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Internal Model Control for Rudder Roll Stabilisation and Course Keeping of a Surface Marine Craft⁎
11th 11th IFAC IFAC Conference Conference on on Control Control Applications Applications in in 11th IFAC IFAC Conference on Control Control Applications in in Marine Systems, Robotics, and Vehicles 11th Conference on Applications Marine Systems, Robotics, and Vehicles 11th IFAC Conference on Control Applications Marine Systems, Robotics, Vehicles Opatija, Croatia, September 10-12, 2018 Marine Systems, Robotics, and and Vehicles Availableinonline at www.sciencedirect.com Opatija, Croatia, September 10-12, 2018 Marine Systems, Robotics, and Vehicles 11th IFAC Conference on Control Applications in Opatija, Croatia, September 10-12, 2018 Opatija, Croatia, September 10-12, 2018 Opatija, Croatia, September 10-12, 2018 Marine Systems, Robotics, and Vehicles Opatija, Croatia, September 10-12, 2018
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IFAC PapersOnLine 51-29 (2018) 457–462
Internal Model Control for Rudder Roll Internal Model Control for Rudder Roll Internal Modeland Control forKeeping Rudder of Roll Stabilisation Course a Stabilisation Course of a Internal Modeland Control forKeeping Rudder Roll Stabilisation and CourseCraft Keeping of a Surface Marine Stabilisation and CourseCraft Keeping of a Surface Surface∗ Marine Marine Craft ∗ ∗ Alejandro Donaire ∗ Christina Perez Surface∗∗∗ Tristan Marine Craft Christina Kazantzidou Kazantzidou Tristan Perez Alejandro Donaire ∗∗∗ ∗ ∗ ∗∗
Christina Tristan Perez Alejandro Donaire Christina Kazantzidou Kazantzidou Perez ∗∗ Francis Valentinis ∗ Tristan ∗ Alejandro Donaire ∗ Francis Valentinis ∗∗ Christina Kazantzidou Tristan Perez ∗∗ Alejandro Donaire Francis Valentinis Francis Valentinis ∗ ∗ ∗∗ Alejandro Donaire ∗ Christina Kazantzidou Tristan Perez Francis Valentinis ∗ ∗∗ ∗ Robotics and Autonomous Systems, School of Electrical Engineering Francis Valentinis Systems, School of Electrical Engineering ∗ Robotics and Autonomous ∗ Robotics and Autonomous Systems, School of Electrical Engineering and Autonomous Systems, School of Electrical Engineering and Computer Science and Institute for Future Environments, ∗ Robotics and Computer Science and Institute for Future Environments, Robotics and Autonomous Systems, School of Electrical Engineering and Computer Science and Institute for Future Environments, and Computer Science and Institute for Future Environments, ∗Queensland University of Technology, Brisbane QLD 4000, Australia Queensland University of Technology, Brisbane QLD 4000, Australia Robotics and Autonomous Systems, School of Electrical Engineering and Computer Science and Institute for Future Environments, Queensland University of of Technology, Technology, Brisbane Brisbane QLD 4000, 4000, Australia Queensland University QLD Australia (e-mail: [email protected], [email protected], (e-mail: [email protected], [email protected], and Computer Science and Institute for Future Environments, Queensland University of Technology, Brisbane QLD 4000, Australia (e-mail: [email protected], [email protected], [email protected], (e-mail: [email protected], [email protected]) [email protected]) Queensland University of Technology, Brisbane QLD 4000, Australia (e-mail: [email protected], [email protected], ∗∗ [email protected]) ∗∗ Maritime Division, [email protected]) Defence Science and Technology Group, Maritime Division, Defence Science and Technology Group, (e-mail: [email protected], [email protected], ∗∗ [email protected]) ∗∗ Maritime Division, Defence Science and Technology Group, Division, Defence Science and Technology Group, Bend VIC 3207, Australia ∗∗ Maritime Fishermans Bend VIC 3207, Australia [email protected]) Maritime Fishermans Division, Defence Science and Technology Group, Fishermans Bend VIC 3207, Australia Fishermans Bend VIC 3207, Australia ∗∗ (e-mail: [email protected]) (e-mail: [email protected]) Maritime Division, Defence Science and Technology Fishermans Bend VIC 3207, Australia (e-mail: [email protected]) [email protected]) Group, (e-mail: Fishermans Bend VIC 3207, Australia (e-mail: [email protected]) (e-mail: [email protected]) Abstract: This paper paper presents presents a a novel novel control control design design procedure procedure for for simultaneous simultaneous course course keeping Abstract: This keeping Abstract: This paper aaa novel control design procedure for simultaneous course keeping Abstract: This paper presents presents novel control design procedure for the simultaneous course keeping and rudder roll stabilisation for surface marine craft. We address design using the Internal and rudder roll stabilisation for a surface marine craft. We address the design using the Internal Abstract: This paper presents a novel control design procedure for simultaneous course keeping and rudder roll stabilisation stabilisation for aahas surface marine craft. Wefundamental address the the design design usingofthe the Internal and rudder roll for surface marine craft. We address using Internal Model Principle. This approach been used to analyse limitations rudder roll Model Principle. This approach been used to analyse limitations of rudder roll Abstract: This paper presents novel control design procedure for the simultaneous course keeping and rudder roll stabilisation for aahas surface marine craft. Wefundamental address design using Internal Model Principle. This approach has been used to analyse fundamental limitations ofthe rudder roll Model Principle. This approach has been used to analyse fundamental limitations of rudder roll stabilisation due to unstable zero dynamics, but it has not been exploited for full control system stabilisation due to unstable zero dynamics, but it has not been exploited for full control system and rudder roll stabilisation for a surface marine craft. We address the design using the Internal Model Principle. This approach has been used to analyse fundamental limitations of rudder roll stabilisation due to unstable zero dynamics, but it has not been exploited for full control system stabilisation due to unstable zero dynamics, but it has not been exploited for full control system design. The design is based on roll sensitivity shaping uses simplified model of design. The design is based on roll sensitivity shaping and uses simplified parametric model of Model Principle. This approach has been used toitanalyse limitations of rudder roll stabilisation due to unstable zero dynamics, but has and notfundamental beenaaaexploited forparametric full control system design. The is based on roll sensitivity shaping and uses simplified parametric model of design. The design design isunstable based on rolldynamics, sensitivity shaping and uses aexploited simplified parametric model of the roll-induced wave motion. This design provides opportunities for adapting the performance the roll-induced wave motion. This design provides opportunities for adapting the performance stabilisation due to zero but it has not been for full control system design. The design is based on roll sensitivity shaping and uses a simplified parametric model of the roll-induced wave motion. This design provides opportunities for adapting the performance the roll-induced wave motion. design provides opportunities for adapting theaaperformance due to changes in environmental sailing conditions. need driven by well-known due to changes in environmental and sailing conditions. This is need driven by well-known design. The design is based on This rolland sensitivity shaping andThis usesis a aaasimplified parametric model of the roll-induced wave motion. This design provides opportunities for adapting theaperformance due to changes in environmental and sailing conditions. This is need driven by well-known due to changes in environmental and sailing conditions. This is a need driven by a well-known issue leading to performance degradation in this control problem, which is due to the presence issue leading to performance degradation in this control problem, which is due to the presence the roll-induced wave motion. This design provides opportunities for adapting the performance due to changes in environmental and sailing conditions. This is a need driven by a well-known issue leading to performance degradation in this control problem, which is due to the presence issue leading to in performance degradation in this control problem, whichdriven isof due to the presence of unstable zero dynamics and significant changes in the power spectrum the disturbance. of unstable zero dynamics and significant changes in the power spectrum of the disturbance. due to changes environmental and sailing conditions. This is a need by a well-known issue leadingzero to performance this control which isofdue the presence of unstable dynamics and anddegradation significant in changes in the the problem, power spectrum spectrum theto disturbance. of unstable zero dynamics significant changes in power of the disturbance. issue leading to performance in thisControl) control problem, which isLtd. the presence of dynamicsFederation anddegradation significant changes in theHosting power by spectrum ofdue the disturbance. © unstable 2018, IFACzero (International of Automatic Elsevier Allto rights reserved. Keywords: control systems, rudder roll stabilisation, internal model of unstable Marine zero dynamics and significant changes in the power spectrum of control. the disturbance. Keywords: Marine control systems, rudder roll stabilisation, internal model control. Keywords: Marine Marine control control systems, systems, rudder rudder roll roll stabilisation, stabilisation, internal internal model model control. control. Keywords: Keywords: Marine control systems, rudder roll stabilisation, internal model control. 1. INTRODUCTION the bulk of the of wave-induced spectrum Keywords: Marine control systems, rudder roll stabilisation, model control. 1. INTRODUCTION the bulk of internal the energy energy of the the wave-induced roll roll spectrum 1. the bulk energy the roll spectrum 1. INTRODUCTION INTRODUCTION the bulk of of the theMoreover, energy of of there the wave-induced wave-induced roll spectrum is significant. is only one control input is significant. Moreover, there is only one control input 1. INTRODUCTION the bulk of the energy of the wave-induced roll spectrum is significant. Moreover, there is only one control input is significant. Moreover, there is only one control input (rudder angle) to achieve two control objectives, which The roll motion of marine craft can have undesirable (rudder angle) to achieve two control objectives, which 1. a INTRODUCTION the bulk of the energy of the wave-induced roll spectrum The roll motion of a marine craft can have undesirable is significant. Moreover, there iscontrol only one control which input (rudder angle) to achieve two objectives, The roll motion of aa marine craft can have undesirable (rudder angle) to achieve two control objectives, which imposes further performance limitations (Goodwin et al., The roll motion of marine craft can have undesirable effects on cargo, human performance, and operation of onimposes further performance limitations (Goodwin et al., is significant. Moreover, iscontrol only one control which input effects on cargo, human performance, and operation of on(rudder angle) to achievethere two objectives, The roll motion of a marine craft can have undesirable imposes further performance limitations (Goodwin et al., effects on cargo, human performance, and operation of onimposes further performance limitations (Goodwin et al., 2000). Several methods have been proposed for rudder-rolleffects on cargo, human performance, and operation of onboard equipment. Thus, several control devices have been 2000). Several methods have been proposed for rudder-roll(rudder angle) to achieve two control objectives, which board equipment. Thus, several control devices have been The roll motion of a marine craft can have undesirable imposes further performance limitations (Goodwin et al., effects on cargo, human performance, and operation of on2000). Several methods have been proposed for rudder-rollboard equipment. Thus, several control devices have been 2000). Several methods have been proposed for rudder-rollstabilisation control design, for example model predictive board equipment. Thus, several control devices have been developed to reduce and control roll motion. The most stabilisation control design, for example model predictive imposes further performance limitations (Goodwin et al., developed to reduce and control roll motion. The most effects on cargo, human performance, and operation of on2000). Several methods have been proposed for rudder-rollboard equipment. Thus, several control devices have been stabilisation control design, for example model predictive developed to reduce and control roll motion. The most stabilisation control design, for example model predictive ,,proposed numerical control, sliding mode control, H developed to reduce andseveral control rollfin motion. The most commonly used control devices are stabilisers, anti∞ numerical optimisation, control, sliding mode control, H 2000). Several methods have been foroptimisation, rudder-rollcommonly used control devices are fin stabilisers, anti∞ board equipment. Thus, control devices have been stabilisation control design, for example model predictive developed to reduce and control roll motion. The most , numerical optimisation, control, sliding mode control, H commonly used control devices are fin stabilisers, anti∞ , numerical optimisation, control, sliding mode design, control, Hand quantitative feedback theory, output sensitivity loop commonly used control devices are fin stabilisers, anti∞ roll tanks, gyrostabilisers, trim flaps, for vessels with quantitative feedback theory, output sensitivity loop stabilisation control for example model predictive roll tanks, gyrostabilisers, trim flaps, and, for vessels with developed to reduce anddevices control rolland, motion. The most , numerical optimisation, control, sliding mode control, Hand commonly used control are fin stabilisers, anti∞ quantitative feedback theory, and output sensitivity loop roll tanks, gyrostabilisers, trim flaps, and, for vessels with quantitative feedback theory, and output sensitivity loop shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal roll tanks, gyrostabilisers, trim flaps, and, for vessels with particular characteristics, the rudder—see Perez (2005) shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal , numerical optimisation, control, sliding mode control, H particular characteristics, the rudder—see Perez (2005) commonly used control devices are fin stabilisers, antiquantitative feedback theory, and output sensitivity loop ∞ roll tanks, gyrostabilisers, trim flaps, and, for vessels with shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal particular characteristics, the rudder—see Perez (2005) shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal and Fossen (1997), Stoustrup et al. (1994), Blanke et al. particular characteristics, the