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Rudder Roll Stabilization of Ships Subject to Input Rate Saturation Using a Gain Scheduled Control Law
Copyright @ IFAC.Control Applications in Marine Systems, Fukuoka, Japan, 1998
Rudder Roll Stabilization of Ships Subject to Input Rate Saturation Using a Gain Scheduled Control Law Trygve Lauvdal
Thor I. Fossen
ABB Industri AS Haslev. 50 N-0501 Oslo, NORWAY [email protected]
Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7034 Trondheim, NORWAY. [email protected]
Abstract. This paper describes a new gain scheduling algorithm for handel~ng the pro?l~m of hard nonlinearities like input rate and magnitude saturations. The proposed algonthm was ongmally designed for stabilization of integrator chains with input rate saturation. I:Iere it is shown that the gain scheduler also is effective for rudder roll control systems ..The approac.h IS to .reduce the contro~ler gains of a linear controller on-line in an computational effective way. A simulatIOn s~u~y companng the algorithm with the Automatic Gain Controller (AGC) of Van der Klugt (1987) IS mcluded.
Copyright
(j)
19981FAC
Key Words. Input nonlinearities, Gain scheduling, Rudder roll damping
ing machine. It has hard nonlinearities like magnitude and rate saturations, and is therefore not a high bandwidth control device. Moreover, the nonlinearities of the rudder imply that the maximum damping ratio is difficult to obtain over a wide range of working conditions by means of linear control laws. This fact was addressed in Van Amerongen et al. (1990), where two possible solutions where proposed; adaptation and gain scheduling. Gain scheduling is particularly attractive if it is possible to find a computational effective gain scheduling algorithm. This problem was addressed by Van der Klugt (1987), who proposed the Automatic Gain Controller (AGC). Later, the algorithm proved its usefulness in full scale experiments reported in Van Amerongen et al. (1990). In fact, it was claimed by the authors that the AGC was an important reason for the success of the rudder roll stabilizer. Although the AGC has shown its effectiveness in applications, it has some drawbacks; numerical derivation of the commanded input is required and the maximum rate limit is assumed to be constant and perfectly known. The latter can, in most steering machines, not be satisfied since the friction will increase as a function of e.g. rudder angle and ship speed. In this paper we propose to use a the timevarying gain reduction (TGR) algorithm presented in Lauvdal and Murray (1997), where it showed its effectiveness on the Caltech ducted fan experiment. This algorithm is robust to varya-
1. INTRODUCTION
Besides controlling the heading of a ship it is often desirable to reduce the rolling motion induced by waves. The main reasons for introducing a roll damping system are to prevent cargo damage and to increase crew effectiveness and passenger comfort. Criteria for the maximum roll angle are given in Faltinsen (1990), and they suggest that the root mean square of the roll angle should be less than 6 degrees for light manual work and 3 degrees for intellectual work. There are several approaches that can be used to reduce the roll motion, e.g. bilge keels, antirolling tanks, fin stabilizers and rudder roll stabilizers (RRS). For a detailed discussion of the different roll stabilizers, see Fossen (1994) and the references therein. Here we discuss the rudder roll damping approach which is attractive, since existing equipment can be used and thus it is a relative inexpensive solution. The drawbacks of a RRS are the requirement of a high-speed steering machine and the ineffectiveness of the rudder at low ship speed. Successful implementation of full-scale RRS system were first reported in Baitis (1980) and later in e.g. Kallstr6m (1987) and Van Amerongen et al. (1990). A nonlinear stability analysis of ships in coupled sway, roll and yaw is given by Fossen and Lauvdal (1994), whereas the design of a nonlinear RRS is discussed by Lauvdal and Fossen (1996) and Lauvdal and Fossen (1997). In rudder roll stabilization of ships, the main limiting factor is the nonlinear nature of the steer-
111
2.3. Nonlinear Model of the Steering Machine
tions of the maximum rudder rate and does not require numerical derivation. A simulation study is used to compare the two algorithms.
The steering machine or the ship actuator is highly nonlinear, and in RRS the dominating nonlinearities are magnitude and rate saturations. Van Amerongen (1982) propose the following nonlinear state-space model:
2. MODELS
(8)
In this section we describe the ship equations of motion, the steering machine and the wave disturbance model.
The saturation functions sati (.), i = 1,2 are defined as
(9)
2.1. Ship Model
where (31 = 8max and (32 = 8max . The model of the steering machine is shown in Figure 1.
The linear ship model used throughout this paper is taken from Van der Klugt (1987), who propose to define a new state variable, v', given by
dv v'(S) = ( K ) 8(s), 1 + TvS
(1)
nom aUlopilot
where 8 is the rudder angle. Thus , v' represent the sway velocity induced by the rudder motion alone. Using this state variable, we get the following ship dynamics
cI>(S) = hq,(s)[KdpO(S) 'ljJ(s) = h1j;(s)[Kdr 8(s)
+ Kvpv'(s) + wq,(s)], + Kvrv'(s) + w1j;(s)],
(4)
8max E [25, 35J (deg),
h ( ) Kdv '" S = (1 + TrS)S
.---0
f-- -
I
0
S
L--
rudderrale limiter
In the linear region the actuator has a unity time constant and the nonlinear nature is described by the maximum magnitude and rate of the rudder, see Van Amerongen (1982) for details. For most commercial ships, the maximum rudder rate and angle will typically satisfy
2
n S2 + 2(nwns + w~ ,
-
....
