder roll stabilization for ships. Automatica26, 679--690.
Appendix A. SHIP DATA In this appendix, the continuous ship model matrices for a constant forward speed of a naval vessel of 9 m/sec are given. The Ship parameters of the model were taken from (Blanke and Christensen, 1993). -0.1795 -0.8404 0.2115 0.9665 0] -0.0159 -0.4492 0.0053 0.0151 0 A = 0.0354 -1.5594 -0.1714 -0.7883 0 o 0 1 0 0 [ o 1 000
In this paper, it has been assumed that the disturbance model is known. An aspect that still needs to be investigated is the inclusion of an adaptive mechanism for tracking the sea conditions for accurate disturbance estimation and prediction. Similarly, additional work is needed to improve the robustness of the algorithm to the assumed ship model.
(A.l)
0'2784] -0.0334
B =
- 0'r94 [
50
(A.2)
MODEL PREDICTIVE RUDDER ROLL STABILIZATION CONTROL FOR SHIPS
Tristan Perez' Ching-YawTzeng" Graham C. Goodwin'"
• Department 0} Electricd And Computer Engineering, The University 0} Newcastle, Austrdia •• Institute D) Marine Technology, Nationd Taiwan Ocean University, Taiwan . ••• Department 0} Electricd And Computer Engineering, The University 0} Newcastle, Austrdia
Abstract: In this paper, the problem of rudder roll stabilization for ships is addressed using model predictive control. The problem of rudder roll stabilization arises from using the rudder produced rolling motion of the ship to reject the undesirable wave produced roll. The main limiting factor of this problem is the highly non linear behaviour of the mechanical devices that command the rudder. The rudder mechanisms impose constraints on the maximum slew rate and magnitude attainable for the rudder movement, which directly affect the roll stabilization performance. The application of model predictive control is motivated by its essential ability to handle these kinds of constraints. Copyright© 2000 IFAC Keywords: Ship Control, Model Predictive Control.
1. INTRODUCTION
and excursion attainable for the rudder movement. In order to deal with the saturation and slew rate problems among other things, methods such as automatic gain control, (van Amerongen et d ., 1990), or gain scheduling algorithms, see (Laudval and Fossen, 1998) have been proposed.
Besides controlling the heading of a ship, it is also desirable to reduce the rolling motion produced by the waves so as to prevent cargo damage and improve crew efficiency and passenger comfort. Conventional methods for ship roll stabilization include water tanks, stabilizing fins, and bilge keels (Fossen, 1994).
In this paper, the application of Model Predictive Control (MPC) is proposed for the problem of RRS. The use of the MPC strategy is motivated by its essential ability to directly handle the constraints mentioned above.
Rudder Roll Stabilization (RRS) is attractive because no extra equipment needs to be added to the ship. Conversely, the fundamental drawbacks of RRS are the loss of effectiveness it presents at slow speeds of the ship, and the interference it produces with heading control.
The paper proceeds as follows. In section 2, the models that describe the ship dynamics, the wave disturbances, and the steering machinery are presented. In section 3, the fundamentals of the MPC formulation are described and the design of the Model Predictive Rudder Roll Stabilization Control (MPRRSc) is developed. Section 4, contains simulation results. Finally, in section 5 the paper is concluded.
From the point of view of control design, the main limiting factor of the RRS problem is the highly non linear behaviour of the mechanical devices that command the rudder. These devices impose constraints on the maximum slew rate
45
Roll
The roll motion due to the rudder presents a faster dynamic response than the one of the rudder produced yaw motion due to the different moments of inertia of the hull associated with each direction. This property is exploited in RRS.