rudder—see Perez (2005) and Perez and Blanke (2012) for a comprehensive review. and Fossen (1997), Stoustrup et al. (1994), Blanke et al. quantitative feedback theory, and output sensitivity and Perez and Blanke (2012) for a comprehensive review. roll tanks, gyrostabilisers, trim flaps, and, for vessels with shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal particular characteristics, the rudder—see Perez (2005) and Fossen Fossen (1997), Stoustrup et al. al.Tzeng (1994), Blanke etloop al. and Perez and Blanke (2012) for aa comprehensive review. and (1997), Stoustrup et (1994), Blanke et al. (2000), Hearns and Blanke (1998), et al. (2001). and Perez and Blanke (2012) for comprehensive review. (2000), Hearns and Blanke (1998), Tzeng et al. (2001). shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal particular characteristics, thefor rudder—see Perez (2005) (2000), and Fossen (1997), Stoustrup et al.Tzeng (1994), Blanke et al. and Perez and Blanke (2012) a capability comprehensive review. The potential for harnessing the of a rudder Hearns and Blanke (1998), et al. (2001). The potential for harnessing the capability of a rudder (2000), Hearns and Blanke (1998), Tzeng et al. (2001). and Fossen (1997), Stoustrup et al.Tzeng (1994), et al. The potential for harnessing the capability of rudder and Perez and through Blanke (2012) for amotion comprehensive review. Hearns and Blanke (1998), et Blanke al. (2001). In this paper, we approach the problem of roll stabilisation Thereduce potential for harnessing the capability of aasystems rudder (2000), to roll the use of control this paper, we approach the problem of roll stabilisation to roll through the use of motion control Thereduce potential for harnessing the capability of asystems rudder In In this paper, we approach the problem of roll stabilisation (2000), Hearns and Blanke (1998), Tzeng et al. (2001). to reduce roll through the use of motion control systems In this paper, we approach the problem of roll stabilisation and course keeping using Internal Model Control (IMC) to reduce roll through the use of motion control systems started to roll be investigated investigated in the early 1970s (Perez and and course keeping using Internal Model Control (IMC) started to be early 1970s and The potential for harnessing the capability of(Perez asystems rudder In this paper, we approach the problem of roll stabilisation to reduce through the in usethe of motion control and course keeping using Internal Model Control (IMC) started to be in the early 1970s (Perez and and course keeping using Internal Model Control (IMC) design. The approach is based on the Youla parameteristarted to be investigated investigated in the earlyaft 1970s (Perez and Blanke, 2012). The rudder is located and below the design. The approach is based on the Youla parameteriIn this paper, we approach the problem of roll stabilisation Blanke, 2012). The rudder is located aft and below the to reduce roll through the use of motion control systems and course keeping using Internal Model Control (IMC) started to be investigated in the early 1970s (Perez and design. The approach is based on the Youla parameteriBlanke, 2012). The rudder is located aft and below the design. The approach is based on the Youla parameterisation of all stabilising controllers and the shaping of the Blanke, 2012). The rudder is located aft and below the centre of mass of the marine vessel. The rudder therefore sation of all stabilising controllers and the shaping of the and course keeping using Internal Model Control (IMC) centre of mass of the marine vessel. The rudder therefore started to be investigated in the early 1970s (Perez and design. The approach is based on the Youla parameteriBlanke, 2012). The rudder is located aft and below the sation of all all stabilising stabilising controllers and the the shaping shaping ofwork the centre of mass of the marine vessel. The rudder therefore sation of controllers and of the roll sensitivity. This approach is motivated by the centre of mass of the marine vessel. The rudder therefore imparts both yaw and roll moments and can be used as roll sensitivity. This approach is motivated by the work design. The approach is based on the Youla parameteriimparts both yaw and roll moments and can be used as Blanke, 2012). The rudder is located aft and below the sation of all stabilising controllers and the shaping of the centre of mass of the marine vessel. The rudder therefore roll sensitivity. This approach is motivated by the work imparts both yaw and roll moments and can be used as roll sensitivity. This approach is motivated by the work of Goodwin et al. (2000), who used a simplified version imparts both yaw and roll moments and can be used as an inexpensive anti-rolling device. Using the rudder for of Goodwin et al. (2000), who used simplified version sation of all stabilising controllers andaa the shaping ofwork the an inexpensive anti-rolling device. Using the rudder for centre of mass of the marine vessel. The rudder therefore roll sensitivity. This approach is motivated by the imparts both yaw and roll moments and can be used as of Goodwin et al. (2000), who used simplified version an anti-rolling device. Using the rudder for of Goodwin et al. (2000), who used a simplified version this approach to analyse fundamental limitations due an inexpensive inexpensive anti-rolling device.control Using the rudder for roll stabilisation is a challenging problem, since of this approach to analyse fundamental limitations due roll sensitivity. This approach is motivated by the work roll stabilisation is a challenging control problem, since imparts both yaw and roll moments and can be used as Goodwin et al. (2000), who used a simplified version an inexpensive anti-rolling device. Using the rudder for of this approach to analyse fundamental limitations due roll stabilisation aa challenging control problem, since of this approach to(2000), analysewho fundamental limitations due the presence phase zero dynamics, roll inexpensive stabilisation is is challenging control problem, since the rudder-to-roll dynamics has non-minimum phase zero, to the presence of non-minimum phase zero dynamics, Goodwin et al.of used a simplified version the rudder-to-roll dynamics has non-minimum phase zero, an device. Using the ruddersince for to of this approach to non-minimum analyse fundamental limitations due roll stabilisationanti-rolling is a challenging control problem, to the presence of non-minimum phase zero dynamics, the rudder-to-roll dynamics has non-minimum phase zero, to the presence of non-minimum phase zero dynamics, and the trade-off between roll reduction and yaw interthe rudder-to-roll dynamics has non-minimum phase zero, and the power spectrum of the roll-induced wave motion and the trade-off between roll reduction and yaw interof this approach to analyse fundamental limitations due and the power spectrum of the roll-induced wave motion roll stabilisation is a challenging control problem, since to the presence of non-minimum phase zero dynamics, the rudder-to-roll dynamics has non-minimum phase zero, and the trade-off between roll reduction and yaw interand the spectrum of the roll-induced wave motion and thepresence trade-off between reduction and yaw interference. Its potential for aaroll full control design was not, and rudder-to-roll the power power spectrum of has the roll-induced wave motion can change significantly due to changes in the sea state ference. Its potential for full control design was not, to the of non-minimum phase zero dynamics, can change significantly due to changes in the sea state the dynamics non-minimum phase zero, and the trade-off between roll reduction and yaw interand change the power spectrum due of the roll-induced waveseamotion ference. Its potential for aapropose full control design was not, can significantly to in ference. potential for fullreduction control design wasinternot, however investigated. We aa control can change significantly due to changes changes in the the sea state state and the sailing conditions and heading relative to however investigated. We control design that and the Its trade-off between roll anddesign yaw and the sailing conditions (speed and heading relative to power spectrum of(speed the wave ference. Its potential for apropose full control design was that not, can change significantly due to roll-induced changes in the seamotion state however investigated. We propose a control design that and the sailing conditions (speed and heading relative to however investigated. We propose a control design that uses a simplified roll disturbance model based on a secondand the sailing conditions (speed and heading relative to the waves). It has been shown in Perez (2005) that there uses a simplified roll disturbance model based on a secondference. Its potential for a full control design was not, the waves). It has been shown in Perez (2005) that there can change significantly due to changes in the sea state however investigated. We propose a control design that and the sailing conditions (speed and heading relative to uses aa shaping simplified roll disturbance disturbance model based based on a aaction secondthe It been in Perez (2005) that there uses simplified roll model on secondorder filter, and incorporates integral in the waves). waves). It has has been shown shown inthe Perez (2005) relative that there are fundamental limitations in achievable extent of order shaping filter, and it incorporates integral action in however investigated. Weit propose a control design that are fundamental limitations in the achievable extent of and the sailing conditions (speed and heading to uses a simplified roll disturbance model based on a secondthe waves). It has been shown in Perez (2005) that there order shaping filter, and it incorporates integral action in are fundamental limitations in the achievable extent of order shaping filter, and it incorporates integral action in yaw. This design provides opportunities for adapting the are fundamental limitations in the achievable extent of roll reduction that depends on the frequency of the nonyaw. This design provides opportunities for adapting the uses a simplified roll disturbance model based on a secondroll reduction that depends on the frequency of the nonthe waves). It has been shown in Perez (2005) that there order shaping filter, and it incorporates integral action in are reduction fundamental limitations in the thefrequency achievableof extent of yaw. This design provides opportunities for adapting the roll that depends on the nonyaw. This design provides opportunities for adapting the performance due to changes in environmental and sailing roll reduction that depends on the frequency of the nonminimum phase zero and the range of frequencies at which performance due to changes in environmental and sailing order shaping filter, and it incorporates integral action in minimum phase zero and the range of frequencies at which are fundamental limitations in the achievable extent of yaw. This design provides opportunities for adapting the roll reduction that depends on the frequency of the nonperformance due changes in environmental and minimum phase zero and range of frequencies at performance due to to changes in is environmental and sailing sailing conditions. However, the latter outside the scope of this minimum phase zerodepends and the the on range offrequency frequencies at which which conditions. However, the latter is outside the scope of this yaw. This design provides opportunities for adapting the roll reduction that the of the nonperformance due to changes in environmental and sailing This work has been supported by the Australian Department of minimum phase zero and the range of frequencies at which conditions. However, However, the the latter latter is is outside outside the the scope scope of of this this This work has been supported by the Australian Department of conditions. paper. This has been the Australian Department of paper. performance due to changes in is environmental and sailing Defence through research agreement with the Maritime Division of minimum phase zerosupported and the by range of frequencies at which conditions. However, the latter outside the scope of this This work work has a supported by the Australian Department Defence through abeen research agreement with the Maritime Division of paper. This work has been supported by the Australian Department of paper. Defence through a research research agreementGroup. with the the Maritime Maritime Division Division of of the Defence Science and Technology conditions. However, the latter is outside the scope of this Defence through a agreement with paper. the Defence Science andsupported Technology This Defence through research agreement with the Maritime Division of work has abeen by Group. the Australian Department the the Defence Defence Science Science and and Technology Technology Group. Group. paper. the Defence Science and Technology Defence through a research agreementGroup. with the Maritime Division of
Copyright © Science 2018 457 the Defence and (International Technology Group. 2405-8963 © 2018, IFAC IFAC Federation of Automatic Control) Copyright 2018 IFAC 457 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 457 Copyright © under 2018 IFAC IFAC 457 Control. Peer review© responsibility of International Federation of Automatic Copyright © 2018 IFAC 457 10.1016/j.ifacol.2018.09.445 Copyright © 2018 IFAC 457
IFAC CAMS 2018 458 Christina Kazantzidou et al. / IFAC PapersOnLine 51-29 (2018) 457–462 Opatija, Croatia, September 10-12, 2018
2. CONTROL-TO-MOTION MODEL We consider the model structure from Perez (2005) for the problem of rudder roll stabilisation and course keeping of a naval patrol boat. The linear model of the response of a marine craft to the control action in restricted degrees of freedom (DOF) is described by (Perez, 2005): η˙ ≈ ν, (1a) (1b) M ν˙ = D(U ) ν + G η + τc , where ν [ v, p, r ] represents the generalised body-fixed velocity, and η [ φ, ψ ] gives the attitude. The vector τc [ Y (α, U ), K(α, U ), N (α, U ) ] has as components the sway force and the roll and yaw moments generated by the rudder motion. These components depend on the rudder angle α and the average forward speed U . The matrices M, D(U ), G are the total mass, damping, and restoring matrices, respectively. Let us define the following state, control, and output variables: x [ v, p, r, φ, ψ ] , u α, y [ φ, ψ ] . Then the state-space representation in (1) becomes: x˙ = A x + B u, (2a) y = C x, (2b) where A Ms−1 F (U ), B Ms−1 H(U ), C 00 00 00 10 01 , Ms
M 03,2 , 02,3 I2
m − Yv˙ −(m zG + Yp˙ ) m xG − Yr˙ −Kr˙ M −(m zG + Kv˙ ) Ixx − Kp˙ , m xG − Nv˙ −Np˙ Izz − Nr˙ f11 (U ) 0 f13 (U ) f14 (U ) 0 f21 (U ) f22 (U ) f23 (U ) f24 (U ) 0 F (U ) f31 (U ) f32 (U ) f33 (U ) f34 (U ) 0 , 0 1 0 0 0 0 0 1 0 0
f11 (U ) Y|u|v |U |, f21 (U ) K|u|v |U |, f31 (U ) N|u|v |U |, f22 (U ) Kp + K|u|p |U |, f32 (U ) Np + N|u|p |U |,
nφ (s) Kroll (s − q1 ) (s + q2 ), nψ (s) Kyaw (s + q3 ) s2 + 2 ξq ωq s + ωq2 , d(s) (s + p1 ) (s + p2 ) s2 + 2 ξp ωp s + ωp2 .
