- Email: [email protected]
A Fuzzy Controller for Integrated Ship Motion Control
14th World Congress ofIFAC
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT ...
Q-8c-02-3
Copyright © 1999 IFAC
14th Triennial World Congress, Beijing, P.R. China
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONTROL
G,N, Roberts\ A. Zirilli l , A. Tiano 2 and R Sutto03 I Mechatronics Research Centre, University afWales College, Newport Allt-yr-yn Campus, P.OBax 180, Ne .....port, NP9 5XR, UK. ([email protected])
lDepartment ofTriformation and Systems, University ofPavia, Pavia, Italy Via Ferrata 1,1-27JOO Pavia, Italy. Also: Institute ofShip Automation C.NR Via de Marini 6, [-16149 Genova, Italy. 3
Department of Mechanical Engineering, University ofPlymouth, Drake's Circus, Plymouth, PL48AA, UK.
Abstract: This paper describes the development of fuzzy controller for integrated control of the coupled yaw-roll motion of a container ship. The basic fuzzy rules used for designing an integrated multivariable rudder roll damping and yaw autopilot are discussed, A random search learning mechanism for automatically tuning the fuzzy controller parameters is presented. The efficiency and robustness of the controller are evaluated by a number of simulation tests conducted at different sea states. The results are used to evaluate the performance of the fuzzy controller in comparison with a conventional LQ design. Copyright © 19991FAC Keywords: Ship Motion Control, Integrated Control, Fuzzy Logic, Fuzzyneural control
1. INTRODUCTION
Many studies have been conducted in the recent years in order to design and put into operation a new generation of ship motion control systems, such as autopilots, stabilizers and dynamic posltlOning systems, capable to efficiently and safely carry out demanding navigation tasks in a wide range of environmental conditions. One of the most interesting research areas concerns Rudder Roll Damping (RRD) or Rudder Roll Stabilization (RRS). The philosophy behind this approach to ship roll stabilization is that rudder may be used as the only actuator to reduce roll motion and at the same time to maintain the ship's course. This result can be in principle achieved by compensating the disturbing effects induced by sea waves through a lowfrequency rudder action for the yaw and a highfrequency rudder action for the roll. The main advantage of such an approach, with respect to other
ship stabilization systems based on fins or on compensating tanks, is given by the reduction of investment costs as well as by the expected safety improvement during navigation in rough sea conditions. Among the papers dedicated to RRD design it is possible to mention a number of contributions where different solutions to the problem have been proposed. Such solutions include a PID controller (Amerongen et al., 1987), a frequency domain based autopilot (Roberts, 1992), a LQG controller (Kallstrom and Schultz, 1990), an artificial neural network controller (Tiano et al., 1994) and an H~ controller (Yang and Blanke, 1997). Tt should be noted, however, that most of such results have been obtained for linearized and weakly coupled ship models generally characterised by a simplified modelling of sea waves induced motion. This simplification is motivated by the need to avoid as far as possible the complications deJ"iving from the
8279
Copyright 1999 IF AC
ISBN: 008 0432484
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT ...
use of a completely coupled non-linear ship model. On the other hand, as evidenced by Son and Nomoto (1982) and Blanke and Jenson (1997), the dynamic behaviour of several ships characterized by a low metacentric height cannot be adequately described in terms of the simplified models of the linear and decoupled type, which are generally used in the hydrodynamic and naval architecture literature (Lewis, 1988)_ It has in fact been clearly demonstrated that for such vessels, including containerships, RORO ships, high-speed ferries, naval vessels and fishing boats, non-linear coupled surge-yaw-sway-roll can supply a better knowledge about the precise interaction between roll and other motions. In this paper a neurofuzzy integrated controller for a container ship will be presented_ The work is based on a design suggested in Sutton et al. (1997) for the yaw motion only. After a concise presentation of the ship mathematical model and of the fuzzy controller, some simulation results will be presented and discussed_
2. MATHEMATICAL MODELLING The mathematical model of a container ship used in this study is described in detail in Son and Nomoto (1982) and Blanke and Jenson (I997). It is herein considered a stochastic extension of such models capable to describe the ship response in irregular sea waves, and is expressed by the following non-linear equations (see Fig 1):
m(it - vr - xur2 + zr;pr)
Terms X and Y denote the deterministic forces acting along x and y while Nand K are the deterministic moments around z and x, which take into account the hydrodynamic effects from the hull movements and forces exerted on the ship by the rudder and by the propUlsion system. Such forces and moments are usually described (Lewis, 1988) by regarding X,Y,N,K as polynomial expansion in terms of state variables, control actions and hydrodynamic coefficients. The effects of external disturbances, i.e_ wind and waves, consist of related forces X w , Y wand moments Nw, Kw acting as perturbation terms in the corresponding right hand parts of equation 1. Such terms, owing to their intrinsica1ly random nature, are generally quite difficuJt to be characterised through explicit mathematical relations. For example, as to the waves, they should be calculated by integrating the wave pressure over the immersed surface of the hull, on the assumption that the pressure within the waves is unaffected by the presence of the ship (Lewis, 1988), As it has been shown in Lewis (1988) and Price and Bishop (1974), a reasonable simplifying assumption consists in applying a linear superposition principle, which makes it possible to separate the ship motion due to the environment from the motion induced by the rudder and by the propeller thrust_ According to this modelling approach, waves and wind are regarded as finite order linear realizations of stochastic processes characterised by known spectral densities. By limiting attention to sea waves, which are by far
= X + X",
m~+w-~p+~~=y+~
14th World Congress ofIFAC
ro
Izzr + m'(c; (ur - v) = N + Nw lzzr - mz(,(ur + v) = K + Kif' - pgDRz (cp) The above equations describe the coupled surge, sway, yaw and roll motions, where 0 is the displacement, g the gravity constant, p the water mass density, Rz(
the dominant disturbance, it is possible to regard a long crested irregular sea height at time t, t;(t), described by a one-dimensional amplitude spectrum, the main parameters of which are the significant wave height, h and the average wave period T. The "ISSC spectrum" is given by:
G,..(m) = ~
- 691 5 4 exp(-4-4) wT TO}
173h 2
(3)
The relation between the response of each individual component of the wave induced ship state vector x.,. = [u w Vw rw PwJ T, can be obtained in terms of the spectrum:
G xw(i)(O}'X,U) = IR xw (i)(co,x,U)1 2 G,(m) (4) i=l.4
where GM is the ship metacentric height and BM is the distance from the centre of buoyancy to the metacentre.