Fig. 1. Schematic drawing of the rudder control loop (Van Amerongen 1982).
(2) (3)
tion and W
.....
':t max
:'
rudder limirer
where 'ljJ, cl> are the yaw and roll angle, respectively, w1j;, wq, are colored noise describing the wave mo-
hq,(s) =
't max
(5)
8max
E [2,7J (degjs). (10)
For efficient roll reduction, larger actuation limits are required and recently steering machines with rudder speeds up to 15- 20 (degjs) have been designed.
For a more detailed explanation of the different parameters, see Van der Klugt (1987).
2.2. Wave Model 3. LINEAR CONTROL DESIGN
The ship motion is determined by the rudder and environmental disturbances. For rudder roll damping, only high-frequency roll motion can be reduced. Consequently, only 1st-order wave disturbances are included in the simulations. These disturbances can e.g. be simulated using a 2nd-order linear approximation of the PiersonMoskowitz spectral density function, see Fossen (1994). Thus
hi(s)
=
s2
2(00"i
2'
+ 2(owos + Wo
i
= 1,2,
In a rudder roll stabilizing system, it is desirable to increase the natural frequency and damping ratio in roll in addition to controlling the heading. Hence, it is appropriate to rewrite the commanded rudder according to (11)
where 8roll is used to roll damping and 8yaw maintains the desired yaw angle. The control structure is shown in Figure 2.
(6)
where O"i, i = 1,2, are constants describing the wave intensity, (0 is the relative damping coefficient and Wo is the dominating wave frequency. Then, the disturbances, w'" and wq" are given by
The linear control law design is not considered in this paper since we are concerned with the performance improvements obtained by introducing a gain scheduling algorithm. Several design methods are available using linear theory, see e.g. Fossen (1994).
where W1(S),W2(S) are Gaussian white noise. 112
"'._-_ ..._... __.__ ... _._-_... _...•._-_...__.__... _...- "'-"'--"--"-".--.---.---.
1--·...,nll.....,..
the output of a memory function . Moreover, (13)
0"",
where A is a forgetting factor, typically in the range 0.95 ::; A < 1. The time derivative of the autopilot output is computed by numerical derivation, and due to the sensitivity of this operation an estimate should be used. The AGC can be explained as follows; if 18(tdl is larger than 8max the gain A(td is instantaneously decreased. When 18(tdl has decreased, the memory function brings the gain A(tk) slowly back to its desired value. The slow increase of A( tk) reduces the phase lag in the system. For a more detailed description, see Van der Klugt (1987).
Fig. 2. Figure showing inner-loop roll controller (fast) and outer-loop yaw controller (slow).
4. GAIN SCHEDULING ALGORITHMS Using a linear controller, considerable damping is obtained if the rudder stays within the linear area. However, changing environments will make this assumption impossible to satisfy unless the controller gains are very small, which again will lead to reduced damping. Thus, it is desirable to reduce the gains only when necessary, and in the following two gain scheduling algorithms are presented.
4.2. Time-varying Gain Reduction (TGR)
The AGC presented above suggest an instantaneous reduction of the controller gains, and that all gains are multiplied with the same gain, A(td. In Lauvdal and Murray (1997) an algorithm for stabilization of an integrator chain in the presence of rate saturation is derived. This is based on individual reduction of the controller gains and time-dependent gain reduction. The possibility for individual gain reduction is not used in this paper for comparison reasons and thus the commanded signal to the ship actuator is simply given by (12). In the definition of the gain scheduling algorithm, a discontinuous function of time, h(t), given by
4.1. Automatic Gain Controller (AGC)
The Automatic Gain Controller (AGC) proposed by Van der Klugt (1987) is shown in Figure 3. The main idea is to calculate a gain A(td, to sleenng machine
8 ::;~, h(t) = 1 if 8 > ~, (14) where 8 = 16 - 6e l and 6, 6e is the actual and comh(t) = 0 if
manded inputs, respectively. To utilize the properties of h(t), we define hM(t) to be the maximum value of h(t) , backwards in time, i.e.
Fig. 3. The automatic gain controller (AGC), (Van der Klugt, 1987) and (Van Amerongen et al., 1990).