(4a) (4b) (4c)
The rudder-to-roll transfer function Gφα (s) is stable with a non-minimum phase zero at q1 and the rudder-to-yaw transfer function Gψα (s) is marginally stable. 3. INTERNAL MODEL CONTROL (IMC) The IMC design considers the two transfer functions Gφα (s) and Gψα (s) plus a first-order model for the rudder machinery dynamics Gα (s) in the configuration shown in Fig. 1. We assume that the pole of the first-order system of the steering machinery Gα (s) is 5 times larger than the dominant poles of the system and therefore it can be neglected for initial design, i.e., Gα (s) = 1 (Perez, 2005). The desired roll angle is 0 and the desired yaw angle is denoted by ψd . The input disturbance is denoted by δi , and the output disturbances by δφ and δψ . In this paper, we consider both sinusoidal disturbances of the form δ (t) = A sin(ω t), with = φ, ψ, and stochastic output disturbances with rational spectrum approximations. The stochastic output disturbances can be described by δ (s) = H (s) w(s), K 2 ξ ω σ ,
f13 (U ) (Yur − m) U, f23 (U ) (Kur + m zG ) U,
H (s) = φ, ψ,
s2
K s , + 2 ξ ω s + ω 2 (5)
where H (s) is a second-order shaping filter and w(s) represents the Laplace transform of unit-variance widebanded noise.
f33 (U ) N|u|r |U | − m xG U, f14 (U ) Yφuu U 2 ,
f24 (U ) Kφuu U 2 − ρ g ∇ GM t, f34 (U ) Nφu|u| U |U |, 1 −rr 1 ∂CL , H(U ) −LCG ρ ur (U )2 Ar ∂α α=0 0 2 0
where Ar is the area of the rudder, CL is the lift coefficient, ur (U ) is the flow speed over the rudder, LCG is the lateral centre of gravity of the vessel, and rr is the roll lever arm, see Perez (2005) for more details on the notation. The transfer function matrix is computed by: nφ (s) G (s) d(s) φα G(s) C (s I5 − A)−1 B = nψ (s) Gψα (s) s d(s) where
Fig. 1. Simplified rudder stabiliser control scheme.
, (3) 458
The control objective is to construct a controller C(s) [ Cφ (s) Cψ (s) ], as shown in Fig. 1, to reduce the roll amplitude and control the heading of the marine craft. The transfer functions Cφ (s) and Cψ (s) represent the roll and yaw controllers, respectively. The class of controllers C(s) giving rise to an internally stable closed-loop system can be parameterised as: C(s) = (1 − Q(s) G(s))−1 Q(s), (6)
where Q(s) [ Qφ (s) Qψ (s) ] is a stable and proper rational row vector, see e.g. Goodwin et al. (2000). The plant G(s) is not stable due to the integrator in Gψα (s) and to ensure internal stability further interpolation constraints must be satisfied for Q(s) (Goodwin et al., 2000): (i) Qφ (0) = Qψ (0) = 0; and (ii) Qψ (s) Gψα (s) = 1 at s = 0.
IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018 Christina Kazantzidou et al. / IFAC PapersOnLine 51-29 (2018) 457–462
4. MAIN RESULT In this section, we seek to extend the work of Goodwin et al. (2000) by constructing Q(s) in (6) based on a secondorder shaping filter Hφ (s) that characterises the roll disturbance, and roll sensitivity shaping. In addition, the proposed design has integral action in the yaw controller to maintain the course of the marine craft in the presence of constant input disturbance. For the design of the controller C(s) in (6), we propose the following form for Q(s): γ1 s2 d(s) γ2 s (s + ζ) d(s) Q(s) , (7) dQ (s) dQ (s) where dQ (s) is a stable (yet unknown) polynomial of degree 6, so that Q(s) is proper, and γ1 , γ2 and ζ are parameters to be chosen by the designer. The selection of Q(s) as in (7) ensures that interpolation constraint (i) is satisfied by construction, and interpolation constraint (ii) is satisfied provided that dQ (0) = γ2 ζ nψ (0). With this choice of Q(s), the controller is computed by C(s) = [ Cφ (s) Cψ (s) ], where γ1 s2 d(s) , dQ (s) − γ1 nφ (s) − γ2 (s + ζ) nψ (s) γ2 s (s + ζ) d(s) . Cψ (s) = dQ (s) − γ1 s2 nφ (s) − γ2 (s + ζ) nψ (s) The roll sensitivity is computed by 1 + Gψα (s) Cψ (s) Sφφ (s) = , 1 + Gφα (s) Cφ (s) + Gψα Cψ (s) and, after some computations, we obtain Cφ (s) =
s2
(8a) (8b)
(9)
nφ (s) γ1 s2 d(s) Sφφ (s) = 1 − Gφα (s) Qφ (s) = 1 − d(s) dQ (s) Kroll (s − q1 ) (s + q2 ) γ1 s2 =1− . (10) dQ (s) Let us choose dQ (s) = Kroll (s + q1 ) (s + q2 ) a(s),
(11)
where a(s) s + a3 s + a2 s + a1 s + a0 is a stable (yet unknown) polynomial. With this choice of dQ (s), the roll sensitivity is given by: (s − q1 ) γ1 s2 . (12) Sφφ (s) = 1 − (s + q1 ) a(s) 4
3
2
According to the Internal Model Principle, steady state disturbance compensation requires that the generating polynomial, i.e., the denominator of the disturbance shaping filter, is included in the controller denominator polynomial (Goodwin et al., 2001). Therefore the desired denominator of C(s) has to include the dynamics of the roll disturbance in order to reduce the effect of it. Thus, the desired denominator of C(s) is chosen, using the denominator of Hφ (s), as follows: dC (s) Kroll s2 d˜C (s) = Kroll s2 s2 + 2 ξφ ωφ s + ωφ2 b(s), (13) where b(s) s2 + b1 s + b0 is a stable (yet unknown) polynomial multiplied by Kroll to have the same leading coefficient as dQ (s). The factor s2 is included in dC (s) 459
459
to incorporate integral action in yaw, as it can be seen by using (13) in the controllers Cφ (s) and Cψ (s) given in (8). Expanding the polynomials dC (s) and dQ (s) − γ1 s2 nφ (s) − γ2 (s + ζ) nψ (s) and equating the coefficients corresponding to the same degrees, we may solve the following system of six linear equations with six unknowns: x ˆ [ a3 , a 2 , a 1 , a 0 , b1 , b 0 ] Px ˆ = c,
(14)
where
1 0 0 0 −1 0 1 0 0 −2 ξφ ωφ −1 q1 + q2 1 0 −ωφ2 −2 ξφ ωφ q1 q2 q1 + q2 P , 2 q1 q2 q1 + q 2 1 0 −ωφ 0 0 0 q 1 q2 q1 + q2 0 0 0 0 0 q 1 q2 0 0 c [ c1 , c2 , c3 , c4 , c5 , c6 ] ,
c1 2 ξφ ωφ − (q1 + q2 ), γ2 Kyaw , c2 ωφ2 − q1 q2 + γ1 + Kroll γ2 Kyaw (2 ξq ωq + q3 + ζ), c3 γ1 (q2 − q1 ) + Kroll γ2 Kyaw 2 ωq + 2 ξq ωq (q3 + ζ) + q3 ζ , c4 −γ1 q1 q2 + Kroll γ2 Kyaw 2 ωq (q3 + ζ) + 2 ξq ωq q3 ζ , c5 Kroll γ2 Kyaw q3 ζ ωq2 c6 . Kroll It can be easily shown that P is non-singular since its columns are linearly independent. Hence, the solution of (14) is given by: x ˆ = P −1 c. (15) The solution is acceptable provided that a(s) in (11) is stable and b1 , b0 > 0. The parameters γ1 , γ2 and ζ can be chosen such that the solution is acceptable. Alternatively, we can compute a0 , a1 by c6 c5 (q1 + q2 ) c6 a0 = , a1 = − (16) q1 q2 q1 q 2 q12 q22 and then compute the remaining four unknowns by: −1 1 0 −1 0 a3 c1 1 −2 ξφ ωφ −1 q1 + q 2 a2 c2 b = q q q +q −ωφ2 −2 ξφ ωφ c7 , 1 2 1 2 1 b0 c8 0 q1 q 2 0 −ωφ2
where c7 c3 − a1 , c8 c4 − (q1 + q2 ) a1 − a0 . Notice that interpolation constraint (ii) is satisfied since dQ (0) = Kroll q1 q2 a0 = γ2 ζ Kyaw q3 ωq2 = γ2 ζ nψ (0). Finding the solution x ˆ, the controller can be written as: γ1 d(s) γ2 (s + ζ) d(s) . (17) C(s) = Kroll d˜C (s) Kroll s d˜C (s) To assess the performance of a roll stabilising controller, the commonly used measures are the percentage of reduction of roll variance: 100 (1 − var[φcl ]/var[φol ]), and the
IFAC CAMS 2018 460 Christina Kazantzidou et al. / IFAC PapersOnLine 51-29 (2018) 457–462 Opatija, Croatia, September 10-12, 2018
percentage of reduction of roll root mean square (RMS) value: 100 (1 − var[φcl ]/ var[φol ]), where φol and φcl denote the open-loop and closed-loop roll angles, respectively.
is Aφ = 0.26 for roll and Aψ = 0.014 for yaw. The parameters of the shaping filters that characterise the stochastic disturbances are σφ = 0.275, ξφ = 0.13, ωφ = 0.89,
5. CASE STUDY
σψ = 0.0147, ξψ = 0.13, ωψ = 0.911.