where X is the angle of encounter between ship and waves, U is the ship velocity and RvliJ is the receptance operator, which is assumed to be known from experimental tests, describing the response of the ship i-th motion to the waves (Blanke and Jenson, 1997). In order to obtain the corresponding spectrum relative to the ship centre of mass, it is finally
8280
Copyright 1999 IF AC
ISBN: 008 0432484
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT ...
necessary to express the spectrum given by equation (4) as a function of the frequency of encounter behveen ship and waves According to this approach, it is possible to implement an accurate and numerically reliable simulation of sea wave induced ship motions.
3. FUZZY CONTROLLER
14th World Congress ofIFAC
1.
2.
3.
4.
The problem of designing an efficient integrated control system is complicated by the complex multivariable structure of the ship mathematical model. The main difficulties derive from the nonlinear, time varying and quite often unknown dynamical characteristics which may prevent conventional control techniques, based on linearized models with known parameters, from achieving optimal or even satisfactory performances. Fuzzy controllers (Fe) have been recognized in the recent years as valid alternative tools for control system design. In fact such controllers are not critically dependent on the knowledge of the ship mathematical model and a/so can cope well with non-linear plants. As it has been demonstrated (Sutton, 1987) and (Sutton et al., 1997), it is possible to design auto pilots for the course-keeping and course-changing process, capable of achieving significant improvements when compared with conventional PID autopilots. The main components of the fuzzy controller (Fe) are fuzzification, inference and defu=ification. Fuzzification is the procedure by which deterministic variables are transformed into a fuzzy format suitable to be utilised within the Fe. This task is carried out by means of a family of fuzzy sets, which in our context defme the linguistic terms Positive (P), Negative (N) and Zero (Z) for the ship measured state variables, which constitute the input to the Fe. According to Sutton et al. (1997), it is convenient to assume for the fuzzy sets a generalized bell curve expressed by:
(5)
i = 1,4 The inference procedure is based on a suitable rule base system, which generally for single input single output systems can be easily deduced from the apriori knowledge. In the case of a multivariable strongly coupled system, like the ship, this task is complicated. The number of possible rules, in fact, may be subjected to an exponential growth with respect to the input/output dimensions. It is therefore important to prune the rules tree, by eliminating all possible redundancies. In this case it was decided to carry out inference by means of Sugeno-type rules (Sugeno, 1985) to deduced as a multivariable extension of the those proposed for a yaw autopilot in Sutton et al. (1997):
is Nand r is N then 6 1 = al E '1' + blr + + d i P + g, if B 'f' is Nand r is Z then 32 = a2 e 'v + b2 r + C2" '" + d z p + g2 if E 'V is Nand r is P then 8 3 = a3 E ~, + b 3 r + C3E C(l + d, P + g3 if e ~I is Z and r is N then 8 4 = a4 E 'v + b 4 r + C4E 'fI + d4 p + B4 if e ~I is Z and r is Z then Os = as E ~I + bsr + CSG 'fI + ds p + Bs if E ~I is Z and r is P then 06 =
'"
C,E'I'
5. 6
7.
8~ 9.
C7E Lp + d 7 P + g7 if E ~I is P and r is Z then Ss = asE '" -+- bar + CsE 'P + dg P + gH if E '" is P and r is P then ()9 = a9 E ~, + b 9r + C9E W + d9 p + g9
ifE C(l is Nand pis N then 010 = alOG 'I' + blOr + CwE C(l + dlO p + glO 11 ife" is Nand pis Z then S]] = all S 'V + bur + CIIS 'fI + d]] p -+- gll 12. ifE'I' is N and p is Pthen 8 12 = a12 E 'I' + b l2 T + CUE C(l + d '2 p + gl2 Ll ifE '" is Z and p is N then 613 = al3 E ~I + b J3 r + Cl3 E C(l + d '4 p + gl3 /4 if £ 'i' is Z and p is Z then bl4 = a14 £ 'I' + b 14r + CI4 E 4' + d 14 P + g14 15. if E 'I' is Z and p is P then 015 = a15 E \I' + b 15r + C'5G ", + d '5 p + g15 16 ifE '" is P and pis N then 8 16 = a16 E 'I' + b 16r + CluE", + dIu P + gIG 17 ifE", is P and pis Z then b17 = aJ7E 'I' + b17r + CJ7S '" + d17 p + gJ7 18 ifE C(l is P and pis P then 8 1a = alaE 'v + b 18r 10.