(15) Then, the scheduling variable is reduced by the following algorithm
where tk is a discrete time instance, such that the phase lag introduced by input rate saturation is reduced if the commanded input signal is multiplied with this gain. The new commanded input signal is
6
~T
1'(t) = max(hM(t), ~T)'
(16)
where ~T > 0 and hM(t) is given by (15). In the simulations A( td is taken as the discrete time version of (16), that is
(12) where 6e is the output of the autopilot and A(td = 8max /y(td· The signal y(tk) is defined as the maximum of three signals; (1) the maximum rudder rate, (2) the absolute value of the time derivative of the autopilot output and (3)
(17) In order to bring A(td back to unity value when approriate, we modify the definition of hM(t), 113
poles remain in the left half plane for all A E (0, 1J. It should be noted that this does not guarantee stability, since the system is non-autonomous. It is, however, a neccesary condition for stability.
yielding: ~
hM(td = h(td, { >..hM(tk-d, where>.. < l. The TGR works as follows ; if the commanded input signal differs from the real input, and it does so for a time larger than D..T, the control gains are reduced. However, the gain is not reduced instantaneously, but as a function of time. When the gains are small enough, A( tk) is slowly brought back to unit value. For further details, see Lauvdal (1998).
Root locus
o.• ,----~--,------=:..:::;..~=..:;~-~-~-__, 0 .6
0 .4
0 .2
E ......
0
-0.2
-0.4
-0.6
5. COMPARISON STUDY
-o .• L--~-~-~-~~-~-~---' -0.7
In this section we compare the two algorithms described above. The parameters for the ship model (1)- (5) are taken from Van der Klugt (1987) and are as follows :
Kdp Kdv Kdr K vp (n
=
0.0014U2, Kvr = -O.OlU, Tv = 0.0027U, Tr 0.21U, Wn = 0.064 + 0.0038U,
(Ji
-0.4
Re
-0.3
-0.2
-0. 1
-0.46,
78/U, 13/U, 0.63,
= 10, i = 1,2,
In order to compare the two algorithms, the damping ratio is determined by the formula suggested by Oda et al. (1992): . ReductlOn (%)
bmax = 25 (deg).
(20)
The wave frequency Wo in (6) and the rate limit in (8) will be changed in the case studies. The linear controller is taken from Example 6.6, page 302 in Fossen (1994) and is given by
8max
b = -7.9r - 9.7'l/J - 17.2p - 2.84>,
AP -RRCS AP ,
>.. = 0.99, D..T = 0.5 , D.. = O.
(23)
(24)
An illustrative simulation is shown in Figure 58, when 8max = 15 (deg/s), U = 10 (m/s) and Wo = 0.7 (rad/s). The heading controller and RRS with TGR is active the first 500 (s) and the RRS is turned off after 500 (s) . Figure 5 show roll angle and it is seen that considerable roll damping is obtained. Using (23) a damping ratio of approximately 70 % is the result of the RRS. Figure 6 show the rudder action used to obtain roll damping, and Figure 7 show a typical behavior the scaling factor. It is redused when necessary and slowly brought back to unity. Finally, in Figure 8 the yaw angle is plotted, and it is seen that the RRS increase the error in heading.
(21)
where it is assumed that 'l/Jdes = O. For simlicity, this controller is used in all of the simulations, and it should be noted that better damping ratio could be obtained by e.g. using speed scaling. However, by fixing the linear controller gains, we can investigate robutness of the gain scheduling algorithms under different conditions. Altough the TGR allows for individual gain reduction, the same gain reduction is used for both algorithms for comparison reasons. Hence, the gain scheduled control law is
+ 17.2p+ 2.84>1
= 100 x
where RRCS and AP are the standard deviation of roll rate with and without the roll damping system. All of the simulations where done over a 10 minute time periode and with the following parameters for the AGC and TGR:
(19)
whereas the steering machine is given by (8) with
b = -A(td[7.9r + 9.7'l/J
-0.5
Fig. 4. Root locus for A E (0,1], where '0' and 'x' are used to indicate the open and closed loop poles, respectively.
where U > 0 is the cruise speed. The parameters in the 1st-order wave disturbance model (6) are chosen as (0 = 0.1,
-0.6
(22)
and the poles of the closed loop system as a function of A are shown in Figure 4. It is seen that the 114
4> (deg)
1/J (deg)
3.----------------------------
20r-----------~----------~
2
10
o -1
-10 -2 _3L---~--------~----------~
-20~------------~----------~
o
.50q )
time
~s
o
1000
Fig. 5. Plot of the ship roll angle.
Parameters U Wo 10 8 0.7 20 8 0.7 15 6 0.7 15 8 0.7 15 10 0.7 15 8 0.5 15 8 0.9
bmax
30r-------------------------~
10
o -10 -20 500
Reductions (%) AGC TGR LIN 32.2 33.0 -13.4 58.8 41.6 53.8 25.2 4.5 28.3 15.7 45.5 48.6 47.6 70.6 70.4 63 .8 59.5 62.9 28.1 32.1 8.2
Table 1: Simulation results with constant and known rate limits. It is seen that the AGC and TGR gives approximatly the same damping ratio. Comparing with the case of no gain scheduling, (LIN), shows that the AGC and TGR are superior.
-30~----------~-----------J
time (s)
1000
Fig. 8. Plot of ship heading angle.
c5 roll (deg)
o
.50Q) time (s
1000
Fig. 6. Plot of Droll-contribution to the rudder angle, 15 = Droll + Dyaw (the RRS is turned off after 500 (s)).