We consider the benchmark example of a naval vessel. Using the values of Appendix B in Perez (2005) and the Marine GNC Toolbox by Fossen and Perez (2004), the state-space model is described by the triple (A, B, C), where −0.1472 0.182 −1.2881 1.1565 0 0.0276 −0.2031 −1.1976 −1.2962 0 A = −0.0115 0.0037 −0.3481 0.0176 0 , 0 1 0 0 0 0 0 1 0 0 0.3267 −0.1339 00010 B = −0.0288 , C = . 00001 0 0
We choose γ1 = 3, γ2 = 1, ζ = 0.2. The polynomial dQ (s) is given in (11), where
The rudder-to-roll and rudder-to-yaw transfer functions nφ (s) n (s) , Gψα (s) = s ψd(s) , where nφ (s), nψ (s), are Gφα (s) = d(s) and d(s) are given in (4) with
a(s) = s4 + 2.0447 s3 + 4.0215 s2 + 0.3597 s + 0.1772. The controller C(s) is given in (17), where d˜C (s) = (s2 + 0.2314 s + 0.7921) (s2 + 2.3552 s + 0.6433). 5.3 Bow seas In the case of bow seas, the encounter angle is assumed to be χ = 135◦ . The amplitude of the sinusoidal disturbances is Aφ = 0.14 for roll and Aψ = 0.013 for yaw. The parameters of the shaping filters that characterise the stochastic disturbances are σφ = 0.1954, ξφ = 0.08, ωφ = 0.9395, σψ = 0.018, ξψ = 0.08, ωψ = 0.9395. We choose γ1 = 3, γ2 = 1, ζ = 0.1. The polynomial dQ (s) is given in (11), where a(s) = s4 + 2.049 s3 + 3.9322 s2 + 0.6229 s + 0.0886. The controller C(s) is given in (17), where d˜C (s) = (s2 + 0.1503 s + 0.8827) (s2 + 2.4406 s + 0.6439).
Kroll = −0.1339, Kyaw = −0.02883, q1 = 0.186,
q2 = 0.3559, q3 = 0.2001, ξq = 0.1275, ωq = 1.1668,
p1 = 0.0415, p2 = 0.4403, ξp = 0.0946, ωp = 1.1445. We design IMC controllers for quartering seas, beam seas, and bow seas, and analyse the performances based on numerical simulations. We assume the following wave conditions: significant wave height H1/3 = 2.5 m, and average wave period T1 = 7.5 s. The rudder angle α cannot exceed 36◦ , and the average forward speed U is assumed to be 8 m/s. 5.1 Quartering seas In the case of quartering seas, the encounter angle is assumed to be χ = 45◦ . The amplitude of the sinusoidal disturbances is Aφ = 0.075 for roll and Aψ = 0.0045 for yaw. The parameters of the shaping filters that characterise the stochastic disturbances are σφ = 0.313, ξφ = 0.03, ωφ = 0.43, σψ = 0.018, ξψ = 0.03, ωψ = 0.43. We choose γ1 = 1.5, γ2 = 0.1, ζ = 0.1. The polynomial dQ (s) is given in (11), where a(s) = s4 + 2.8349 s3 + 1.2254 s2 + 0.0623 s + 0.0089. The controller C(s) is given in (17), where d˜C (s) = (s2 + 0.0258 s + 0.1849) (s2 + 3.351 s + 1.035). 5.2 Beam seas In the case of beam seas, the encounter angle is assumed to be χ = 90◦ . The amplitude of the sinusoidal disturbances 460
The parameters γ1 and γ2 were chosen such that γ1 > γ2 and for quartering seas the choice of small γ2 increased the roll reduction. The parameter ζ was chosen such that the polynomials a(s) and b(s) are stable. For values of ζ between 0.1 and 0.2 the stability requirement is satisfied and the integral action in yaw is effective. 5.4 Performance analysis The roll sensitivities for quartering, beam, and bow seas using the designed controllers are shown in Fig. 2. For quartering seas, the roll sensitivity is below 1 for frequencies approximately between 0.253 and 1.224 with minimum value 0.027 at frequency 0.43 rad/s. There is at least 50% attenuation in |Sφφ | for frequencies between 0.324 to 0.616 rad/s. For beam seas, the roll sensitivity is below 1 for frequencies approximately between 0.382 and 1.783 with minimum value 0.0977 at frequency 0.864 rad/s. There is at least 50% attenuation in |Sφφ | for frequencies from 0.545 to 1.288 rad/s. For bow seas, the roll sensitivity is below 1 for frequencies approximately between 0.4 and 1.751 with minimum value 0.0658 at frequency 0.933 rad/s. There is at least 50% attenuation in |Sφφ | for frequencies from 0.611 to 1.309 rad/s. The controller will be switched off for frequencies below 0.25 rad/s to avoid large disturbance amplification. The open and closed-loop roll angle, yaw angle and rudder angle for quartering, beam, bow seas and sinusoidal output disturbances are shown in Fig. 3, 4, 5, respectively. The open and closed-loop roll angle, yaw angle and rudder
IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018 Christina Kazantzidou et al. / IFAC PapersOnLine 51-29 (2018) 457–462
Roll Angle [°]
S 4.5 Quartering Seas Beam Seas Bow Seas
4
Yaw Angle [°]
3 2.5 2
0 -5
0
20
20
40
60
80
100
120
140
160
180
200
1 0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Open loop
Closed loop (input dist)
Closed loop
0 -20
1.5
0
Quartering Seas (Sinusoidal Disturbances)
5
Time [s]
Rudder Angle [°]
Magnitude (abs)
3.5
461
0
20
40
60
80
100
120
140
160
180
200
120
140
160
180
200
Time [s] 50 0 -50
Frequency (rad/s)
0
20
40
60
80
100
Time [s]
Fig. 2. Roll sensitivity (χ = 45◦ , 90◦ , 135◦ ).
Table 1. Roll reduction (δi = 0).
Sinusoidal
Stochastic
χ=
90◦
χ=
135◦
RR
χ=
Var
96.40 %
91.22 %
99.14 %
RMS
80.98 %
70.35 %
90.70 %
Var
95.61 %
85.41 %
91.85 %
RMS
79.05 %
61.81 %
71.45 %
Roll Angle [°]
Beam Seas (Sinusoidal Disturbances)
20 0 -20
0
20
40
60
80
100
120
140
160
180
200
Time [s]
Rudder Angle [°]
Disturbance
45◦
Fig. 3. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 45◦ , sinusoidal disturbances).
Yaw Angle [°]
angle for quartering, beam, bow seas and stochastic disturbances are shown in Fig. 6, 7, 8, respectively. We run two simulations for each case, assuming that there is no input disturbance and a 3◦ input disturbance, which represents an equivalent yaw disturbance torque induced by wind. The percentages of roll reduction with respect to variance and RMS are shown in Tables 1 and 2. In the presence of constant input disturbance, it is observed that the roll reduction is smaller, yet we have course keeping due to the integral action in the yaw controller.
10
Open loop
Closed loop (input dist)
Closed loop
0 -10
0
20
40
60
80
100
120
140
160
180
200
120
140
160
180
200
Time [s] 50 0 -50
0
20
40
60
80
100
Time [s]
Table 2. Roll reduction (δi = 3◦ ).
Sinusoidal
Stochastic
RR
χ = 45◦
χ = 90◦
χ = 135◦
Var
93.41 %
89.46%
99.20 %
RMS
74.28 %
67.52%
91.03 %
Var
93.78 %
84.81 %
91.80 %
RMS
75.06 %
61.03 %
71.37 %
Fig. 4. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 90◦ , sinusoidal disturbances). Roll Angle [°]
Disturbance
Bow Seas (Sinusoidal Disturbances)
10 0 -10
0
20
40
60
80
100
120
140
160
180
200
In this paper, we design a controller for rudder roll stabilisation and course keeping via Internal Model Control. The approach is based on the Youla parameterisation, roll sensitivity shaping, and integral action in the yaw controller for course keeping. A procedure is given for the construction of Q(s) to include integral action in yaw and the computation of the controller gains. A demonstration of the method is provided in the form of a case study of the benchmark example (see Perez (2005)) where the proposed controller is tested in different sailing conditions. In particular, three controllers are designed for the cases of quartering, beam, and bow seas with 461
Rudder Angle [°]
6. DISCUSSION AND CONCLUSIONS
Yaw Angle [°]
Time [s] 5
Open loop
Closed loop (input dist)
Closed loop
0 -5
0
20
40
60
80
100
120
140
160
180
200
120
140
160
180
200
Time [s] 50 0 -50
0
20
40
60
80
100
Time [s]
Fig. 5. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 135◦ , sinusoidal disturbances).
Roll Angle [°]
IFAC CAMS 2018 462 Christina Kazantzidou et al. / IFAC PapersOnLine 51-29 (2018) 457–462 Opatija, Croatia, September 10-12, 2018
encounter angles χ = 45◦ , 90◦ , 135◦ , respectively. We consider sinusoidal and stochastic disturbances for the roll and yaw angles and run simulations assuming there is no input disturbance and small input disturbance. There is satisfactory roll reduction and the course keeping is maintained in the presence of constant input disturbance.
Quartering Seas (Stochastic Disturbances)
5 0 -5
0
20
40
60
80
100
120
140
160
180
200
Rudder Angle [°]
Yaw Angle [°]
Time [s] 20
Open loop
Closed loop (input dist)
Future research will consider an optimisation algorithm for the computation of the parameters γ1 , γ2 and ζ. A selection of a larger degree for the denominator dQ (s) of Q(s) will be also considered. To increase the effectiveness of the roll stabilisation control over a wide envelope of sea states and sailing conditions, an adaptive scheme to track the changes in the parameters of the disturbance shaping filter as well as a switched system will be considered. Finally, a natural topic for future work is the design of automatic gain control scheme to reduce the control effort for the proposed method.