..L
CISS 'I'
+ d ls P + g18
is Nand p is N then 019 = al9 F. 'I' + b 19T + d)9 p + gl9 20 if S \I' is Nand p is Z then 8 20 = a20 E Cl + b 20T + CzoE (jl + d 20 p + g20 21. if E 'i' is Nand p is P then 821 = a2J E I)! + bZ1r + C21!: 'I' + d 2 ] p + gn 22 if81jl is Z and pis N then 5 22 = a22S 'I' + bnf + C22E q> + d22 P + g22 23. if E '" is Z and p is Z then 8 23 = a23 8 v + b 23 f + CZ)E + d24 P + g24 25. if G~, is P and p is N then chs = a25 I: "' + b 25 f 19
if E
+
..L
'I'
C19E "'
C2SE
C(l
+ d Z5 p + g25
is P and p is Z then 026 = a2 6 8 C) + b 26r + C261;;; "' + d 2G p + g26 27. if E \I' is P and p is P then S27 = a27 e '" + bZ7r + en!:: 'l' + d27 P + g27 28. if E 'I' is Nand r is N then 028 = a2SC w + h28r + Clse 'I' + d 28 P + g28
26.
if I:
4'
8281
Copyright 1999 IF AC
ISBN: 008 0432484
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT ...
29.
if I; C29E
3()
if E
'P q1
is Nand r is Z then 029 = + d 29 P + g29 is Nand r is P then ':),0 =
a2910 4f
+ b 29r +
a30 G '"
+ b3~r
14th World Congress ofIFAC
whi1e and [x(j)]~~r are the corresponding values as given by the FC. In order to cope with the quite high dimension of such vector, differently from Jang (1993), it is been chosen a learning of the random search type based on Simulated Annealing (A arts and Korst, 1989). The main advantage is given by the fact that the algorithm does not require the computation of Jacobian matrix, which may detract from obtaining a global minimum for the preset cost function.
+ C3DE «J + d30 P + g3() 31.
ifG,~
J2.
+ C'IE 'I' + d31 p + g31 ifE q1 is Z and r is Z then
31 .
34
35
36.
isZandrisNtheno31=a3Is,1' +b31 r
0]2 = a3210 w + bJ2r + C32G '" + d)2 P + g32 if E 'P is z and r is P then 033 = a33 e 'I' + b]3r + CJJE 'l1 + d)3 p + g3) ifE", is P and r is N then 8 34 = a34E~, + b 34 r + C34G 'I' + d 34 p + g34 if E 'I' is P and r is Z then 035 = a35 G V' + b 3Sr + C3S& 'P + d 35 p + g35 if E 'P is P and r is P then 8 36 = a36 E ~I + b 36 r + C36& + d 36 p + g36
In fact the rules 1-9 have exactly the same fonn as the ones proposed in SuUon et al. (1997) for the purely yaw autopilot, while the rules IO-18 are an equivalent extension for a purely roll autopilot. Rules 19-27 and 28-36 are introduced in order to take into account the cross coupling between yaw and roll motions. Defuzzification of the output variable, constituted by
the rudder angle /) is obtained as a weighted linear combination of all the single-valued outputs OJ supplied by the equation: 36
:L LW;
WiD;
o =-':\C::(j--
(6)
where the coefficients Wj are the weighting factors of the antecedent of the i-th rule having taken into account the fuzzy set intersection operator.
In order to assure that an acceptable performance of the Fe may be achieved and maintained across a wide range of operating conditions, it is. a necessary to develop a learning procedure, according to which all the parameters of equation.(5) and in the 36 rules can be adapted on-line. The resulting controller is of the nemo-fuzzy type (Jang, 1993) and incorporates the learning procedure into the Fe design. After an initial assessment of the parameter vector 9, which contains all the relevant infonnation:
The detennination of the parameter vector .9 is carried out according to a leaming algorithm which iteratively minimizes a cost function ofthe type: where
[x(})];_r is a time record of a selected
training vector containing the ship variables to be controlled as obtained by a reference controller,
4. SIMULATION RESULTS In order to evaluate the behaviour of above presented RRD controller a number of simulations have been carried out. For this purpose, the ship described in Son and Nomoto (1982) has been chosen, owing to the extremely low metacentric height of such fast container ship, which exhibits strong coupling between roll and the other motions. The waveinduced disturbances were simulated according to the procedure described in section 2. The receptance operator described in Blanke and Jenson (1997) for a similar type of ship has been adapted for use in this study. A wide range of sea conditions have been simulated, with a particular attention to the situations when the wave effect on roll is maximum, i.e. in beam sea characterised by an angle of encounter between ship and prevailing wave direction X = 90°. As a reference control a Linear Quadratic (LQ) autopilot has been designed, based on a Iinearized ship equations of motion. This autopilot, which minimized a quadratic cost function weighting the ship state variables and rudder input, is described in more detail ill Tiano et a1. (1994). In this way many time series of input/output variables related to the LQ autopilot at different sea state conditions were collected. 5. RESULTS Simulation results are presented for the ship travelling at 13ms-1 in a sea condition defined by h=5m and T= 10 seconds, with sea encounter angles of 45° and 90°. Flg 2 shows the roll and yaw angles together with the rudder activity for an encounter angle of 45 with the LQ and fuzzyneural controllers Fig 3 shows the equivalent results for an encounter angle of 90°. Q
6. CONCLUDING REMARKS The control parameters of the FC controller have then been detennined, after a suitable initialisation on the basis of the results presented in Sutton (1987) by the learning algorithm based on Simulated Annealing discussed in section 3. It has been verified that, owing to the extremely high number of parameters to be adapted on-line, the convergence of the learning procedure was quite slow and in many cases it had to be stopped before reaching the global minimum.