(0)
Rudder Angle
Surge
(0)
Pitch Sway
(8)
tU Tu
Yaw
(x)
(
The common method of modelling the ship motion is based on the use of Newton's equations in surge, sway and roll, (Blanke and Christensen, 1993). Using this approach, a non linear model is obtained from which a linearised state space model can be derived by defining the following vector of state space variables:
Heave (z)
(y)
Fig. 1. Magnitudes and Conventions for Ship Motion Description 2. MATHEMATICAL MODELS
X :=[V Tp cfJ1jJJT
2.1 Magnitudes and Conventions Jor Ship Motion Descnption
(1 )
Thus, the linearised model around the equilibrium point x = [0000 O]T, and at a forward ship speed U is of the form:
In order to describe the motion of a ship, six independent coordinates are necessary. The first three coordinates and their time derivatives correspond to the position and translational motion while the other three coordinates and their time derivatives correspond to orientation and rotational motion descriptio~ , (Fossen, 1994). For marine vehicles, the six different motion components are called: su'8e, SUXl1j, heave, roll, pitch, and yaw Accordingly, the most generally used notation for theses quantities are: x, y, Z. q), B, and 'l/J respectively, while their time derivatives are denoted u, v, Uj p, q, and rrespectively. Figure 1 shows the 6 coordinate definitions and the most generally adopted reference frame.
x=Ax+ B 6
(2)
The model depends on the ship speed, but this is not of much concern as the model is quasiconstant over a reasonable speed range. This allows the use of gain scheduling to update the model according to different operating speeds. For more detailed explanation of the model see (Blanke and Christensen, 1993).
2.3 Wave Disturbance lv10del Undesired variations of the roll and yaw angles are mainly generated by waves. These wave disturbances cannot be simply modeled as forces and moments or simply included in (2). The reason is that wave forces act over the entire hull, (Blanke and Christensen, 1993). As a result, the disturbances are considered at the output, and the complete model of the ship dynamics including the output equation is of the form:
A part from the mentioned magnitudes, figure 1 also shows the definition and positive convention for the Rudder Angl~ which is usually denoted 6.
x =Ax +B6 , y = C x + d wa ve
2.2 Rudder Generated Motion
(3)
,
where y := [rJ1 'l/JV is defined as the vector of the output variables of interest in the RRS problem under consideration and dwa ve is the wave induced disturbance on the output variables.
When the rudder moves away from the central position, a lift force, called rudder Jorc~ appears on the rudder. This rudder force is due to the non symmetric pressure distribution over the rudder surface. Since the size of the rudder is relatively small, the rudder force is small. However, the roll moment it produces is large as the rudder is located at the bottom of the hull. This moment produces rolling motion. At the same time, the rudder force also produces a yaw moment. This moment is small but once the ship has a small turning deviation, there exists an angle of attack between the hull and the water flow which produces a lift force acting on the hull. This force, called the hull hydrodynamic Jorc~ supplies the centripetal force necessary for the turning motion, and also produces a roll moment as the hull hydrodynamic force acts halfway along the draft. This moment is opposite and larger than the one produced by the rudder.
The wave disturbances can be characterized in terms of their frequency spectrum, for instance, using the Pierson-Moskowits spectrum, see (Fossen, 1994). This frequency spectrum can be simulated by either a series of added sinusoidal signals, or using filtered white noise. In the latter case, the filter used to approximate the spectrum is a second order one of the form R (s)
=
Kw S
s2
+ 2 ~ w. S + w;
(4)
in which Kw is a constant describing the wave intensity, ~ is a damping coefficient, and We is the encounter frequency, which depends on the wave frequency, forward speed of the ship, and the angle between the heading and the direction of the wave . For more details see (Fossen, 1994).