5.2. Constant and unknown rate limits The AGC uses the rate limit in the calculation of the gain A(td. Thus, it is of interest to see the importance of an error in the rate limit used in the algorithm . The physical rate limit is assumed to be 15 (deg/s) and in the AGC simulations an error off ±30 (%) was used , see Table 2.
5.1. Constant and known rate limits First , it is assumed that the rate limit of the steering machine is constant and perfectly known . To investigate the robustness to changing conditions, simulations were done at different rate limits bmax , dominating wave frequency Wo and ship speed U. The results are summarized in Table 1.
Parameters
bmax 10.5 15 19.5
1.2.---------------------------,
U 8 8 8
Wo
0.7 0.7 0.7
Reductions (%) AGC 44.2 45.5 37.9
TGR 48.6 48.6 48.6
LIN 15.7 15.7 15.7
Table 2: Simulation results with constant and unknown rate limits. The simulation results show that an error in the rate limit gives a significant reduction in performance for the AGC, whereas the the TGR is robust to parameter changes.
0 .8 0.6 0.4
0.2
5.3. Varying rate limit,
O~------------~----------~
o
500
time (s)
1000
bmax
The rate limit in a steering machine will change with ship speed and rudder angle. To see the robustness to varying rate limits, consider the fol-
Fig. 7. Plot of the scaling factor , A(h). 115
ference on Marine Craft Maneuvering and Control (MCMC '94), Southampton, UK. pp. 113124. Kiillstrom, C. G. (1987) . Improved Operational Effectiveness of Naval Ships by Rudder Roll Stabilization. In: NAVAL '87, Asian Pacific Naval Exhibition and Conference. Singapore. Lauvdal, T. (1998). Stabilization of Linear Systems with Input Magnitude and Rate Saturations. PhD thesis. Norwegian University of Science and Technology. Lauvdal, T. and R. M. Murray (1997) . A TimeVarying Controller for Stabilization in the Presence of Rate Saturation. In: Proceedings of the AIAA Conference on Guidance, Navigation and Control. New Orleans, USA. Lauvdal, T . and T. I. Fossen (1996). Nonlinear Non-Minimum Phase Rudder-Roll Damping System for Ships Using Sliding Mode Control. Presented at the IFAC World Congress (IFAC'96), San Francisco, 1996. Lauvdal, T. and T . I. Fossen (1997). Rudder-roll damping system for a nonlinear ship model using sliding mode control. In: Proceedings of European Control Conference (ECC'97) . Oda et al., H. (1992) . Rudder Roll Stabilization Control System through Multivariable Auto Regressive Model. In: Proceedings of the 3rd IFAC Workshop on Control Applications in Marine Systems (CAMS'92). Genova, Italy. pp. 113- 127. Van Amerongen, J . (1982). Adaptive Steering of Ships - A Model Reference Approach to Improved Maneuvering and Economical Course Keeping. PhD thesis. Delft University of Technology, The Netherlands. Van Amerongen, J ., P. G. M. Van der Klugt and H. R. Van Nauta Lempke (1990). Rudder Roll Stabilization for Ships. Automatica AUT26(4) , 679- 690. Van der Klugt, P. G. M. (1987). Rudder Roll Stabilization. PhD thesis. Delft University of Technology, The Netherlands.
lowing model
8max (8) = { . 8m (ax, / if) 181 ~ 8max 8
-1 5 ,
1, if 181 > 1.
(25)
Thus, the rudder rate is decreasing with increasing rudder angle magnitude. The simulation results are given in Table 3. Parameters
8max 15 15 15
U 6 8 10
Wo
0.7 0.7 0.7
Reductions (%) AGC 10.1 22.0 42.1
TGR 21.3 38.1 57.0
LIN -11.1 -13.8 -8.8
Table 3: Simulation results with rate limits given by formula (25) . Again it is seen that the TGR perform better than the AGC under parameter uncertainty. Notice that the linear controller (no gain scheduling) is increasing the roll motion. 5.4. Comments It should be noted that the TGR has more flexibility than illustrated in the above simulations. Although it is highly effective with the fixed parameters, better damping are obtained by carefull tuning of the parameters and individual gain reduction .
6. CONCLUSION In this paper a new gain scheduling algorithm that reduces the negative effects of limited actuator rate in rudder roll damping systems have been presented. Simulation results indicate that the resulting time-varying controller is robust to changing working conditions and varying rate limits. Comparison with the automatic gain Controller of Van der Klugt (1987) show that the proposed gain scheduling algorithm is more robust to steering machine uncertainties.