8282
Copyright 1999 IF AC
ISBN: 008 0432484
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT...
14th World Congress ofIFAC
Roberts, G.N., (1992), 'Ship roll damping using rudder and stabilizing fins', Proceedings of IFAC Workshop on Control Applications in Marine Systems, Genoa, pp 234-248. Son, KH and Nomoto,K", (1982), 'On the coupled motion of steering and rolling of a high speed container ship', Journal of Naval Architecture and Ocean Engineering. Vo1.20, pp 73-83. Tiano, A., Mort, N., Derradji, Cuneo,Ranzi, A . and Zhou, W.W., (1994), ' Rudder roll stabilization by neural network-based control systems' , Proceedings of 3'd International Conference Manoeuvring and Control of Marine Craft, Southampton, pp 33-44. Price, W .G., and Bishop, R , (1974) , 'Probabilistic Theory of Ship Dynamics', Chapman and Hall, London Sugeno, M., (1985), 'Industrial applications a/fuzzy control', North Holland, The Netherlands Sutton, R., (1987), 'Fuzzy set models of the helmsman steering a ship in course-keeping and course-changing', Ph.D. Thesis, University of Wales. Sutton, R., Roberts, G.N. and Taylor, S.D.H., (1997), 'Tuning fuzzy ship autopilot using artificial neural networks', Transactions of The Institute of Measurement and Control, Vo1.19, No.2, pp 94-106. Yang, C. and Blanke, M., (1997), 'A robust roll damping controller', Proceedings of 4 th Internal Conference Manoeuvring and Control of Marine Craft, Brijuni, pp 60-64.
Nevertheless, the performances of the FC autopilot was quite good, and in any case better than the LQ autopilot. For example, in both Fig 1 and Fig 2 the fuzzyneural controller produces less roll motion that of the LQ controller.
7. REFERENCES Aarts, E. and Korst, J., (1989), 'Simulated Annealing and Boltzmann Machines " J. Wiley & Sons, New York. Van Amerongen, J, Van der Klugt, P.G.M. and Pieffets, J.B.M., (1987), 'Rudder roll stabilisation-controller design', Proceedings of 8th Ship Control Systems Symposium, The Hague, pp 120-142. Blanke, M., and Jensen, A.G., (1997), 'Dynamic properties of container vessel with low metacentric height' , Transactions of The Institute of Measurement and Control, Vo1.19, No.2, pp 78-93 Jang, J-S. R., (1993), 'ANFIS: Adaptive network based fuzzy inference system', IEEE Transactions on Systems, Man and Cybernetics, Vol.23, No.], pp 665-685. Kallstrom, e.G. and Schultz, W.L., (1990), 'An integrated rudder control system for roll damping and maintenance' > Proceedings of 9 th Ship Control Systems Symposium, Bethesda, Vol.3, pp 278-296. Lewis, E.W., (1988), 'Principles of Naval Architecture', Society of Naval Architecture and Marine Engineers, New York.
Earth Fixed Axes Xo
Roll p,K
Surge Ship Fixed Axes
Pitch q ,M
Yaw
r,N
Y
Sway
Heave
z Figure I: Ship fixed and reference coordinate system
8283
Copyright 1999 IF AC
ISBN: 0 08 043248 4
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT.. .
Roll Angle .
14th World Congress oflFAC
Roll Angle
--- ~~--~--~--~~
-.. .. ......
- · ·····1:·v··I·~···:··J· .:· ._ . ..· .·~.·.·· ... . .
_-
•. , . •. .. .•.• _,.. • · ·· · _.• " ..• · · ··· ·0· 0· ...• · ••
. .
.. .
.
•
( .
.
, ,, .•.. ___ . .. .. . .__ .•. _ .. . .. .••
I
Yaw Angle
Yaw Angle
::~ . "'1.'. ':' : \..:'~'.' ... ':hA' ::':::X:: ..
. ' ~~1 ··\,J· . ·I,ftV ··· .
", . .
. -t' V .'"1'..
.
\t·!·l ·~[..~ .. :w ..\' 4 -··
Rudder Angle
Rudder Angle
LQ Controller
FUZZY'neural Controller
Figure 2: h =5m, T=IO seconds, encounter angle = 45 u
Roll Angle
Roll Angle
Yaw Angle
Yaw Angle
Rudder Angle
Rudder Angle
LQ Controller
Fuzzyneural Controller
Figure 3: h =5m, T=lO seconds, encounter angle = 90°
Copyright 1999 IFAC
8284
ISBN: 0 08 043248 4
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT ...