Closed loop
0 -20
0
20
40
60
80
100
120
140
160
180
200
120
140
160
180
200
Time [s] 50 0 -50
0
20
40
60
80
100
Time [s]
Roll Angle [°]
Fig. 6. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 45◦ , stochastic disturbances). Beam Seas (Stochastic Disturbances)
20 0 -20
0
20
40
60
80
100
120
140
160
180
200
Rudder Angle [°]
Yaw Angle [°]
Time [s] 5
Open loop
Closed loop (input dist)
Closed loop
0 -5
0
20
40
60
80
100
120
140
160
180
200
120
140
160
180
200
Time [s] 50 0 -50
0
20
40
60
80
100
Time [s]
Roll Angle [°]
Fig. 7. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 90◦ , stochastic disturbances). Bow Seas (Stochastic Disturbances)
10 0 -10
0
20
40
60
80
100
120
140
160
180
200
Rudder Angle [°]
Yaw Angle [°]
Time [s] 5
Open loop
Closed loop (input dist)
Closed loop
0 -5
0
20
40
60
80
100
120
140
160
180
200
120
140
160
180
200
Time [s] 50 0 -50
0
20
40
60
80
100
Time [s]
Fig. 8. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 135◦ , stochastic disturbances). 462
REFERENCES Blanke, M., Adrian, J., Larsen, K., and Bentsen, J. (2000), Rudder roll damping in coastal region sea conditions. In Proceedings of the 5th IFAC Conference on Manoeuvring and Control of Marine Craft, Aalborg, Denmark. Fossen, T.I. (2011), Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley & Sons, Chichester, West Sussex, UK. Fossen, T.I., and Perez, T. (2004), Marine Systems Simulator (MSS),. Goodwin, G.C., Graebe, S., and Salgado, M. (2001), Control System Design. Prentice-Hall, Inc., Upper Saddle River, NJ, USA. Goodwin, G.C., Perez, T., Seron, M., and Tzeng, C.Y. (2000), On fundamental limitations for rudder roll stabilization of ships. In Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia. Hearns, G., and Blanke, M. (1998), Quantitative analysis and design of rudder roll damping controllers. In Proceedings of the IFAC Conference on Control Applications in Marine Systems, Fukuoka, Japan. Lauvdal, T., and Fossen, T.I., (1997), Nonlinear nonminimum phase rudder-roll damping system for ships using sliding mode control. In Proceedings of European Control Conference, Brussels, Belgium. Perez, T. (2005), Ship Motion Control: Course Keeping and Roll Stabilisation Using Rudder and Fins. Advances in Industrial Control, Springer, London, UK. Perez, T., and Blanke, M. (2012), Ship roll damping control. Annual Reviews in Control, 36(1), pp. 129-147. Perez, T., Goodwin, G.C., and Tzeng, C.Y. (2000), Model predictive rudder roll stabilization control for ships. In Proceedings of the 5th IFAC Conference on Manoeuvring and Control of Marine Craft, Aalborg, Denmark. Stoustrup, J., Niemann, H.H., and Blanke, M. (1994), Roll damping by rudder control - a new H∞ approach. In Proceedings of IEEE International Conference on Control and Applications, Glasgow, UK. Tzeng, C.Y., Wu, C.Y., and Chu, Y.L. (2001), A sensitivity function approach to the design of rudder roll stabilization controller. Journal of Marine Science and Technology, 9(2), pp. 100-112.
ScienceDirect
IFAC PapersOnLine 51-29 (2018) 457–462
Internal Model Control for Rudder Roll Internal Model Control for Rudder Roll Internal Modeland Control forKeeping Rudder of Roll Stabilisation Course a Stabilisation Course of a Internal Modeland Control forKeeping Rudder Roll Stabilisation and CourseCraft Keeping of a Surface Marine Stabilisation and CourseCraft Keeping of a Surface Surface∗ Marine Marine Craft ∗ ∗ Alejandro Donaire ∗ Christina Perez Surface∗∗∗ Tristan Marine Craft Christina Kazantzidou Kazantzidou Tristan Perez Alejandro Donaire ∗∗∗ ∗ ∗ ∗∗
Christina Tristan Perez Alejandro Donaire Christina Kazantzidou Kazantzidou Perez ∗∗ Francis Valentinis ∗ Tristan ∗ Alejandro Donaire ∗ Francis Valentinis ∗∗ Christina Kazantzidou Tristan Perez ∗∗ Alejandro Donaire Francis Valentinis Francis Valentinis ∗ ∗ ∗∗ Alejandro Donaire ∗ Christina Kazantzidou Tristan Perez Francis Valentinis ∗ ∗∗ ∗ Robotics and Autonomous Systems, School of Electrical Engineering Francis Valentinis Systems, School of Electrical Engineering ∗ Robotics and Autonomous ∗ Robotics and Autonomous Systems, School of Electrical Engineering and Autonomous Systems, School of Electrical Engineering and Computer Science and Institute for Future Environments, ∗ Robotics and Computer Science and Institute for Future Environments, Robotics and Autonomous Systems, School of Electrical Engineering and Computer Science and Institute for Future Environments, and Computer Science and Institute for Future Environments, ∗Queensland University of Technology, Brisbane QLD 4000, Australia Queensland University of Technology, Brisbane QLD 4000, Australia Robotics and Autonomous Systems, School of Electrical Engineering and Computer Science and Institute for Future Environments, Queensland University of of Technology, Technology, Brisbane Brisbane QLD 4000, 4000, Australia Queensland University QLD Australia (e-mail: [email protected], [email protected], (e-mail: [email protected], [email protected], and Computer Science and Institute for Future Environments, Queensland University of Technology, Brisbane QLD 4000, Australia (e-mail: [email protected], [email protected], [email protected], (e-mail: [email protected], [email protected]) [email protected]) Queensland University of Technology, Brisbane QLD 4000, Australia (e-mail: [email protected], [email protected], ∗∗ [email protected]) ∗∗ Maritime Division, [email protected]) Defence Science and Technology Group, Maritime Division, Defence Science and Technology Group, (e-mail: [email protected], [email protected], ∗∗ [email protected]) ∗∗ Maritime Division, Defence Science and Technology Group, Division, Defence Science and Technology Group, Bend VIC 3207, Australia ∗∗ Maritime Fishermans Bend VIC 3207, Australia [email protected]) Maritime Fishermans Division, Defence Science and Technology Group, Fishermans Bend VIC 3207, Australia Fishermans Bend VIC 3207, Australia ∗∗ (e-mail: [email protected]) (e-mail: [email protected]) Maritime Division, Defence Science and Technology Fishermans Bend VIC 3207, Australia (e-mail: [email protected]) [email protected]) Group, (e-mail: Fishermans Bend VIC 3207, Australia (e-mail: [email protected]) (e-mail: [email protected]) Abstract: This paper paper presents presents a a novel novel control control design design procedure procedure for for simultaneous simultaneous course course keeping Abstract: This keeping Abstract: This paper aaa novel control design procedure for simultaneous course keeping Abstract: This paper presents presents novel control design procedure for the simultaneous course keeping and rudder roll stabilisation for surface marine craft. We address design using the Internal and rudder roll stabilisation for a surface marine craft. We address the design using the Internal Abstract: This paper presents a novel control design procedure for simultaneous course keeping and rudder roll stabilisation stabilisation for aahas surface marine craft. Wefundamental address the the design design usingofthe the Internal and rudder roll for surface marine craft. We address using Internal Model Principle. This approach been used to analyse limitations rudder roll Model Principle. This approach been used to analyse limitations of rudder roll Abstract: This paper presents novel control design procedure for the simultaneous course keeping and rudder roll stabilisation for aahas surface marine craft. Wefundamental address design using Internal Model Principle. This approach has been used to analyse fundamental limitations ofthe rudder roll Model Principle. This approach has been used to analyse fundamental limitations of rudder roll stabilisation due to unstable zero dynamics, but it has not been exploited for full control system stabilisation due to unstable zero dynamics, but it has not been exploited for full control system and rudder roll stabilisation for a surface marine craft. We address the design using the Internal Model Principle. This approach has been used to analyse fundamental limitations of rudder roll stabilisation due to unstable zero dynamics, but it has not been exploited for full control system stabilisation due to unstable zero dynamics, but it has not been exploited for full control system design. The design is based on roll sensitivity shaping uses simplified model of design. The design is based on roll sensitivity shaping and uses simplified parametric model of Model Principle. This approach has been used toitanalyse limitations of rudder roll stabilisation due to unstable zero dynamics, but has and notfundamental beenaaaexploited forparametric full control system design. The is based on roll sensitivity shaping and uses simplified parametric model of design. The design design isunstable based on rolldynamics, sensitivity shaping and uses aexploited simplified parametric model of the roll-induced wave motion. This design provides opportunities for adapting the performance the roll-induced wave motion. This design provides opportunities for adapting the performance stabilisation due to zero but it has not been for full control system design. The design is based on roll sensitivity shaping and uses a simplified parametric model of the roll-induced wave motion. This design provides opportunities for adapting the performance the roll-induced wave motion. design provides opportunities for adapting theaaperformance due to changes in environmental sailing conditions. need driven by well-known due to changes in environmental and sailing conditions. This is need driven by well-known design. The design is based on This rolland sensitivity shaping andThis usesis a aaasimplified parametric model of the roll-induced wave motion. This design provides opportunities for adapting theaperformance due to changes in environmental and sailing conditions. This is need driven by well-known due to changes in environmental and sailing conditions. This is a need driven by a well-known issue leading to performance degradation in this control problem, which is due to the presence issue leading to performance degradation in this control problem, which is due to the presence the roll-induced wave motion. This design provides opportunities for adapting the performance due to changes in environmental and sailing conditions. This is a need driven by a well-known issue leading to performance degradation in this control problem, which is due to the presence issue leading to in performance degradation in this control problem, whichdriven isof due to the presence of unstable zero dynamics and significant changes in the power spectrum the disturbance. of unstable zero dynamics and significant changes in the power spectrum of the disturbance. due to changes environmental and sailing conditions. This is a need by a well-known issue leadingzero to performance this control which isofdue the presence of unstable dynamics and anddegradation significant in changes in the the problem, power spectrum spectrum theto disturbance. of unstable zero dynamics significant changes in power of the disturbance. issue leading to performance in thisControl) control problem, which isLtd. the presence of dynamicsFederation anddegradation significant changes in theHosting power by spectrum ofdue the disturbance. © unstable 2018, IFACzero (International of Automatic Elsevier Allto rights reserved. Keywords: control systems, rudder roll stabilisation, internal model of unstable Marine zero dynamics and significant changes in the power spectrum of control. the disturbance. Keywords: Marine control systems, rudder roll stabilisation, internal model control. Keywords: Marine Marine control control systems, systems, rudder rudder roll roll stabilisation, stabilisation, internal internal model model control. control. Keywords: Keywords: Marine control systems, rudder roll stabilisation, internal model control. 1. INTRODUCTION the bulk of the of wave-induced spectrum Keywords: Marine control systems, rudder roll stabilisation, model control. 