7. REFERENCES Baitis, A. E. (1980) . The Development and Evaluation of a Rudder Roll Stabilization System for the WHEC Hamiltonian Class. Technical Report DTNSRDC. Naval Ship Research and Development Cent er. Bethesda, Md. Faltinsen, O. M. (1990). Sea Loads on Ships and Offshore Structures. Cambridge University Press. Fossen, T . I. (1994). Guidance and Control of Ocean Vehicles. John Wiley and Sons Ltd. Fossen, T. I. and T . Lauvdal (1994) . Nonlinear Stability Analysis of Ship Autopilots in Sway, Roll and Yaw. In: Proceedings of the 3rd Con116
Rudder Roll Stabilization of Ships Subject to Input Rate Saturation Using a Gain Scheduled Control Law Trygve Lauvdal
Thor I. Fossen
ABB Industri AS Haslev. 50 N-0501 Oslo, NORWAY [email protected]
Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7034 Trondheim, NORWAY. [email protected]
Abstract. This paper describes a new gain scheduling algorithm for handel~ng the pro?l~m of hard nonlinearities like input rate and magnitude saturations. The proposed algonthm was ongmally designed for stabilization of integrator chains with input rate saturation. I:Iere it is shown that the gain scheduler also is effective for rudder roll control systems ..The approac.h IS to .reduce the contro~ler gains of a linear controller on-line in an computational effective way. A simulatIOn s~u~y companng the algorithm with the Automatic Gain Controller (AGC) of Van der Klugt (1987) IS mcluded.
Copyright
(j)
19981FAC
Key Words. Input nonlinearities, Gain scheduling, Rudder roll damping
ing machine. It has hard nonlinearities like magnitude and rate saturations, and is therefore not a high bandwidth control device. Moreover, the nonlinearities of the rudder imply that the maximum damping ratio is difficult to obtain over a wide range of working conditions by means of linear control laws. This fact was addressed in Van Amerongen et al. (1990), where two possible solutions where proposed; adaptation and gain scheduling. Gain scheduling is particularly attractive if it is possible to find a computational effective gain scheduling algorithm. This problem was addressed by Van der Klugt (1987), who proposed the Automatic Gain Controller (AGC). Later, the algorithm proved its usefulness in full scale experiments reported in Van Amerongen et al. (1990). In fact, it was claimed by the authors that the AGC was an important reason for the success of the rudder roll stabilizer. Although the AGC has shown its effectiveness in applications, it has some drawbacks; numerical derivation of the commanded input is required and the maximum rate limit is assumed to be constant and perfectly known. The latter can, in most steering machines, not be satisfied since the friction will increase as a function of e.g. rudder angle and ship speed. In this paper we propose to use a the timevarying gain reduction (TGR) algorithm presented in Lauvdal and Murray (1997), where it showed its effectiveness on the Caltech ducted fan experiment. This algorithm is robust to varya-
1. INTRODUCTION
Besides controlling the heading of a ship it is often desirable to reduce the rolling motion induced by waves. The main reasons for introducing a roll damping system are to prevent cargo damage and to increase crew effectiveness and passenger comfort. Criteria for the maximum roll angle are given in Faltinsen (1990), and they suggest that the root mean square of the roll angle should be less than 6 degrees for light manual work and 3 degrees for intellectual work. There are several approaches that can be used to reduce the roll motion, e.g. bilge keels, antirolling tanks, fin stabilizers and rudder roll stabilizers (RRS). For a detailed discussion of the different roll stabilizers, see Fossen (1994) and the references therein. Here we discuss the rudder roll damping approach which is attractive, since existing equipment can be used and thus it is a relative inexpensive solution. The drawbacks of a RRS are the requirement of a high-speed steering machine and the ineffectiveness of the rudder at low ship speed. Successful implementation of full-scale RRS system were first reported in Baitis (1980) and later in e.g. Kallstr6m (1987) and Van Amerongen et al. (1990). A nonlinear stability analysis of ships in coupled sway, roll and yaw is given by Fossen and Lauvdal (1994), whereas the design of a nonlinear RRS is discussed by Lauvdal and Fossen (1996) and Lauvdal and Fossen (1997). In rudder roll stabilization of ships, the main limiting factor is the nonlinear nature of the steer-
111
2.3. Nonlinear Model of the Steering Machine
tions of the maximum rudder rate and does not require numerical derivation. A simulation study is used to compare the two algorithms.
The steering machine or the ship actuator is highly nonlinear, and in RRS the dominating nonlinearities are magnitude and rate saturations. Van Amerongen (1982) propose the following nonlinear state-space model:
2. MODELS
(8)
In this section we describe the ship equations of motion, the steering machine and the wave disturbance model.
The saturation functions sati (.), i = 1,2 are defined as
(9)
2.1. Ship Model
where (31 = 8max and (32 = 8max . The model of the steering machine is shown in Figure 1.
The linear ship model used throughout this paper is taken from Van der Klugt (1987), who propose to define a new state variable, v', given by
dv v'(S) = ( K ) 8(s), 1 + TvS
(1)
nom aUlopilot
where 8 is the rudder angle. Thus , v' represent the sway velocity induced by the rudder motion alone. Using this state variable, we get the following ship dynamics
cI>(S) = hq,(s)[KdpO(S) 'ljJ(s) = h1j;(s)[Kdr 8(s)
+ Kvpv'(s) + wq,(s)], + Kvrv'(s) + w1j;(s)],
(4)
8max E [25, 35J (deg),
h ( ) Kdv '" S = (1 + TrS)S
.---0
f-- -
I
0
S
L--
rudderrale limiter
In the linear region the actuator has a unity time constant and the nonlinear nature is described by the maximum magnitude and rate of the rudder, see Van Amerongen (1982) for details. For most commercial ships, the maximum rudder rate and angle will typically satisfy
2
n S2 + 2(nwns + w~ ,
-
....