46
bC
introduces phase lag if the system is rate saturated at the frequency of interest. A simple rule to determine a suitable rudder rate that leads to good roll reduction, arises from considering the problem at just the natural frequency of the ship. Therefore, given the model of the ship and the maximum rudder excursion limit, it is easy to show that to avoid saturation of the steering machinery rate, and at the same time reach the maximum rudder angle limits at the natural frequency of the ship the maximum slew rate should be:
-{f] Excursion Limiter
Rate Limiter
Fig. 2. Schematic Block Diagram of the Rudder Mechanism (van Amerongen) 2.4
Steering Machinery
The ship steering machinery is typically an hydraulic mechanism. A widely used mathematical model of the steering machinery is the one proposed by van Amerongen, see (Fossen, 1994). Figure 2 shows a block diagram representation of the model. The saturation blocks in the block diagram are defined as sat C ) := si gn( ·) min(I · !' f3) , f3
>0
(6)
For example, if the natural frequency of the ship is 0.5 rad/ sec then for a maximum rudder excursion of 20 deg, a maximum ra te of 10 deg/ sec is necessary to avoid saturation whilst reatining full rudder amplitude. Whereas, a rate of 15 deg/ sec would be necessary if the the maximum rudder excursion were 30 deg.
(5)
For merchant ships the rudder rate is within 3 to 4 deg/ sec while for military vessels, it is in the range of 3 to 8 deg/ sec. The maximum rudder excursion is in the range of 20 to 35 deg.
This simple rule can give an indication of the quality of roll reduction achievable or whether it is desirable to upgrade the rudder rate to achieve good roll reduction. If the rudder slew rate is less than the value given in (6), then the steering machinery will saturate in rate. Thus, it will not be possible to obtain the maximum amplitude for the rudder movement at that frequency, and some capability of roll reduction will be lost. Similar considerations for the applicability of RRS are given in (Roberts, 1993).
In order to achieve better roll reduction, new steering machinery systems have been designed, and the rudder rate has been increased to 15-20 deg/sec, (Laudval and Fossen, 1998). 3. CONTROL DESIGN
3.1 Control O~ ectives The design objectives for RRS are:
• Increase the damping and reduce roll amplitude.
3.2 Model Predictive Rudder Roll Stabilization Control
• Control the heading oJ the ship.
Model Predictive Control is an optimization method that selects control inputs by on-line minimization of a performance index. The performance index is defined in terms of both present and predicted system variables and is evaluated using an explicit model to predict fu ture system outputs. The control sequence, which is computed based on an estimate of the current state of the system, is an open loop strategy. However, this is converted into a feedback strategy by continuously re-evaluating the state based on the most recent observations of the output. Typically, the first step of the open-loop control sequence is applied, and the whole procedure is repeated as updated state information becomes available. An essential feature of MPC is the ability to handle constraints directly, including constraints on the control inputs, outputs or internal states. This feature has been one of the keys to its success in industrial applications (Meadows and Rawlings, 1997). In the following, the general MPC design is described, and then applied to the RRS problem.
In order to achieve these control objectives, there must be some frequency separation between the roll and ya w responses to the rudder movement. Basically, high frequency rudder movements are required for roll control while low frequency rudder movements are used to control the yaw or heading. The ship presents a roll natural frequency, and it is within the range of this natural frequency where roll reduction is most desired. On the other hand, the ya w / rudder frequency response is sensitive at low frequency and attenuates quickly at high frequency (Fossen, 1994). In order to achieve high roll reduction, the steering machinery has to be able to generate a rudder movement with enough power spectral content around the ship natural frequency. The power transferred from the rudder to the roll is a function of the maximum rudder excursion and the rudder rate. The maximum rudder angle limits the roll reduction ability directly while the limited rudder speed reduces the amplitude and
The discrete time model of the system is given by 47
x(t + 1) = A x(e) + B ure), y( £) = C x(e),
and,
(7)
where x( €) E IRn , u( £) E IRm, and y( €) E IRP. It is assumed that (A, B, C) is stabilizable and detectable. For this plant, the problem of tracking a constant set-point Ys and rejecting the output disturbance d is considered, i.e., the objective is to regulate the error ere) = (y(e)
+ d(e))
- Ys
Y =
l =k
w
x(k),
(9)
,
.