Q-8c-02-3
Copyright © 1999 IFAC
14th Triennial World Congress, Beijing, P.R. China
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONTROL
G,N, Roberts\ A. Zirilli l , A. Tiano 2 and R Sutto03 I Mechatronics Research Centre, University afWales College, Newport Allt-yr-yn Campus, P.OBax 180, Ne .....port, NP9 5XR, UK. ([email protected])
lDepartment ofTriformation and Systems, University ofPavia, Pavia, Italy Via Ferrata 1,1-27JOO Pavia, Italy. Also: Institute ofShip Automation C.NR Via de Marini 6, [-16149 Genova, Italy. 3
Department of Mechanical Engineering, University ofPlymouth, Drake's Circus, Plymouth, PL48AA, UK.
Abstract: This paper describes the development of fuzzy controller for integrated control of the coupled yaw-roll motion of a container ship. The basic fuzzy rules used for designing an integrated multivariable rudder roll damping and yaw autopilot are discussed, A random search learning mechanism for automatically tuning the fuzzy controller parameters is presented. The efficiency and robustness of the controller are evaluated by a number of simulation tests conducted at different sea states. The results are used to evaluate the performance of the fuzzy controller in comparison with a conventional LQ design. Copyright © 19991FAC Keywords: Ship Motion Control, Integrated Control, Fuzzy Logic, Fuzzyneural control
1. INTRODUCTION
Many studies have been conducted in the recent years in order to design and put into operation a new generation of ship motion control systems, such as autopilots, stabilizers and dynamic posltlOning systems, capable to efficiently and safely carry out demanding navigation tasks in a wide range of environmental conditions. One of the most interesting research areas concerns Rudder Roll Damping (RRD) or Rudder Roll Stabilization (RRS). The philosophy behind this approach to ship roll stabilization is that rudder may be used as the only actuator to reduce roll motion and at the same time to maintain the ship's course. This result can be in principle achieved by compensating the disturbing effects induced by sea waves through a lowfrequency rudder action for the yaw and a highfrequency rudder action for the roll. The main advantage of such an approach, with respect to other
ship stabilization systems based on fins or on compensating tanks, is given by the reduction of investment costs as well as by the expected safety improvement during navigation in rough sea conditions. Among the papers dedicated to RRD design it is possible to mention a number of contributions where different solutions to the problem have been proposed. Such solutions include a PID controller (Amerongen et al., 1987), a frequency domain based autopilot (Roberts, 1992), a LQG controller (Kallstrom and Schultz, 1990), an artificial neural network controller (Tiano et al., 1994) and an H~ controller (Yang and Blanke, 1997). Tt should be noted, however, that most of such results have been obtained for linearized and weakly coupled ship models generally characterised by a simplified modelling of sea waves induced motion. This simplification is motivated by the need to avoid as far as possible the complications deJ"iving from the
8279
Copyright 1999 IF AC
ISBN: 008 0432484
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT ...
use of a completely coupled non-linear ship model. On the other hand, as evidenced by Son and Nomoto (1982) and Blanke and Jenson (1997), the dynamic behaviour of several ships characterized by a low metacentric height cannot be adequately described in terms of the simplified models of the linear and decoupled type, which are generally used in the hydrodynamic and naval architecture literature (Lewis, 1988)_ It has in fact been clearly demonstrated that for such vessels, including containerships, RORO ships, high-speed ferries, naval vessels and fishing boats, non-linear coupled surge-yaw-sway-roll can supply a better knowledge about the precise interaction between roll and other motions. In this paper a neurofuzzy integrated controller for a container ship will be presented_ The work is based on a design suggested in Sutton et al. (1997) for the yaw motion only. After a concise presentation of the ship mathematical model and of the fuzzy controller, some simulation results will be presented and discussed_
2. MATHEMATICAL MODELLING The mathematical model of a container ship used in this study is described in detail in Son and Nomoto (1982) and Blanke and Jenson (I997). It is herein considered a stochastic extension of such models capable to describe the ship response in irregular sea waves, and is expressed by the following non-linear equations (see Fig 1):
m(it - vr - xur2 + zr;pr)
Terms X and Y denote the deterministic forces acting along x and y while Nand K are the deterministic moments around z and x, which take into account the hydrodynamic effects from the hull movements and forces exerted on the ship by the rudder and by the propUlsion system. Such forces and moments are usually described (Lewis, 1988) by regarding X,Y,N,K as polynomial expansion in terms of state variables, control actions and hydrodynamic coefficients. The effects of external disturbances, i.e_ wind and waves, consist of related forces X w , Y wand moments Nw, Kw acting as perturbation terms in the corresponding right hand parts of equation 1. Such terms, owing to their intrinsica1ly random nature, are generally quite difficuJt to be characterised through explicit mathematical relations. For example, as to the waves, they should be calculated by integrating the wave pressure over the immersed surface of the hull, on the assumption that the pressure within the waves is unaffected by the presence of the ship (Lewis, 1988), As it has been shown in Lewis (1988) and Price and Bishop (1974), a reasonable simplifying assumption consists in applying a linear superposition principle, which makes it possible to separate the ship motion due to the environment from the motion induced by the rudder and by the propeller thrust_ According to this modelling approach, waves and wind are regarded as finite order linear realizations of stochastic processes characterised by known spectral densities. By limiting attention to sea waves, which are by far
= X + X",
m~+w-~p+~~=y+~
14th World Congress ofIFAC
ro
Izzr + m'(c; (ur - v) = N + Nw lzzr - mz(,(ur + v) = K + Kif' - pgDRz (cp) The above equations describe the coupled surge, sway, yaw and roll motions, where 0 is the displacement, g the gravity constant, p the water mass density, Rz(
the dominant disturbance, it is possible to regard a long crested irregular sea height at time t, t;(t), described by a one-dimensional amplitude spectrum, the main parameters of which are the significant wave height, h and the average wave period T. The "ISSC spectrum" is given by:
G,..(m) = ~
- 691 5 4 exp(-4-4) wT TO}
173h 2
(3)
The relation between the response of each individual component of the wave induced ship state vector x.,. = [u w Vw rw PwJ T, can be obtained in terms of the spectrum:
G xw(i)(O}'X,U) = IR xw (i)(co,x,U)1 2 G,(m) (4) i=l.4
where GM is the ship metacentric height and BM is the distance from the centre of buoyancy to the metacentre.