Fig. 1. Schematic drawing of the rudder control loop (Van Amerongen 1982).
(2) (3)
tion and W
.....
':t max
:'
rudder limirer
where 'ljJ, cl> are the yaw and roll angle, respectively, w1j;, wq, are colored noise describing the wave mo-
hq,(s) =
't max
(5)
8max
E [2,7J (degjs). (10)
For efficient roll reduction, larger actuation limits are required and recently steering machines with rudder speeds up to 15- 20 (degjs) have been designed.
For a more detailed explanation of the different parameters, see Van der Klugt (1987).
2.2. Wave Model 3. LINEAR CONTROL DESIGN
The ship motion is determined by the rudder and environmental disturbances. For rudder roll damping, only high-frequency roll motion can be reduced. Consequently, only 1st-order wave disturbances are included in the simulations. These disturbances can e.g. be simulated using a 2nd-order linear approximation of the PiersonMoskowitz spectral density function, see Fossen (1994). Thus
hi(s)
=
s2
2(00"i
2'
+ 2(owos + Wo
i
= 1,2,
In a rudder roll stabilizing system, it is desirable to increase the natural frequency and damping ratio in roll in addition to controlling the heading. Hence, it is appropriate to rewrite the commanded rudder according to (11)
where 8roll is used to roll damping and 8yaw maintains the desired yaw angle. The control structure is shown in Figure 2.
(6)
where O"i, i = 1,2, are constants describing the wave intensity, (0 is the relative damping coefficient and Wo is the dominating wave frequency. Then, the disturbances, w'" and wq" are given by
The linear control law design is not considered in this paper since we are concerned with the performance improvements obtained by introducing a gain scheduling algorithm. Several design methods are available using linear theory, see e.g. Fossen (1994).
where W1(S),W2(S) are Gaussian white noise. 112
"'._-_ ..._... __.__ ... _._-_... _...•._-_...__.__... _...- "'-"'--"--"-".--.---.---.
1--·...,nll.....,..
the output of a memory function . Moreover, (13)
0"",
where A is a forgetting factor, typically in the range 0.95 ::; A < 1. The time derivative of the autopilot output is computed by numerical derivation, and due to the sensitivity of this operation an estimate should be used. The AGC can be explained as follows; if 18(tdl is larger than 8max the gain A(td is instantaneously decreased. When 18(tdl has decreased, the memory function brings the gain A(tk) slowly back to its desired value. The slow increase of A( tk) reduces the phase lag in the system. For a more detailed description, see Van der Klugt (1987).
Fig. 2. Figure showing inner-loop roll controller (fast) and outer-loop yaw controller (slow).
4. GAIN SCHEDULING ALGORITHMS Using a linear controller, considerable damping is obtained if the rudder stays within the linear area. However, changing environments will make this assumption impossible to satisfy unless the controller gains are very small, which again will lead to reduced damping. Thus, it is desirable to reduce the gains only when necessary, and in the following two gain scheduling algorithms are presented.
4.2. Time-varying Gain Reduction (TGR)
The AGC presented above suggest an instantaneous reduction of the controller gains, and that all gains are multiplied with the same gain, A(td. In Lauvdal and Murray (1997) an algorithm for stabilization of an integrator chain in the presence of rate saturation is derived. This is based on individual reduction of the controller gains and time-dependent gain reduction. The possibility for individual gain reduction is not used in this paper for comparison reasons and thus the commanded signal to the ship actuator is simply given by (12). In the definition of the gain scheduling algorithm, a discontinuous function of time, h(t), given by
4.1. Automatic Gain Controller (AGC)
The Automatic Gain Controller (AGC) proposed by Van der Klugt (1987) is shown in Figure 3. The main idea is to calculate a gain A(td, to sleenng machine
8 ::;~, h(t) = 1 if 8 > ~, (14) where 8 = 16 - 6e l and 6, 6e is the actual and comh(t) = 0 if
manded inputs, respectively. To utilize the properties of h(t), we define hM(t) to be the maximum value of h(t) , backwards in time, i.e.
Fig. 3. The automatic gain controller (AGC), (Van der Klugt, 1987) and (Van Amerongen et al., 1990).
(15) Then, the scheduling variable is reduced by the following algorithm
where tk is a discrete time instance, such that the phase lag introduced by input rate saturation is reduced if the commanded input signal is multiplied with this gain. The new commanded input signal is
6
~T
1'(t) = max(hM(t), ~T)'
(16)
where ~T > 0 and hM(t) is given by (15). In the simulations A( td is taken as the discrete time version of (16), that is
(12) where 6e is the output of the autopilot and A(td = 8max /y(td· The signal y(tk) is defined as the maximum of three signals; (1) the maximum rudder rate, (2) the absolute value of the time derivative of the autopilot output and (3)
(17) In order to bring A(td back to unity value when approriate, we modify the definition of hM(t), 113
poles remain in the left half plane for all A E (0, 1J. It should be noted that this does not guarantee stability, since the system is non-autonomous. It is, however, a neccesary condition for stability.
yielding: ~
hM(td = h(td, { >..hM(tk-d, where>.. < l. The TGR works as follows ; if the commanded input signal differs from the real input, and it does so for a time larger than D..T, the control gains are reduced. However, the gain is not reduced instantaneously, but as a function of time. When the gains are small enough, A( tk) is slowly brought back to unit value. For further details, see Lauvdal (1998).