<1>=
o
AM-IB A M - 2 B .. . AB AMB AM-IB .. . A2B
diag[C.... , C] ,
Q[CAx(k) - Yds]
u
uTWU
+ 2U TF.
(15)
In many control applications, the spectrum of the disturbances can be approximated by the output of a white noise driven filter. Based on this approximation, a Kalman filter can be used to estimate the state of the augmented system comprising the state of the plant and the state of the noise filter.
o o B AB
In the RRS application problem, the noise filter that approximates the disturbance spectrum is given in (4). Consequently, by converting (4) into a state space form and using an equivalent discrete time model, the disturbance model is given by, xd(l + 1) = Ad Xd(e) + Bd 1)(e), dwave(l) = Cd Xd(e),
Q = diag[Q, ... ,Q],
C=
T
In order to implement the MPC algorithm proposed in subsection 3.2, it is necessary to have an estimate of the state and the disturbance; as well as to predict the disturbance over the prediction horizon N described in 13. These aspects of the design are taken into consideration below.
Then, using (10), (11), the following definitions
diag[R, ... , RJ,
c
3.3 Optimal Estimation and Prediction OJ Sate and Output Disturbance
(11)
R. =
(14)
Going back to the RRS application, the state x is the one defined in (1), the output vector is y = [tP ?jJjT, the disturbance is the wave generated one, i.e., dwave , and the control signal, u, is the rudder angle 6.
(10)
o o
R. u
The first control move uopt(k) is the control applied to the plant (7) at time k, and then the whole procedure is repeated at time k + l.
x(k+N)
B
+ d(k) ] +
LU'SK
U=
B AB
Q [y(k) - Ys
cT QC + R.,
=
U opt = arg min
X(k+l ) ] x(k + 2) [
(13)
Magnitude and rate constraints on the plant input and output can be easily expressed as linear constraints on U of the form LU < K, and the optimal solution can be numerically computed via quadratic programming:
where
x=
+ d(k)?
F =
In the presence of constraints on the input and output, the above dynamic optimization problem amounts to finding the solution of a nondynamic constrained quadratic program (QP) (Muske and Rawlings, 1993). Indeed, letting (7) evolve N steps from the initial condition x(k), the following equality holds:
+A
+ N)
,
Where in (14), ] is independent of U and
In (9), N is the prediction horizon; M :S N is the control horizon; and Q 2: 0 and R > 0 are the state and control weighting matrices, respectively. The minimization of (9) is performed on the assumption that the control reaches its steady state value after M steps. Note that, if the model contains a pure integrator, then the desired steady state input can be zero, i.e., u(€) = 0, ve 2: k + M.
= U
Ys - d(k
[Y - Yds]T Q [Y - Yds] + UT = J + UT.\f U + 2UTF .
i=k
x
+ + 2)
[
J = [y( k ) - Ys
(8)
u(ef Ru(l).
Il]
YS - d(k Ys - d(k .
the functional (9) can be expressed as:
k+M-J
I: eT(f)Qe(l) + I:
l
y(k+ N)
In order to do this, the following standard MPC algorithm is used: given the external signal d(k) and the current state measurement x(k) (these signals will be later replaced by on-line estimates), find the M -move control sequence {u (k) , u(k + l), ... ,u(k + M -I)} that minimizes the finite horizon performance index: k+N
_ Yds -
[
to zero.
J =
Y(k+l ) ] y(k + 2) .
(12)
(16)
Where T/( €) is a discrete time white noise vector.
48
.d"...,es ,
I
Ys--; Diswrbance Predictor . QP "
,
i
0
i Stecrin\!- Machinery :~ i
.:. +
,
I
'p
;
~
.
I
.
I"
V. V i
I
:
~
I, :
"
I
"
I
"
'J
· 1
f}~~~~~~Nv~~
'd
Fig. 3. Schematic Block Diagram of the Control Structure
i: .
