`
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`Ex. PGS 1060
`EX. PGS 1060
`(EXCERPTED)
`(EXCERPTED)
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`
`
`Guidance
`and
`Control
`
`of Ocean
`
`Vehicles
`
`
`Thor I. Eossen
`University of Trondheim
`Norway
`
`JOHN WILEY & SONS
`Chichester - New York - Brisbane - Toronto - Singapore
`
`

`

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`Copyright © 1994 by John Wiley & Sons Ltd.
`Baffins Lane. Chichester
`West Sussex P019 IUD. England
`National Chichester (0243) 779777
`International (+44) 243 779777
`
`All rights reserved.
`
`No part of this publication may be reproduced by any means. or
`transmitted, or translated into a machine language without the written
`permission of the publisher.
`
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`Block B. Union Industrial Building, Singapore 2057
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`British Library Cataloguing in Publication Data
`
`A catalogue record for this book is available from the British Library
`ISBN 0 471 94113 1
`
`Produced from camera-ready copy supplied by the author using LaTeX.
`Printed and bound in Great Britain by Bookoraft (Bath) Ltd.
`
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`286
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`Automatic Control of Ships
`
`This system can be expressed in a more compact form as:
`
`where a: = [W — 114% — 1/J]T and
`
`:3 = Ax + b qu ("9
`
`(6.299)
`
`A=|:_OKm ::l' b=|:;)]!
`
`Tm
`
`T".
`
`U
`
`¢:l¢-¢ra¢7llTl
`
`0=leaKdyKilT
`
`(6.300)
`‘
`Here b0 = K/T and 1%,, = XI, — K1,, Rd 2 K.) — K4 and K1 = Ki — K)- are the
`parameter estimation errors. Hence, the control objective can be expressed as:
`
`t—woo
`lim a:(t) = 0
`
`(6.301)
`
`which simply states that both the heading angle error and heading rate error
`should converge to zero. This is guaranteed by applying the following theorem.
`
`Theorem 6.3 (Model Reference Adaptive Control)
`The adaptive control law:
`
`3: —r —- qS bTPw;
`
`r = FT > o
`
`(6.302)
`
`where P = PT > 0 satisfies the Lyapunov equation:
`
`ATP + PA = —Q;
`
`Q = QT > 0
`
`(6.303)
`
`guarantees that the tracking error :1; —) 0 as t —> 00 and that the parameter esti-
`mation error 0 is bounded.
`
`Proof: Consider the Lyapunov function candidate:
`
`V(m, ("9, t) = mTPa: + |b0| ("fr—16
`
`(6.304)
`
`Diflerentiating V with respect to time yields:
`
`V = wT(ATP + 1mm + 2 [b0| [9T(1~-1 i9 +
`
`1
`
`Bo-l
`
`3 6T P m)
`
`(6.305)
`
`Substituting (6.302) and (6.303) into the expression for V yields:
`
`V = —:cTQ:l: g 0
`
`(6.306)
`
`This implies that V(t) S V(O), and therefore that a: and 9 are bounded. Difi‘er-
`entiating V with respect to time, yields:
`
`(6.307)
`V = —2 mTQd:
`
`
`
`

`

`287
`
`Assuming that (b is bounded implies that is is bounded, sec (6.299). This in turn
`implies that V is bounded. Hence, V must be uniformly continuous. Application
`of Barbdlat’s lemma {see Appendix 0.1) then indicates that a: —> 0 as t —> 00.
`.
`
`To implement the parameter adaptation law (6.302) we have to rewrite the un—
`known term:
`
`bT
`
`m: [0, m] —[o. sgneon
`
`b0
`
`_
`
`(6.308)
`
` 6.4 Turning Controllers
`
`The direct MRAC is based on the assumption that perfect model matching can be
`achieved. Hard nonlinearities like saturation in the rudder angle and the rudder
`rate implies that the linear reference model specifying the desired closed-loop
`dynamics cannot be matched by the system resulting from the ship dynamics
`and the adaptive controller. Instead of introducing nonlinearities in the reference
`model, Van Amerongen (1982, 1984) suggests modifying the commanded input
`11), to the reference model such that the reference model remain linear. This can
`be done by introducing a command generator according to Figure 6.35.
`The command generator should be designed such that the reference model
`remains linear and thus that the parameter estimates remains bounded. This can
`be done by introducing a new mechanism for compensation of rudder angle and
`rudder rate saturation, see Figure 6.36.
`The SAT function in Figure 6.36 is defined as:
`
`This implies that only the sign of the ratio b0 = K/T must be known while the
`magnitude of be not is used. Let F = diag('yl,’yg,'ya). Hence, (6.302) can be
`written in component form as:
`
`Kn = “Y1 sgn(bo) (w — We
`it'd = ——'yz sgn(b0) ibe
`Ki 2 —73 sgn(b0) 6
`
`(6.309)
`(6.310)
`(6.311)
`
`where 7,- > 0 for (i = 1, 2, 3). The error signal is computed as:
`
`where the elements 1921 = p12 and p22 of:
`
`6 = P21 (W — 1P) + P22 (1hr ‘ 1b)
`
`(6.312)
`
`P ___ [P11 p12 ]
`
`P22
`
`P21
`
`(6.313)
`
`are given from (6.303).
`
`Limitations of the Steering Machine
`
`