1. INTRODUCTION the bulk of internal the energy energy of the the wave-induced roll roll spectrum 1. the bulk energy the roll spectrum 1. INTRODUCTION INTRODUCTION the bulk of of the theMoreover, energy of of there the wave-induced wave-induced roll spectrum is significant. is only one control input is significant. Moreover, there is only one control input 1. INTRODUCTION the bulk of the energy of the wave-induced roll spectrum is significant. Moreover, there is only one control input is significant. Moreover, there is only one control input (rudder angle) to achieve two control objectives, which The roll motion of marine craft can have undesirable (rudder angle) to achieve two control objectives, which 1. a INTRODUCTION the bulk of the energy of the wave-induced roll spectrum The roll motion of a marine craft can have undesirable is significant. Moreover, there iscontrol only one control which input (rudder angle) to achieve two objectives, The roll motion of aa marine craft can have undesirable (rudder angle) to achieve two control objectives, which imposes further performance limitations (Goodwin et al., The roll motion of marine craft can have undesirable effects on cargo, human performance, and operation of onimposes further performance limitations (Goodwin et al., is significant. Moreover, iscontrol only one control which input effects on cargo, human performance, and operation of on(rudder angle) to achievethere two objectives, The roll motion of a marine craft can have undesirable imposes further performance limitations (Goodwin et al., effects on cargo, human performance, and operation of onimposes further performance limitations (Goodwin et al., 2000). Several methods have been proposed for rudder-rolleffects on cargo, human performance, and operation of onboard equipment. Thus, several control devices have been 2000). Several methods have been proposed for rudder-roll(rudder angle) to achieve two control objectives, which board equipment. Thus, several control devices have been The roll motion of a marine craft can have undesirable imposes further performance limitations (Goodwin et al., effects on cargo, human performance, and operation of on2000). Several methods have been proposed for rudder-rollboard equipment. Thus, several control devices have been 2000). Several methods have been proposed for rudder-rollstabilisation control design, for example model predictive board equipment. Thus, several control devices have been developed to reduce and control roll motion. The most stabilisation control design, for example model predictive imposes further performance limitations (Goodwin et al., developed to reduce and control roll motion. The most effects on cargo, human performance, and operation of on2000). Several methods have been proposed for rudder-rollboard equipment. Thus, several control devices have been stabilisation control design, for example model predictive developed to reduce and control roll motion. The most stabilisation control design, for example model predictive ,,proposed numerical control, sliding mode control, H developed to reduce andseveral control rollfin motion. The most commonly used control devices are stabilisers, anti∞ numerical optimisation, control, sliding mode control, H 2000). Several methods have been foroptimisation, rudder-rollcommonly used control devices are fin stabilisers, anti∞ board equipment. Thus, control devices have been stabilisation control design, for example model predictive developed to reduce and control roll motion. The most , numerical optimisation, control, sliding mode control, H commonly used control devices are fin stabilisers, anti∞ , numerical optimisation, control, sliding mode design, control, Hand quantitative feedback theory, output sensitivity loop commonly used control devices are fin stabilisers, anti∞ roll tanks, gyrostabilisers, trim flaps, for vessels with quantitative feedback theory, output sensitivity loop stabilisation control for example model predictive roll tanks, gyrostabilisers, trim flaps, and, for vessels with developed to reduce anddevices control rolland, motion. The most , numerical optimisation, control, sliding mode control, Hand commonly used control are fin stabilisers, anti∞ quantitative feedback theory, and output sensitivity loop roll tanks, gyrostabilisers, trim flaps, and, for vessels with quantitative feedback theory, and output sensitivity loop shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal roll tanks, gyrostabilisers, trim flaps, and, for vessels with particular characteristics, the rudder—see Perez (2005) shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal , numerical optimisation, control, sliding mode control, H particular characteristics, the rudder—see Perez (2005) commonly used control devices are fin stabilisers, antiquantitative feedback theory, and output sensitivity loop ∞ roll tanks, gyrostabilisers, trim flaps, and, for vessels with shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal particular characteristics, the rudder—see Perez (2005) shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal and Fossen (1997), Stoustrup et al. (1994), Blanke et al. particular characteristics, the rudder—see Perez (2005) and Perez and Blanke (2012) for a comprehensive review. and Fossen (1997), Stoustrup et al. (1994), Blanke et al. quantitative feedback theory, and output sensitivity and Perez and Blanke (2012) for a comprehensive review. roll tanks, gyrostabilisers, trim flaps, and, for vessels with shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal particular characteristics, the rudder—see Perez (2005) and Fossen Fossen (1997), Stoustrup et al. al.Tzeng (1994), Blanke etloop al. and Perez and Blanke (2012) for aa comprehensive review. and (1997), Stoustrup et (1994), Blanke et al. (2000), Hearns and Blanke (1998), et al. (2001). and Perez and Blanke (2012) for comprehensive review. (2000), Hearns and Blanke (1998), Tzeng et al. (2001). shaping, to name a few, see e.g. Perez et al. (2000), Lauvdal particular characteristics, thefor rudder—see Perez (2005) (2000), and Fossen (1997), Stoustrup et al.Tzeng (1994), Blanke et al. and Perez and Blanke (2012) a capability comprehensive review. The potential for harnessing the of a rudder Hearns and Blanke (1998), et al. (2001). The potential for harnessing the capability of a rudder (2000), Hearns and Blanke (1998), Tzeng et al. (2001). and Fossen (1997), Stoustrup et al.Tzeng (1994), et al. The potential for harnessing the capability of rudder and Perez and through Blanke (2012) for amotion comprehensive review. Hearns and Blanke (1998), et Blanke al. (2001). In this paper, we approach the problem of roll stabilisation Thereduce potential for harnessing the capability of aasystems rudder (2000), to roll the use of control this paper, we approach the problem of roll stabilisation to roll through the use of motion control Thereduce potential for harnessing the capability of asystems rudder In In this paper, we approach the problem of roll stabilisation (2000), Hearns and Blanke (1998), Tzeng et al. (2001). to reduce roll through the use of motion control systems In this paper, we approach the problem of roll stabilisation and course keeping using Internal Model Control (IMC) to reduce roll through the use of motion control systems started to roll be investigated investigated in the early 1970s (Perez and and course keeping using Internal Model Control (IMC) started to be early 1970s and The potential for harnessing the capability of(Perez asystems rudder In this paper, we approach the problem of roll stabilisation to reduce through the in usethe of motion control and course keeping using Internal Model Control (IMC) started to be in the early 1970s (Perez and and course keeping using Internal Model Control (IMC) design. The approach is based on the Youla parameteristarted to be investigated investigated in the earlyaft 1970s (Perez and Blanke, 2012). The rudder is located and below the design. The approach is based on the Youla parameteriIn this paper, we approach the problem of roll stabilisation Blanke, 2012). The rudder is located aft and below the to reduce roll through the use of motion control systems and course keeping using Internal Model Control (IMC) started to be investigated in the early 1970s (Perez and design. The approach is based on the Youla parameteriBlanke, 2012). The rudder is located aft and below the design. The approach is based on the Youla parameterisation of all stabilising controllers and the shaping of the Blanke, 2012). The rudder is located aft and below the centre of mass of the marine vessel. The rudder therefore sation of all stabilising controllers and the shaping of the and course keeping using Internal Model Control (IMC) centre of mass of the marine vessel. The rudder therefore started to be investigated in the early 1970s (Perez and design. The approach is based on the Youla parameteriBlanke, 2012). The rudder is located aft and below the sation of all all stabilising stabilising controllers and the the shaping shaping ofwork the centre of mass of the marine vessel. The rudder therefore sation of controllers and of the roll sensitivity. This approach is motivated by the centre of mass of the marine vessel. The rudder therefore imparts both yaw and roll moments and can be used as roll sensitivity. This approach is motivated by the work design. The approach is based on the Youla parameteriimparts both yaw and roll moments and can be used as Blanke, 2012). The rudder is located aft and below the sation of all stabilising controllers and the shaping of the centre of mass of the marine vessel. The rudder therefore roll sensitivity. This approach is motivated by the work imparts both yaw and roll moments and can be used as roll sensitivity. This approach is motivated by the work of Goodwin et al. (2000), who used a simplified version imparts both yaw and roll moments and can be used as an inexpensive anti-rolling device. Using the rudder for of Goodwin et al. (2000), who used simplified version sation of all stabilising controllers andaa the shaping ofwork the an inexpensive anti-rolling device. Using the rudder for centre of mass of the marine vessel. The rudder therefore roll sensitivity. This approach is motivated by the imparts both yaw and roll moments and can be used as of Goodwin et al. (2000), who used simplified version an anti-rolling device. Using the rudder for of Goodwin et al. (2000), who used a simplified version this approach to analyse fundamental limitations due an inexpensive inexpensive anti-rolling device.control Using the rudder for roll stabilisation is a challenging problem, since of this approach to analyse fundamental limitations due roll sensitivity. This approach is motivated by the work roll stabilisation is a challenging control problem, since imparts both yaw and roll moments and can be used as Goodwin et al. (2000), who used a simplified version an inexpensive anti-rolling device. Using the rudder for of this approach to analyse fundamental limitations due roll stabilisation aa challenging control problem, since of this approach to(2000), analysewho fundamental limitations due the presence phase zero dynamics, roll inexpensive stabilisation is is challenging control problem, since the rudder-to-roll dynamics has non-minimum phase zero, to the presence of non-minimum phase zero dynamics, Goodwin et al.of used a simplified version the rudder-to-roll dynamics has non-minimum phase zero, an device. Using the ruddersince for to of this approach to non-minimum analyse fundamental limitations due roll stabilisationanti-rolling is a challenging control problem, to the presence of non-minimum phase zero dynamics, the rudder-to-roll dynamics has non-minimum phase zero, to the presence of non-minimum phase zero dynamics, and the trade-off between roll reduction and yaw interthe rudder-to-roll dynamics has non-minimum phase zero, and the power spectrum of the roll-induced wave motion and the trade-off between roll reduction and yaw interof this approach to analyse fundamental limitations due and the power spectrum of the roll-induced wave motion roll stabilisation is a challenging control problem, since to the presence of non-minimum phase zero dynamics, the rudder-to-roll dynamics has non-minimum phase zero, and the trade-off between roll reduction and yaw interand the spectrum of the roll-induced wave motion and thepresence trade-off between reduction and yaw interference. Its potential for aaroll full control design was not, and rudder-to-roll the power power spectrum of has the roll-induced wave motion can change significantly due to changes in the sea state ference. Its potential for full control design was not, to the of non-minimum phase zero dynamics, can change significantly due to changes in the sea state the dynamics non-minimum phase zero, and the trade-off between roll reduction and yaw interand change the power spectrum due of the roll-induced waveseamotion ference. Its potential for aapropose full control design was not, can significantly to in ference. potential for fullreduction control design wasinternot, however investigated. We aa control can change significantly due to changes changes in the the sea state state and the sailing conditions and heading relative to however investigated. We control design that and the Its trade-off between roll anddesign yaw and the sailing conditions (speed and heading relative to power spectrum of(speed the wave ference. Its potential for apropose full control design was that not, can change significantly due to roll-induced changes in the seamotion state however investigated. We propose a control design that and the sailing conditions (speed and heading relative to however investigated. We propose a control design that uses a simplified roll disturbance model based on a secondand the sailing conditions (speed and heading relative to the waves). It has been shown in Perez (2005) that there uses a simplified roll disturbance model based on a secondference. Its potential for a full control design was not, the waves). It has been shown in Perez (2005) that there can change significantly due to changes in the sea state however investigated. We propose a control design that and the sailing conditions (speed and heading relative to uses aa shaping simplified roll disturbance disturbance model based based on a aaction secondthe It been in Perez (2005) that there uses simplified roll model on secondorder filter, and incorporates integral