where X is the angle of encounter between ship and waves, U is the ship velocity and RvliJ is the receptance operator, which is assumed to be known from experimental tests, describing the response of the ship i-th motion to the waves (Blanke and Jenson, 1997). In order to obtain the corresponding spectrum relative to the ship centre of mass, it is finally
8280
Copyright 1999 IF AC
ISBN: 008 0432484
A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT ...
necessary to express the spectrum given by equation (4) as a function of the frequency of encounter behveen ship and waves According to this approach, it is possible to implement an accurate and numerically reliable simulation of sea wave induced ship motions.
3. FUZZY CONTROLLER
14th World Congress ofIFAC
1.
2.
3.
4.
The problem of designing an efficient integrated control system is complicated by the complex multivariable structure of the ship mathematical model. The main difficulties derive from the nonlinear, time varying and quite often unknown dynamical characteristics which may prevent conventional control techniques, based on linearized models with known parameters, from achieving optimal or even satisfactory performances. Fuzzy controllers (Fe) have been recognized in the recent years as valid alternative tools for control system design. In fact such controllers are not critically dependent on the knowledge of the ship mathematical model and a/so can cope well with non-linear plants. As it has been demonstrated (Sutton, 1987) and (Sutton et al., 1997), it is possible to design auto pilots for the course-keeping and course-changing process, capable of achieving significant improvements when compared with conventional PID autopilots. The main components of the fuzzy controller (Fe) are fuzzification, inference and defu=ification. Fuzzification is the procedure by which deterministic variables are transformed into a fuzzy format suitable to be utilised within the Fe. This task is carried out by means of a family of fuzzy sets, which in our context defme the linguistic terms Positive (P), Negative (N) and Zero (Z) for the ship measured state variables, which constitute the input to the Fe. According to Sutton et al. (1997), it is convenient to assume for the fuzzy sets a generalized bell curve expressed by:
(5)
i = 1,4 The inference procedure is based on a suitable rule base system, which generally for single input single output systems can be easily deduced from the apriori knowledge. In the case of a multivariable strongly coupled system, like the ship, this task is complicated. The number of possible rules, in fact, may be subjected to an exponential growth with respect to the input/output dimensions. It is therefore important to prune the rules tree, by eliminating all possible redundancies. In this case it was decided to carry out inference by means of Sugeno-type rules (Sugeno, 1985) to deduced as a multivariable extension of the those proposed for a yaw autopilot in Sutton et al. (1997):
is Nand r is N then 6 1 = al E '1' + blr + + d i P + g, if B 'f' is Nand r is Z then 32 = a2 e 'v + b2 r + C2" '" + d z p + g2 if E 'V is Nand r is P then 8 3 = a3 E ~, + b 3 r + C3E C(l + d, P + g3 if e ~I is Z and r is N then 8 4 = a4 E 'v + b 4 r + C4E 'fI + d4 p + B4 if e ~I is Z and r is Z then Os = as E ~I + bsr + CSG 'fI + ds p + Bs if E ~I is Z and r is P then 06 =
'"
C,E'I'
5. 6
7.
8~ 9.
C7E Lp + d 7 P + g7 if E ~I is P and r is Z then Ss = asE '" -+- bar + CsE 'P + dg P + gH if E '" is P and r is P then ()9 = a9 E ~, + b 9r + C9E W + d9 p + g9
ifE C(l is Nand pis N then 010 = alOG 'I' + blOr + CwE C(l + dlO p + glO 11 ife" is Nand pis Z then S]] = all S 'V + bur + CIIS 'fI + d]] p -+- gll 12. ifE'I' is N and p is Pthen 8 12 = a12 E 'I' + b l2 T + CUE C(l + d '2 p + gl2 Ll ifE '" is Z and p is N then 613 = al3 E ~I + b J3 r + Cl3 E C(l + d '4 p + gl3 /4 if £ 'i' is Z and p is Z then bl4 = a14 £ 'I' + b 14r + CI4 E 4' + d 14 P + g14 15. if E 'I' is Z and p is P then 015 = a15 E \I' + b 15r + C'5G ", + d '5 p + g15 16 ifE '" is P and pis N then 8 16 = a16 E 'I' + b 16r + CluE", + dIu P + gIG 17 ifE", is P and pis Z then b17 = aJ7E 'I' + b17r + CJ7S '" + d17 p + gJ7 18 ifE C(l is P and pis P then 8 1a = alaE 'v + b 18r 10.