02040
The plant model is given in (7), but for the sake of filtering purposes, state and measurement noise are considered, i.e.,
+ 1)
" ~ ~ ~ :'
"
,
Xp(£
"
,) y, ~" '"
"
I
MPRRSCoff MPRRSCon
I
.~~ ~' "
,"
!
I K.. lrnanFiller
i
Ship DvnamiCS~Or)
I
I, I
:
= Ap Xp(£)
y(£) = C p xp(£)
+ Bp u(£) + v(e) , + d wa ve (£) + v(£) ,
I
n
60
a>
I
I
60
80
120
I~
160
I
.
1
.
.
120
14()
160
100
1~
r""'lsecl
~
~
f~~~~!v ifvVVVvVVVV\AJ
(I 7)
er: -'0
o
"
20
40
100
180
'
200
rtmelsocl
Now, by defining the augmented state vector
1,
Fig. 4. MPRRSC Performance. a- Roll Angle; bYaw Angle; and c- Rudder Angle.
(18)
4. SIMULATION RESULTS the total evolution and output generation equations are
In this section, simulation results will be presented to verify the performance of the proposed MPRRSC scheme. The ship model data was taken from (Blanke and Christensen, 1993) at a ship speed of 9 mj sec. The continuous time model matrices are given in the appendix. The discrete time model is obtained using the zero order hold eqUivalent, See (Fralnklin et d., 1998), for a 0.5 sec sampling period. The prediction and control horizons are 20 and 18 samples respectively. During the first simulation, both output references are kept to zero. Figure 4 shows the results obtained. Figure 4a shows the roll angle when MPRRSC is both on and oJl respectively. When MPRRSC is off, the rudder is kept at the central position. Therefore, the roll depicted is that produced by wave action. Figure 4b shows the yaw angle for the same situation, while 4c shows the rudder angle when the MPRRSC is on. For this case, the maximum rudder excursion is 30 deg. while the slew ra te is 15 deg j sec. In the second simulation, a step-wise refrence for the yaw of ±30 deg. is generated , while the MPRRSC is on. The results are shown in figure 5.
(19)
These last equations, are in the form where one can immediately apply standard state estimation procedures, i.e., Xa(£
+ 1)
= Aa xa(£) + Ba u(£) + r w w(£),
y(e ) = Ca Xa(£)
+ v(£) ,
(20)
Now using the stationary Kalman filter, an estimation of the plant state and wave disturbances, say xp(e) and dwave(e), at the instant e = k are obtained given u(k) and y(k) . However, the problem of predicting d wave over the prediction horizon N still remains. The optimal n-step-ahead predictor given xd(k) for the system (16) is of the form (Goodwin and Sin, 1984): (21)
Thus, the prediction of the disturbance is given by d(k d(k
In order to measure the roll reduction produced by MPRRSC, the following percent quantity is used, (Fossen, 1994):
1)]
+ + 2)
RR%
(22)
rd(k~N) pl~t
and
x 100,
(23)
in which AP is the standard deviation of the measured roll when the Roll stabilization controller is oJt while ReS is the standard deviation of the measured roll when the Roll stabilization controller is on The results obtained for the simulation shown in figure 4, and as well as for other simulations with different maximum rudder excursions and slew rates are summarized in the following table:
Finally, a schematic block diagram of the control structure is shown in figure 3.
I The subindices p and d refer to state variables respectively.
- RCS = AP AP
disturb~ce -}ilter
49
Finally, even though the results presented are based on digital simulation, the algorithm could be easily implemented in real-time. Indeed, the simulation required less than real time using MATLAB on a 300Mhz Pentium II standard Pc.
6. REFERENCES
Bitmed, R, M. Gevers and V. Wertz (1990). Adap-
tive Optimal Contra: the thinking's man GPC _ 401 ' , ~; 201I "l (I 1\~ i" \ ;C~ 1 i ?