in the waves). waves). It has has been shown shown inthe Perez (2005) relative that there are fundamental limitations in achievable extent of order shaping filter, and it incorporates integral action in however investigated. Weit propose a control design that are fundamental limitations in the achievable extent of and the sailing conditions (speed and heading to uses a simplified roll disturbance model based on a secondthe waves). It has been shown in Perez (2005) that there order shaping filter, and it incorporates integral action in are fundamental limitations in the achievable extent of order shaping filter, and it incorporates integral action in yaw. This design provides opportunities for adapting the are fundamental limitations in the achievable extent of roll reduction that depends on the frequency of the nonyaw. This design provides opportunities for adapting the uses a simplified roll disturbance model based on a secondroll reduction that depends on the frequency of the nonthe waves). It has been shown in Perez (2005) that there order shaping filter, and it incorporates integral action in are reduction fundamental limitations in the thefrequency achievableof extent of yaw. This design provides opportunities for adapting the roll that depends on the nonyaw. This design provides opportunities for adapting the performance due to changes in environmental and sailing roll reduction that depends on the frequency of the nonminimum phase zero and the range of frequencies at which performance due to changes in environmental and sailing order shaping filter, and it incorporates integral action in minimum phase zero and the range of frequencies at which are fundamental limitations in the achievable extent of yaw. This design provides opportunities for adapting the roll reduction that depends on the frequency of the nonperformance due changes in environmental and minimum phase zero and range of frequencies at performance due to to changes in is environmental and sailing sailing conditions. However, the latter outside the scope of this minimum phase zerodepends and the the on range offrequency frequencies at which which conditions. However, the latter is outside the scope of this yaw. This design provides opportunities for adapting the roll reduction that the of the nonperformance due to changes in environmental and sailing This work has been supported by the Australian Department of minimum phase zero and the range of frequencies at which conditions. However, However, the the latter latter is is outside outside the the scope scope of of this this This work has been supported by the Australian Department of conditions. paper. This has been the Australian Department of paper. performance due to changes in is environmental and sailing Defence through research agreement with the Maritime Division of minimum phase zerosupported and the by range of frequencies at which conditions. However, the latter outside the scope of this This work work has a supported by the Australian Department Defence through abeen research agreement with the Maritime Division of paper. This work has been supported by the Australian Department of paper. Defence through a research research agreementGroup. with the the Maritime Maritime Division Division of of the Defence Science and Technology conditions. However, the latter is outside the scope of this Defence through a agreement with paper. the Defence Science andsupported Technology This Defence through research agreement with the Maritime Division of work has abeen by Group. the Australian Department the the Defence Defence Science Science and and Technology Technology Group. Group. paper. the Defence Science and Technology Defence through a research agreementGroup. with the Maritime Division of
Copyright © Science 2018 457 the Defence and (International Technology Group. 2405-8963 © 2018, IFAC IFAC Federation of Automatic Control) Copyright 2018 IFAC 457 Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 457 Copyright © under 2018 IFAC IFAC 457 Control. Peer review© responsibility of International Federation of Automatic Copyright © 2018 IFAC 457 10.1016/j.ifacol.2018.09.445 Copyright © 2018 IFAC 457
IFAC CAMS 2018 458 Christina Kazantzidou et al. / IFAC PapersOnLine 51-29 (2018) 457–462 Opatija, Croatia, September 10-12, 2018
2. CONTROL-TO-MOTION MODEL We consider the model structure from Perez (2005) for the problem of rudder roll stabilisation and course keeping of a naval patrol boat. The linear model of the response of a marine craft to the control action in restricted degrees of freedom (DOF) is described by (Perez, 2005): η˙ ≈ ν, (1a) (1b) M ν˙ = D(U ) ν + G η + τc , where ν [ v, p, r ] represents the generalised body-fixed velocity, and η [ φ, ψ ] gives the attitude. The vector τc [ Y (α, U ), K(α, U ), N (α, U ) ] has as components the sway force and the roll and yaw moments generated by the rudder motion. These components depend on the rudder angle α and the average forward speed U . The matrices M, D(U ), G are the total mass, damping, and restoring matrices, respectively. Let us define the following state, control, and output variables: x [ v, p, r, φ, ψ ] , u α, y [ φ, ψ ] . Then the state-space representation in (1) becomes: x˙ = A x + B u, (2a) y = C x, (2b) where A Ms−1 F (U ), B Ms−1 H(U ), C 00 00 00 10 01 , Ms
M 03,2 , 02,3 I2
m − Yv˙ −(m zG + Yp˙ ) m xG − Yr˙ −Kr˙ M −(m zG + Kv˙ ) Ixx − Kp˙ , m xG − Nv˙ −Np˙ Izz − Nr˙ f11 (U ) 0 f13 (U ) f14 (U ) 0 f21 (U ) f22 (U ) f23 (U ) f24 (U ) 0 F (U ) f31 (U ) f32 (U ) f33 (U ) f34 (U ) 0 , 0 1 0 0 0 0 0 1 0 0
f11 (U ) Y|u|v |U |, f21 (U ) K|u|v |U |, f31 (U ) N|u|v |U |, f22 (U ) Kp + K|u|p |U |, f32 (U ) Np + N|u|p |U |,
nφ (s) Kroll (s − q1 ) (s + q2 ), nψ (s) Kyaw (s + q3 ) s2 + 2 ξq ωq s + ωq2 , d(s) (s + p1 ) (s + p2 ) s2 + 2 ξp ωp s + ωp2 .
(4a) (4b) (4c)
The rudder-to-roll transfer function Gφα (s) is stable with a non-minimum phase zero at q1 and the rudder-to-yaw transfer function Gψα (s) is marginally stable. 3. INTERNAL MODEL CONTROL (IMC) The IMC design considers the two transfer functions Gφα (s) and Gψα (s) plus a first-order model for the rudder machinery dynamics Gα (s) in the configuration shown in Fig. 1. We assume that the pole of the first-order system of the steering machinery Gα (s) is 5 times larger than the dominant poles of the system and therefore it can be neglected for initial design, i.e., Gα (s) = 1 (Perez, 2005). The desired roll angle is 0 and the desired yaw angle is denoted by ψd . The input disturbance is denoted by δi , and the output disturbances by δφ and δψ . In this paper, we consider both sinusoidal disturbances of the form δ (t) = A sin(ω t), with = φ, ψ, and stochastic output disturbances with rational spectrum approximations. The stochastic output disturbances can be described by δ (s) = H (s) w(s), K 2 ξ ω σ ,
f13 (U ) (Yur − m) U, f23 (U ) (Kur + m zG ) U,
H (s) = φ, ψ,
s2
K s , + 2 ξ ω s + ω 2 (5)
where H (s) is a second-order shaping filter and w(s) represents the Laplace transform of unit-variance widebanded noise.
f33 (U ) N|u|r |U | − m xG U, f14 (U ) Yφuu U 2 ,
f24 (U ) Kφuu U 2 − ρ g ∇ GM t, f34 (U ) Nφu|u| U |U |, 1 −rr 1 ∂CL , H(U ) −LCG ρ ur (U )2 Ar ∂α α=0 0 2 0
where Ar is the area of the rudder, CL is the lift coefficient, ur (U ) is the flow speed over the rudder, LCG is the lateral centre of gravity of the vessel, and rr is the roll lever arm, see Perez (2005) for more details on the notation. The transfer function matrix is computed by: nφ (s) G (s) d(s) φα G(s) C (s I5 − A)−1 B = nψ (s) Gψα (s) s d(s) where
Fig. 1. Simplified rudder stabiliser control scheme.
, (3) 458
The control objective is to construct a controller C(s) [ Cφ (s) Cψ (s) ], as shown in Fig. 1, to reduce the roll amplitude and control the heading of the marine craft. The transfer functions Cφ (s) and Cψ (s) represent the roll and yaw controllers, respectively. The class of controllers C(s) giving rise to an internally stable closed-loop system can be parameterised as: C(s) = (1 − Q(s) G(s))−1 Q(s), (6)
where Q(s) [ Qφ (s) Qψ (s) ] is a stable and proper rational row vector, see e.g. Goodwin et al. (2000). The plant G(s) is not stable due to the integrator in Gψα (s) and to ensure internal stability further interpolation constraints must be satisfied for Q(s) (Goodwin et al., 2000): (i) Qφ (0) = Qψ (0) = 0; and (ii) Qψ (s) Gψα (s) = 1 at s = 0.
IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018 Christina Kazantzidou et al. / IFAC PapersOnLine 51-29 (2018) 457–462
4. MAIN RESULT In this section, we seek to extend the work of Goodwin et al. (2000) by constructing Q(s) in (6) based on a secondorder shaping filter Hφ (s) that characterises the roll disturbance, and roll sensitivity shaping. In addition, the proposed design has integral action in the yaw controller to maintain the course of the marine craft in the presence of constant input disturbance. For the design of the controller C(s) in (6), we propose the following form for Q(s): γ1 s2 d(s) γ2 s (s + ζ) d(s) Q(s) , (7) dQ (s) dQ (s) where dQ (s) is a stable (yet unknown) polynomial of degree 6, so that Q(s) is proper, and γ1 , γ2 and ζ are parameters to be chosen by the designer. The selection of Q(s) as in (7) ensures that interpolation constraint (i) is satisfied by construction, and interpolation constraint (ii) is satisfied provided that dQ (0) = γ2 ζ nψ (0). With this choice of Q(s), the controller is computed by C(s) = [ Cφ (s) Cψ (s) ], where γ1 s2 d(s) , dQ (s) − γ1 nφ (s) − γ2 (s + ζ) nψ (s) γ2 s (s + ζ) d(s) . Cψ (s) = dQ (s) − γ1 s2 nφ (s) − γ2 (s + ζ) nψ (s) The roll sensitivity is computed by 1 + Gψα (s) Cψ (s) Sφφ (s) = , 1 + Gφα (s) Cφ (s) + Gψα Cψ (s) and, after some computations, we obtain Cφ (s) =
s2
(8a) (8b)
(9)
nφ (s) γ1 s2 d(s) Sφφ (s) = 1 − Gφα (s) Qφ (s) = 1 − d(s) dQ (s) Kroll (s − q1 ) (s + q2 ) γ1 s2 =1− . (10) dQ (s) Let us choose dQ (s) = Kroll (s + q1 ) (s + q2 ) a(s),
(11)
where a(s) s + a3 s + a2 s + a1 s + a0 is a stable (yet unknown) polynomial. With this choice of dQ (s), the roll sensitivity is given by: (s − q1 ) γ1 s2 . (12) Sφφ (s) = 1 − (s + q1 ) a(s) 4
3
2
According to the Internal Model Principle, steady state disturbance compensation requires that the generating polynomial, i.e., the denominator of the disturbance shaping filter, is included in the controller denominator polynomial (Goodwin et al., 2001). Therefore the desired denominator of C(s) has to include the dynamics of the roll disturbance in order to reduce the effect of it. Thus, the desired denominator of C(s) is chosen, using the denominator of Hφ (s), as follows: dC (s) Kroll s2 d˜C (s) = Kroll s2 s2 + 2 ξφ ωφ s + ωφ2 b(s), (13) where b(s) s2 + b1 s + b0 is a stable (yet unknown) polynomial multiplied by Kroll to have the same leading coefficient as dQ (s). The factor s2 is included in dC (s) 459
459
to incorporate integral action in yaw, as it can be seen by using (13) in the controllers Cφ (s) and Cψ (s) given in (8). Expanding the polynomials dC (s) and dQ (s) − γ1 s2 nφ (s) − γ2 (s + ζ) nψ (s) and equating the coefficients corresponding to the same degrees, we may solve the following system of six linear equations with six unknowns: x ˆ [ a3 , a 2 , a 1 , a 0 , b1 , b 0 ] Px ˆ = c,
(14)
where
1 0 0 0 −1 0 1 0 0 −2 ξφ ωφ −1 q1 + q2 1 0 −ωφ2 −2 ξφ ωφ q1 q2 q1 + q2 P , 2 q1 q2 q1 + q 2 1 0 −ωφ 0 0 0 q 1 q2 q1 + q2 0 0 0 0 0 q 1 q2 0 0 c [ c1 , c2 , c3 , c4 , c5 , c6 ] ,
c1 2 ξφ ωφ − (q1 + q2 ), γ2 Kyaw , c2 ωφ2 − q1 q2 + γ1 + Kroll γ2 Kyaw (2 ξq ωq + q3 + ζ), c3 γ1 (q2 − q1 ) + Kroll γ2 Kyaw 2 ωq + 2 ξq ωq (q3 + ζ) + q3 ζ , c4 −γ1 q1 q2 + Kroll γ2 Kyaw 2 ωq (q3 + ζ) + 2 ξq ωq q3 ζ , c5 Kroll γ2 Kyaw q3 ζ ωq2 c6 . Kroll It can be easily shown that P is non-singular since its columns are linearly independent. Hence, the solution of (14) is given by: x ˆ = P −1 c. (15) The solution is acceptable provided that a(s) in (11) is stable and b1 , b0 > 0. The parameters γ1 , γ2 and ζ can be chosen such that the solution is acceptable. Alternatively, we can compute a0 , a1 by c6 c5 (q1 + q2 ) c6 a0 = , a1 = − (16) q1 q2 q1 q 2 q12 q22 and then compute the remaining four unknowns by: −1 1 0 −1 0 a3 c1 1 −2 ξφ ωφ −1 q1 + q 2 a2 c2 b = q q q +q −ωφ2 −2 ξφ ωφ c7 , 1 2 1 2 1 b0 c8 0 q1 q 2 0 −ωφ2
where c7 c3 − a1 , c8 c4 − (q1 + q2 ) a1 − a0 . Notice that interpolation constraint (ii) is satisfied since dQ (0) = Kroll q1 q2 a0 = γ2 ζ Kyaw q3 ωq2 = γ2 ζ nψ (0). Finding the solution x ˆ, the controller can be written as: γ1 d(s) γ2 (s + ζ) d(s) . (17) C(s) = Kroll d˜C (s) Kroll s d˜C (s) To assess the performance of a roll stabilising controller, the commonly used measures are the percentage of reduction of roll variance: 100 (1 − var[φcl ]/var[φol ]), and the
IFAC CAMS 2018 460 Christina Kazantzidou et al. / IFAC PapersOnLine 51-29 (2018) 457–462 Opatija, Croatia, September 10-12, 2018
percentage of reduction of roll root mean square (RMS) value: 100 (1 − var[φcl ]/ var[φol ]), where φol and φcl denote the open-loop and closed-loop roll angles, respectively.
is Aφ = 0.26 for roll and Aψ = 0.014 for yaw. The parameters of the shaping filters that characterise the stochastic disturbances are σφ = 0.275, ξφ = 0.13, ωφ = 0.89,
5. CASE STUDY
σψ = 0.0147, ξψ = 0.13, ωψ = 0.911.