..L
CISS 'I'
+ d ls P + g18
is Nand p is N then 019 = al9 F. 'I' + b 19T + d)9 p + gl9 20 if S \I' is Nand p is Z then 8 20 = a20 E Cl + b 20T + CzoE (jl + d 20 p + g20 21. if E 'i' is Nand p is P then 821 = a2J E I)! + bZ1r + C21!: 'I' + d 2 ] p + gn 22 if81jl is Z and pis N then 5 22 = a22S 'I' + bnf + C22E q> + d22 P + g22 23. if E '" is Z and p is Z then 8 23 = a23 8 v + b 23 f + CZ)E
if E
+
..L
'I'
C19E "'
C2SE
C(l
+ d Z5 p + g25
is P and p is Z then 026 = a2 6 8 C) + b 26r + C261;;; "' + d 2G p + g26 27. if E \I' is P and p is P then S27 = a27 e '" + bZ7r + en!:: 'l' + d27 P + g27 28. if E 'I' is Nand r is N then 028 = a2SC w + h28r + Clse 'I' + d 28 P + g28
26.
if I:
4'
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A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT ...
29.
if I; C29E
3()
if E
'P q1
is Nand r is Z then 029 = + d 29 P + g29 is Nand r is P then ':),0 =
a2910 4f
+ b 29r +
a30 G '"
+ b3~r
14th World Congress ofIFAC
whi1e and [x(j)]~~r are the corresponding values as given by the FC. In order to cope with the quite high dimension of such vector, differently from Jang (1993), it is been chosen a learning of the random search type based on Simulated Annealing (A arts and Korst, 1989). The main advantage is given by the fact that the algorithm does not require the computation of Jacobian matrix, which may detract from obtaining a global minimum for the preset cost function.
+ C3DE «J + d30 P + g3() 31.
ifG,~
J2.
+ C'IE 'I' + d31 p + g31 ifE q1 is Z and r is Z then
31 .
34
35
36.
isZandrisNtheno31=a3Is,1' +b31 r
0]2 = a3210 w + bJ2r + C32G '" + d)2 P + g32 if E 'P is z and r is P then 033 = a33 e 'I' + b]3r + CJJE 'l1 + d)3 p + g3) ifE", is P and r is N then 8 34 = a34E~, + b 34 r + C34G 'I' + d 34 p + g34 if E 'I' is P and r is Z then 035 = a35 G V' + b 3Sr + C3S& 'P + d 35 p + g35 if E 'P is P and r is P then 8 36 = a36 E ~I + b 36 r + C36& + d 36 p + g36
In fact the rules 1-9 have exactly the same fonn as the ones proposed in SuUon et al. (1997) for the purely yaw autopilot, while the rules IO-18 are an equivalent extension for a purely roll autopilot. Rules 19-27 and 28-36 are introduced in order to take into account the cross coupling between yaw and roll motions. Defuzzification of the output variable, constituted by
the rudder angle /) is obtained as a weighted linear combination of all the single-valued outputs OJ supplied by the equation: 36
:L LW;
WiD;
o =-':\C::(j--
(6)
where the coefficients Wj are the weighting factors of the antecedent of the i-th rule having taken into account the fuzzy set intersection operator.
In order to assure that an acceptable performance of the Fe may be achieved and maintained across a wide range of operating conditions, it is. a necessary to develop a learning procedure, according to which all the parameters of equation.(5) and in the 36 rules can be adapted on-line. The resulting controller is of the nemo-fuzzy type (Jang, 1993) and incorporates the learning procedure into the Fe design. After an initial assessment of the parameter vector 9, which contains all the relevant infonnation:
The detennination of the parameter vector .9 is carried out according to a leaming algorithm which iteratively minimizes a cost function ofthe type: where
[x(})];_r is a time record of a selected
training vector containing the ship variables to be controlled as obtained by a reference controller,
4. SIMULATION RESULTS In order to evaluate the behaviour of above presented RRD controller a number of simulations have been carried out. For this purpose, the ship described in Son and Nomoto (1982) has been chosen, owing to the extremely low metacentric height of such fast container ship, which exhibits strong coupling between roll and the other motions. The waveinduced disturbances were simulated according to the procedure described in section 2. The receptance operator described in Blanke and Jenson (1997) for a similar type of ship has been adapted for use in this study. A wide range of sea conditions have been simulated, with a particular attention to the situations when the wave effect on roll is maximum, i.e. in beam sea characterised by an angle of encounter between ship and prevailing wave direction X = 90°. As a reference control a Linear Quadratic (LQ) autopilot has been designed, based on a Iinearized ship equations of motion. This autopilot, which minimized a quadratic cost function weighting the ship state variables and rudder input, is described in more detail ill Tiano et a1. (1994). In this way many time series of input/output variables related to the LQ autopilot at different sea state conditions were collected. 5. RESULTS Simulation results are presented for the ship travelling at 13ms-1 in a sea condition defined by h=5m and T= 10 seconds, with sea encounter angles of 45° and 90°. Flg 2 shows the roll and yaw angles together with the rudder activity for an encounter angle of 45 with the LQ and fuzzyneural controllers Fig 3 shows the equivalent results for an encounter angle of 90°. Q
6. CONCLUDING REMARKS The control parameters of the FC controller have then been detennined, after a suitable initialisation on the basis of the results presented in Sutton (1987) by the learning algorithm based on Simulated Annealing discussed in section 3. It has been verified that, owing to the extremely high number of parameters to be adapted on-line, the convergence of the learning procedure was quite slow and in many cases it had to be stopped before reaching the global minimum.