Prentice-Hall, Inc. Blanke, M. and A. Christensen (1993). Rudder roll dumping autopilot robustness to swayyaw-roll couplings. In: Proceedingj DJ 10th
1~' '".nr" !'. ; 1 \ I· 1\1\ \'1 11 \I i rI ) \I ! 1 I '\ \' 1 ' I.; \ ) \: I I i UJ V LI V U
l~,!\I :!\!\I; "I\ fW,
;; a., \' \!
~ -20~
I 1 i.1 ! ' !\ I \ i \! " ! I , ! \1
l! ~ V U V
-40 '
o
/ \
V
\
, I
j
\
SCSS, Otauxl, Canadapp. 93--119.
,
20
4C
60
80
100
12C
140
160
180
Fossen, Thor 1. (1994). Guidance and Control DJ Ocean Marine Vehicles John Wiley and Sons Ltd. New York. Fralnklin, G. E, J. D. Powell and M. Workman (1998) . Digital Control DJ Dynamic Systems 3rd ed .. Addison Wesley. Good win, G. C. and K. S. Sin (1984). Adaptive filtering, Presiction, and Control Prentice-Hall. Laudval, Trygve and Thor 1. Fossen (1998). Rudder roll stabilization for ships subject to input rate saturation using a gain scheduling control law. In: Proceeding DJ IFAC CAMS'98 pp. 121--126. Meadows, E. S. and J. B. Rawlings (1997). Non Linear Process Control Prentice-Hall. Muske, K. Rand J. B. Rawlings (19 93). Model predictive control with linear models. AIChe
200
TIme jsecj
Fig. 5. MPRRSC Performance. a- Roll Angle; bYaw Angle; and c- Rudder Angle. Max. ExcurSIon I Max. Slew Rote 30deg/15deg/sec 30 deg /12 deg /sec 30 deg /8 deg / sec 20 deg /15 deg / sec 20 deg /12 deg /sec 20 deg /8 deg /sec
RR"Io 59 53 41 49 47 42
Table 1. Roll Reduction Obtained for Different Maximum Excursions and Slew Rates.
5. CONCLUSIONS
lournal3~2),
In this paper, a solution to the problem of RRS using MPC was proposed. The results obtained are very encouraging. The rudder constraints are very well managed by the MPC controller. However, some other aspects of the problem still remain the subject of on-going research. For instance, while tuning the controller, a tradeoff was faced when choosing values for the matrices Q and R (in the functional given in (9» since more roll reduction results in more interference in the heading angle. This tradeoff is essentially produced by the single input two outputs (SITO) characteristic of the plant. Therefore, an interesting research question is to obtain a better understanding of the fundamental limitations that the SITO nature of the problem imposes regardless of the control strategy being used . It would also be interesting to link these limitations to the specific solution proposed in this paper.
262--2872.
Roberts, G. N. (1993). A note on the applicability of rudder roll stabilization for ships. In Proceedings DJ ACe, San Francisco, CaliJornia, USA. pp. 2403--2407. van Amerongen, J., P. van der Klugt and H. van Nauta Lemke (1990). Rudder roll stabilization for ships. Automatica26, 679--690.
Appendix A. SHIP DATA In this appendix, the continuous ship model matrices for a constant forward speed of a naval vessel of 9 m/sec are given. The Ship parameters of the model were taken from (Blanke and Christensen, 1993). -0.1795 -0.8404 0.2115 0.9665 0] -0.0159 -0.4492 0.0053 0.0151 0 A = 0.0354 -1.5594 -0.1714 -0.7883 0 o 0 1 0 0 [ o 1 000
In this paper, it has been assumed that the disturbance model is known. An aspect that still needs to be investigated is the inclusion of an adaptive mechanism for tracking the sea conditions for accurate disturbance estimation and prediction. Similarly, additional work is needed to improve the robustness of the algorithm to the assumed ship model.
(A.l)
0'2784] -0.0334
B =
- 0'r94 [
50
(A.2)