We consider the benchmark example of a naval vessel. Using the values of Appendix B in Perez (2005) and the Marine GNC Toolbox by Fossen and Perez (2004), the state-space model is described by the triple (A, B, C), where −0.1472 0.182 −1.2881 1.1565 0 0.0276 −0.2031 −1.1976 −1.2962 0 A = −0.0115 0.0037 −0.3481 0.0176 0 , 0 1 0 0 0 0 0 1 0 0 0.3267 −0.1339 00010 B = −0.0288 , C = . 00001 0 0
We choose γ1 = 3, γ2 = 1, ζ = 0.2. The polynomial dQ (s) is given in (11), where
The rudder-to-roll and rudder-to-yaw transfer functions nφ (s) n (s) , Gψα (s) = s ψd(s) , where nφ (s), nψ (s), are Gφα (s) = d(s) and d(s) are given in (4) with
a(s) = s4 + 2.0447 s3 + 4.0215 s2 + 0.3597 s + 0.1772. The controller C(s) is given in (17), where d˜C (s) = (s2 + 0.2314 s + 0.7921) (s2 + 2.3552 s + 0.6433). 5.3 Bow seas In the case of bow seas, the encounter angle is assumed to be χ = 135◦ . The amplitude of the sinusoidal disturbances is Aφ = 0.14 for roll and Aψ = 0.013 for yaw. The parameters of the shaping filters that characterise the stochastic disturbances are σφ = 0.1954, ξφ = 0.08, ωφ = 0.9395, σψ = 0.018, ξψ = 0.08, ωψ = 0.9395. We choose γ1 = 3, γ2 = 1, ζ = 0.1. The polynomial dQ (s) is given in (11), where a(s) = s4 + 2.049 s3 + 3.9322 s2 + 0.6229 s + 0.0886. The controller C(s) is given in (17), where d˜C (s) = (s2 + 0.1503 s + 0.8827) (s2 + 2.4406 s + 0.6439).
Kroll = −0.1339, Kyaw = −0.02883, q1 = 0.186,
q2 = 0.3559, q3 = 0.2001, ξq = 0.1275, ωq = 1.1668,
p1 = 0.0415, p2 = 0.4403, ξp = 0.0946, ωp = 1.1445. We design IMC controllers for quartering seas, beam seas, and bow seas, and analyse the performances based on numerical simulations. We assume the following wave conditions: significant wave height H1/3 = 2.5 m, and average wave period T1 = 7.5 s. The rudder angle α cannot exceed 36◦ , and the average forward speed U is assumed to be 8 m/s. 5.1 Quartering seas In the case of quartering seas, the encounter angle is assumed to be χ = 45◦ . The amplitude of the sinusoidal disturbances is Aφ = 0.075 for roll and Aψ = 0.0045 for yaw. The parameters of the shaping filters that characterise the stochastic disturbances are σφ = 0.313, ξφ = 0.03, ωφ = 0.43, σψ = 0.018, ξψ = 0.03, ωψ = 0.43. We choose γ1 = 1.5, γ2 = 0.1, ζ = 0.1. The polynomial dQ (s) is given in (11), where a(s) = s4 + 2.8349 s3 + 1.2254 s2 + 0.0623 s + 0.0089. The controller C(s) is given in (17), where d˜C (s) = (s2 + 0.0258 s + 0.1849) (s2 + 3.351 s + 1.035). 5.2 Beam seas In the case of beam seas, the encounter angle is assumed to be χ = 90◦ . The amplitude of the sinusoidal disturbances 460
The parameters γ1 and γ2 were chosen such that γ1 > γ2 and for quartering seas the choice of small γ2 increased the roll reduction. The parameter ζ was chosen such that the polynomials a(s) and b(s) are stable. For values of ζ between 0.1 and 0.2 the stability requirement is satisfied and the integral action in yaw is effective. 5.4 Performance analysis The roll sensitivities for quartering, beam, and bow seas using the designed controllers are shown in Fig. 2. For quartering seas, the roll sensitivity is below 1 for frequencies approximately between 0.253 and 1.224 with minimum value 0.027 at frequency 0.43 rad/s. There is at least 50% attenuation in |Sφφ | for frequencies between 0.324 to 0.616 rad/s. For beam seas, the roll sensitivity is below 1 for frequencies approximately between 0.382 and 1.783 with minimum value 0.0977 at frequency 0.864 rad/s. There is at least 50% attenuation in |Sφφ | for frequencies from 0.545 to 1.288 rad/s. For bow seas, the roll sensitivity is below 1 for frequencies approximately between 0.4 and 1.751 with minimum value 0.0658 at frequency 0.933 rad/s. There is at least 50% attenuation in |Sφφ | for frequencies from 0.611 to 1.309 rad/s. The controller will be switched off for frequencies below 0.25 rad/s to avoid large disturbance amplification. The open and closed-loop roll angle, yaw angle and rudder angle for quartering, beam, bow seas and sinusoidal output disturbances are shown in Fig. 3, 4, 5, respectively. The open and closed-loop roll angle, yaw angle and rudder
IFAC CAMS 2018 Opatija, Croatia, September 10-12, 2018 Christina Kazantzidou et al. / IFAC PapersOnLine 51-29 (2018) 457–462
Roll Angle [°]
S 4.5 Quartering Seas Beam Seas Bow Seas
4
Yaw Angle [°]
3 2.5 2
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1.5
0
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5
Time [s]
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Magnitude (abs)
3.5
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Fig. 2. Roll sensitivity (χ = 45◦ , 90◦ , 135◦ ).
Table 1. Roll reduction (δi = 0).
Sinusoidal
Stochastic
χ=
90◦
χ=
135◦
RR
χ=
Var
96.40 %
91.22 %
99.14 %
RMS
80.98 %
70.35 %
90.70 %
Var
95.61 %
85.41 %
91.85 %
RMS
79.05 %
61.81 %
71.45 %
Roll Angle [°]
Beam Seas (Sinusoidal Disturbances)
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Time [s]
Rudder Angle [°]
Disturbance
45◦
Fig. 3. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 45◦ , sinusoidal disturbances).
Yaw Angle [°]
angle for quartering, beam, bow seas and stochastic disturbances are shown in Fig. 6, 7, 8, respectively. We run two simulations for each case, assuming that there is no input disturbance and a 3◦ input disturbance, which represents an equivalent yaw disturbance torque induced by wind. The percentages of roll reduction with respect to variance and RMS are shown in Tables 1 and 2. In the presence of constant input disturbance, it is observed that the roll reduction is smaller, yet we have course keeping due to the integral action in the yaw controller.
10
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Table 2. Roll reduction (δi = 3◦ ).
Sinusoidal
Stochastic
RR
χ = 45◦
χ = 90◦
χ = 135◦
Var
93.41 %
89.46%
99.20 %
RMS
74.28 %
67.52%
91.03 %
Var
93.78 %
84.81 %
91.80 %
RMS
75.06 %
61.03 %
71.37 %
Fig. 4. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 90◦ , sinusoidal disturbances). Roll Angle [°]
Disturbance
Bow Seas (Sinusoidal Disturbances)
10 0 -10
0
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In this paper, we design a controller for rudder roll stabilisation and course keeping via Internal Model Control. The approach is based on the Youla parameterisation, roll sensitivity shaping, and integral action in the yaw controller for course keeping. A procedure is given for the construction of Q(s) to include integral action in yaw and the computation of the controller gains. A demonstration of the method is provided in the form of a case study of the benchmark example (see Perez (2005)) where the proposed controller is tested in different sailing conditions. In particular, three controllers are designed for the cases of quartering, beam, and bow seas with 461
Rudder Angle [°]
6. DISCUSSION AND CONCLUSIONS
Yaw Angle [°]
Time [s] 5
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Fig. 5. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 135◦ , sinusoidal disturbances).
Roll Angle [°]
IFAC CAMS 2018 462 Christina Kazantzidou et al. / IFAC PapersOnLine 51-29 (2018) 457–462 Opatija, Croatia, September 10-12, 2018
encounter angles χ = 45◦ , 90◦ , 135◦ , respectively. We consider sinusoidal and stochastic disturbances for the roll and yaw angles and run simulations assuming there is no input disturbance and small input disturbance. There is satisfactory roll reduction and the course keeping is maintained in the presence of constant input disturbance.
Quartering Seas (Stochastic Disturbances)
5 0 -5
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Rudder Angle [°]
Yaw Angle [°]
Time [s] 20
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Closed loop (input dist)
Future research will consider an optimisation algorithm for the computation of the parameters γ1 , γ2 and ζ. A selection of a larger degree for the denominator dQ (s) of Q(s) will be also considered. To increase the effectiveness of the roll stabilisation control over a wide envelope of sea states and sailing conditions, an adaptive scheme to track the changes in the parameters of the disturbance shaping filter as well as a switched system will be considered. Finally, a natural topic for future work is the design of automatic gain control scheme to reduce the control effort for the proposed method.
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Roll Angle [°]
Fig. 6. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 45◦ , stochastic disturbances). Beam Seas (Stochastic Disturbances)
20 0 -20
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200
Rudder Angle [°]
Yaw Angle [°]
Time [s] 5
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Fig. 7. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 90◦ , stochastic disturbances). Bow Seas (Stochastic Disturbances)
10 0 -10
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Rudder Angle [°]
Yaw Angle [°]
Time [s] 5
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Fig. 8. Open and closed-loop roll angle, yaw angle and rudder angle (χ = 135◦ , stochastic disturbances). 462
REFERENCES Blanke, M., Adrian, J., Larsen, K., and Bentsen, J. (2000), Rudder roll damping in coastal region sea conditions. In Proceedings of the 5th IFAC Conference on Manoeuvring and Control of Marine Craft, Aalborg, Denmark. Fossen, T.I. (2011), Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley & Sons, Chichester, West Sussex, UK. Fossen, T.I., and Perez, T. (2004), Marine Systems Simulator (MSS),