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Roberts, G.N., (1992), 'Ship roll damping using rudder and stabilizing fins', Proceedings of IFAC Workshop on Control Applications in Marine Systems, Genoa, pp 234-248. Son, KH and Nomoto,K", (1982), 'On the coupled motion of steering and rolling of a high speed container ship', Journal of Naval Architecture and Ocean Engineering. Vo1.20, pp 73-83. Tiano, A., Mort, N., Derradji, Cuneo,Ranzi, A . and Zhou, W.W., (1994), ' Rudder roll stabilization by neural network-based control systems' , Proceedings of 3'd International Conference Manoeuvring and Control of Marine Craft, Southampton, pp 33-44. Price, W .G., and Bishop, R , (1974) , 'Probabilistic Theory of Ship Dynamics', Chapman and Hall, London Sugeno, M., (1985), 'Industrial applications a/fuzzy control', North Holland, The Netherlands Sutton, R., (1987), 'Fuzzy set models of the helmsman steering a ship in course-keeping and course-changing', Ph.D. Thesis, University of Wales. Sutton, R., Roberts, G.N. and Taylor, S.D.H., (1997), 'Tuning fuzzy ship autopilot using artificial neural networks', Transactions of The Institute of Measurement and Control, Vo1.19, No.2, pp 94-106. Yang, C. and Blanke, M., (1997), 'A robust roll damping controller', Proceedings of 4 th Internal Conference Manoeuvring and Control of Marine Craft, Brijuni, pp 60-64.
Nevertheless, the performances of the FC autopilot was quite good, and in any case better than the LQ autopilot. For example, in both Fig 1 and Fig 2 the fuzzyneural controller produces less roll motion that of the LQ controller.
7. REFERENCES Aarts, E. and Korst, J., (1989), 'Simulated Annealing and Boltzmann Machines " J. Wiley & Sons, New York. Van Amerongen, J, Van der Klugt, P.G.M. and Pieffets, J.B.M., (1987), 'Rudder roll stabilisation-controller design', Proceedings of 8th Ship Control Systems Symposium, The Hague, pp 120-142. Blanke, M., and Jensen, A.G., (1997), 'Dynamic properties of container vessel with low metacentric height' , Transactions of The Institute of Measurement and Control, Vo1.19, No.2, pp 78-93 Jang, J-S. R., (1993), 'ANFIS: Adaptive network based fuzzy inference system', IEEE Transactions on Systems, Man and Cybernetics, Vol.23, No.], pp 665-685. Kallstrom, e.G. and Schultz, W.L., (1990), 'An integrated rudder control system for roll damping and maintenance' > Proceedings of 9 th Ship Control Systems Symposium, Bethesda, Vol.3, pp 278-296. Lewis, E.W., (1988), 'Principles of Naval Architecture', Society of Naval Architecture and Marine Engineers, New York.
Earth Fixed Axes Xo
Roll p,K
Surge Ship Fixed Axes
Pitch q ,M
Yaw
r,N
Y
Sway
Heave
z Figure I: Ship fixed and reference coordinate system
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A FUZZY CONTROLLER FOR INTEGRATED SHIP MOTION CONT.. .
Roll Angle .
14th World Congress oflFAC
Roll Angle
--- ~~--~--~--~~
-.. .. ......
- · ·····1:·v··I·~···:··J· .:· ._ . ..· .·~.·.·· ... . .
_-
•. , . •. .. .•.• _,.. • · ·· · _.• " ..• · · ··· ·0· 0· ...• · ••
. .
.. .
.
•
( .
.
, ,, .•.. ___ . .. .. . .__ .•. _ .. . .. .••
I
Yaw Angle
Yaw Angle
::~ . "'1.'. ':' : \..:'~'.' ... ':hA' ::':::X:: ..
. ' ~~1 ··\,J· . ·I,ftV ··· .
", . .
. -t' V .'"1'..
.
\t·!·l ·~[..~ .. :w ..\' 4 -··
Rudder Angle
Rudder Angle
LQ Controller
FUZZY'neural Controller
Figure 2: h =5m, T=IO seconds, encounter angle = 45 u
Roll Angle
Roll Angle
Yaw Angle
Yaw Angle
Rudder Angle
Rudder Angle
LQ Controller
Fuzzyneural Controller
Figure 3: h =5m, T=lO seconds, encounter angle = 90°
Copyright 1999 IFAC
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