Root locus
o.• ,----~--,------=:..:::;..~=..:;~-~-~-__, 0 .6
0 .4
0 .2
E ......
0
-0.2
-0.4
-0.6
5. COMPARISON STUDY
-o .• L--~-~-~-~~-~-~---' -0.7
In this section we compare the two algorithms described above. The parameters for the ship model (1)- (5) are taken from Van der Klugt (1987) and are as follows :
Kdp Kdv Kdr K vp (n
=
0.0014U2, Kvr = -O.OlU, Tv = 0.0027U, Tr 0.21U, Wn = 0.064 + 0.0038U,
(Ji
-0.4
Re
-0.3
-0.2
-0. 1
-0.46,
78/U, 13/U, 0.63,
= 10, i = 1,2,
In order to compare the two algorithms, the damping ratio is determined by the formula suggested by Oda et al. (1992): . ReductlOn (%)
bmax = 25 (deg).
(20)
The wave frequency Wo in (6) and the rate limit in (8) will be changed in the case studies. The linear controller is taken from Example 6.6, page 302 in Fossen (1994) and is given by
8max
b = -7.9r - 9.7'l/J - 17.2p - 2.84>,
AP -RRCS AP ,
>.. = 0.99, D..T = 0.5 , D.. = O.
(23)
(24)
An illustrative simulation is shown in Figure 58, when 8max = 15 (deg/s), U = 10 (m/s) and Wo = 0.7 (rad/s). The heading controller and RRS with TGR is active the first 500 (s) and the RRS is turned off after 500 (s) . Figure 5 show roll angle and it is seen that considerable roll damping is obtained. Using (23) a damping ratio of approximately 70 % is the result of the RRS. Figure 6 show the rudder action used to obtain roll damping, and Figure 7 show a typical behavior the scaling factor. It is redused when necessary and slowly brought back to unity. Finally, in Figure 8 the yaw angle is plotted, and it is seen that the RRS increase the error in heading.
(21)
where it is assumed that 'l/Jdes = O. For simlicity, this controller is used in all of the simulations, and it should be noted that better damping ratio could be obtained by e.g. using speed scaling. However, by fixing the linear controller gains, we can investigate robutness of the gain scheduling algorithms under different conditions. Altough the TGR allows for individual gain reduction, the same gain reduction is used for both algorithms for comparison reasons. Hence, the gain scheduled control law is
+ 17.2p+ 2.84>1
= 100 x
where RRCS and AP are the standard deviation of roll rate with and without the roll damping system. All of the simulations where done over a 10 minute time periode and with the following parameters for the AGC and TGR:
(19)
whereas the steering machine is given by (8) with
b = -A(td[7.9r + 9.7'l/J
-0.5
Fig. 4. Root locus for A E (0,1], where '0' and 'x' are used to indicate the open and closed loop poles, respectively.
where U > 0 is the cruise speed. The parameters in the 1st-order wave disturbance model (6) are chosen as (0 = 0.1,
-0.6
(22)
and the poles of the closed loop system as a function of A are shown in Figure 4. It is seen that the 114
4> (deg)
1/J (deg)
3.----------------------------
20r-----------~----------~
2
10
o -1
-10 -2 _3L---~--------~----------~
-20~------------~----------~
o
.50q )
time
~s
o
1000
Fig. 5. Plot of the ship roll angle.
Parameters U Wo 10 8 0.7 20 8 0.7 15 6 0.7 15 8 0.7 15 10 0.7 15 8 0.5 15 8 0.9
bmax
30r-------------------------~
10
o -10 -20 500
Reductions (%) AGC TGR LIN 32.2 33.0 -13.4 58.8 41.6 53.8 25.2 4.5 28.3 15.7 45.5 48.6 47.6 70.6 70.4 63 .8 59.5 62.9 28.1 32.1 8.2
Table 1: Simulation results with constant and known rate limits. It is seen that the AGC and TGR gives approximatly the same damping ratio. Comparing with the case of no gain scheduling, (LIN), shows that the AGC and TGR are superior.
-30~----------~-----------J
time (s)
1000
Fig. 8. Plot of ship heading angle.
c5 roll (deg)
o
.50Q) time (s
1000
Fig. 6. Plot of Droll-contribution to the rudder angle, 15 = Droll + Dyaw (the RRS is turned off after 500 (s)).