`

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`288
`
`Automatic Control of Ships
`
`Reference
`
`
`model
`
`generator maxmax
`
`Command
`
`Figure 6.35: MRAC structure with reference model and command generator in series.
`
`
`
`
`
`-f»-—-u,l-3—‘-a
`
`
`u—na.a—AJ.....¢L.-
`
`
`
`_
`
`SAT _{
`
`
`9m 1
`|6c| ‘1+TAs
`
`1+TA-‘1
`
`1
`
`if
`
`if
`
`|6c|>6max
`
`|6 |< 6
`
`c
`
`_. max
`
`(6.314)
`
`where 60 is the commanded rudder angle from the autOpilot and 6mm is the
`maximum allowed rudder angle. Rudder rate limitations are avoided by selecting
`the time constant TA in the low-pass filter large enough, for instance by manually
`increasing the value of TA until 6(t) tracks 60(t). In fact, this simple modification
`implies that the MRAC scheme remains stable since nonlinearities in the steering
`machine will not affect the perfect model matching conditions.
`
`
`
`Figure 6.36: Command generator
`
`The last element in the command generator, the yaw rate limiter, is motivated
`by the desire to describe a course-changing maneuver by three phases:
`
`
`
`
`
`
`
`
`Hence, the user can specify the maximum allowed turning rate during the second
`phase of the turn. The location of the yawing rate limiter is shown in Figure
`
`6.36.
`
`1) Start of turn
`
`2) Stationary turn (7* = rm“ and r = 0)
`
`3) End of turn
`
`
`
`

`

`6.5 Track-Keeping Systems
`
`Recently, more sophisticated high—precision track controllers have been de—
`signed. These systems are based on optimal control theory utilizing Navstar
`GPS (Global Positioning System). Navstar GPS consists of 21 satellites in six or-
`bital planes, with three or four satellites in each plane, together with three active
`spares. By measuring the distance to the satellite, the global position (27, y, z) of
`the vessel can be computed by application of the Kalman filter algorithm. A more
`detailed description of the GPS receiver and the Kalman filter implementation is
`given by Bardal and @rpen (1983).
`
`Kinematics
`
`For the design of the track controller it is convenient to describe the kinematics
`of the ship according to Figure 6.37. From the figure it is seen that (assuming
`that 9: ¢=0);
`
`Classical autopilot control of ships involves controlling the course angle 1,0. How-
`ever, by including an additional control-loop in the control system with position
`feedback a ship guidance system can be designed. This system is usually designed
`such that the ship can move forward with constant speed U at the same time as
`the sway position :1; is controlled. Hence, the ship can be made to track a pre—
`defined reference path which again can be generated by some route management
`system. The desired route is most easily specified by way points. If weather data
`are available, the optimal route can be generated such that the ship’s wind and
`water resistance is minimized. Hence, fuel can be saved.
`Many track controllers are based on low-accuracy positioning systems like
`Decca, Omega and Loren-C (Forssell 1991). These systems are usually combined
`with a low-gain PI-controller in cascade with the autopilot. The output from
`the autopilot will then represent the desired course angle. Unfortunately such
`systems result in tracking errors up to 300 m, which are only satisfactory in open
`
` 6.5 Track-Keeping Systems
`
`2': = UCOS'l/J — USln1/)
`g = wéin 1p + 1) cos 1/)
`1/} = r
`
`(6.315)
`(6.316)
`(6.317)
`
`Unfortunately, these equations are nonlinear in the states 11., v and 1/). However
`a linear approximation can be derived under the assumption that the earth-fixed
`coordinate system can be rotated such that the desired heading is W = 0. We
`can also move the origin of the coordinate system such that it coincides with the
`starting point [xd(t0),yd(t0)]. Hence, the heading angle 1/) will be small during
`track control such that:
`
`