5.2. Constant and unknown rate limits The AGC uses the rate limit in the calculation of the gain A(td. Thus, it is of interest to see the importance of an error in the rate limit used in the algorithm . The physical rate limit is assumed to be 15 (deg/s) and in the AGC simulations an error off ±30 (%) was used , see Table 2.
5.1. Constant and known rate limits First , it is assumed that the rate limit of the steering machine is constant and perfectly known . To investigate the robustness to changing conditions, simulations were done at different rate limits bmax , dominating wave frequency Wo and ship speed U. The results are summarized in Table 1.
Parameters
bmax 10.5 15 19.5
1.2.---------------------------,
U 8 8 8
Wo
0.7 0.7 0.7
Reductions (%) AGC 44.2 45.5 37.9
TGR 48.6 48.6 48.6
LIN 15.7 15.7 15.7
Table 2: Simulation results with constant and unknown rate limits. The simulation results show that an error in the rate limit gives a significant reduction in performance for the AGC, whereas the the TGR is robust to parameter changes.
0 .8 0.6 0.4
0.2
5.3. Varying rate limit,
O~------------~----------~
o
500
time (s)
1000
bmax
The rate limit in a steering machine will change with ship speed and rudder angle. To see the robustness to varying rate limits, consider the fol-
Fig. 7. Plot of the scaling factor , A(h). 115
ference on Marine Craft Maneuvering and Control (MCMC '94), Southampton, UK. pp. 113124. Kiillstrom, C. G. (1987) . Improved Operational Effectiveness of Naval Ships by Rudder Roll Stabilization. In: NAVAL '87, Asian Pacific Naval Exhibition and Conference. Singapore. Lauvdal, T. (1998). Stabilization of Linear Systems with Input Magnitude and Rate Saturations. PhD thesis. Norwegian University of Science and Technology. Lauvdal, T. and R. M. Murray (1997) . A TimeVarying Controller for Stabilization in the Presence of Rate Saturation. In: Proceedings of the AIAA Conference on Guidance, Navigation and Control. New Orleans, USA. Lauvdal, T . and T. I. Fossen (1996). Nonlinear Non-Minimum Phase Rudder-Roll Damping System for Ships Using Sliding Mode Control. Presented at the IFAC World Congress (IFAC'96), San Francisco, 1996. Lauvdal, T. and T . I. Fossen (1997). Rudder-roll damping system for a nonlinear ship model using sliding mode control. In: Proceedings of European Control Conference (ECC'97) . Oda et al., H. (1992) . Rudder Roll Stabilization Control System through Multivariable Auto Regressive Model. In: Proceedings of the 3rd IFAC Workshop on Control Applications in Marine Systems (CAMS'92). Genova, Italy. pp. 113- 127. Van Amerongen, J . (1982). Adaptive Steering of Ships - A Model Reference Approach to Improved Maneuvering and Economical Course Keeping. PhD thesis. Delft University of Technology, The Netherlands. Van Amerongen, J ., P. G. M. Van der Klugt and H. R. Van Nauta Lempke (1990). Rudder Roll Stabilization for Ships. Automatica AUT26(4) , 679- 690. Van der Klugt, P. G. M. (1987). Rudder Roll Stabilization. PhD thesis. Delft University of Technology, The Netherlands.
lowing model
8max (8) = { . 8m (ax, / if) 181 ~ 8max 8
-1 5 ,
1, if 181 > 1.
(25)
Thus, the rudder rate is decreasing with increasing rudder angle magnitude. The simulation results are given in Table 3. Parameters
8max 15 15 15
U 6 8 10
Wo
0.7 0.7 0.7
Reductions (%) AGC 10.1 22.0 42.1
TGR 21.3 38.1 57.0
LIN -11.1 -13.8 -8.8
Table 3: Simulation results with rate limits given by formula (25) . Again it is seen that the TGR perform better than the AGC under parameter uncertainty. Notice that the linear controller (no gain scheduling) is increasing the roll motion. 5.4. Comments It should be noted that the TGR has more flexibility than illustrated in the above simulations. Although it is highly effective with the fixed parameters, better damping are obtained by carefull tuning of the parameters and individual gain reduction .
6. CONCLUSION In this paper a new gain scheduling algorithm that reduces the negative effects of limited actuator rate in rudder roll damping systems have been presented. Simulation results indicate that the resulting time-varying controller is robust to changing working conditions and varying rate limits. Comparison with the automatic gain Controller of Van der Klugt (1987) show that the proposed gain scheduling algorithm is more robust to steering machine uncertainties.
7. REFERENCES Baitis, A. E. (1980) . The Development and Evaluation of a Rudder Roll Stabilization System for the WHEC Hamiltonian Class. Technical Report DTNSRDC. Naval Ship Research and Development Cent er. Bethesda, Md. Faltinsen, O. M. (1990). Sea Loads on Ships and Offshore Structures. Cambridge University Press. Fossen, T . I. (1994). Guidance and Control of Ocean Vehicles. John Wiley and Sons Ltd. Fossen, T. I. and T . Lauvdal (1994) . Nonlinear Stability Analysis of Ship Autopilots in Sway, Roll and Yaw. In: Proceedings of the 3rd Con116