Wavelet-based Controller Design for Dynamic Positioning of Vessels

Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th World Congress Control The of Toulouse, France,Federation July 9-14, 2017 Available online at www.sciencedirect.com The International International Federation of Automatic Automatic Control The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

ScienceDirect

IFAC PapersOnLine 50-1 (2017) 1133–1138

Wavelet-based Controller Design for Wavelet-based Controller Design for Wavelet-based Controller Design for Dynamic Positioning of Vessels Dynamic Positioning of Vessels Dynamic Positioning of∗,1 Vessels ∗ ∗

Awantha Jayasiri ∗ Salim Ahmed ∗,1 Syed Imtiaz ∗ ∗ Awantha Jayasiri ∗∗ Salim Salim Ahmed ∗,1 Syed Imtiaz Awantha Awantha Jayasiri Jayasiri Salim Ahmed Ahmed ∗,1 Syed Syed Imtiaz Imtiaz ∗ ∗ Department of Process Engineering ∗ ∗ Department of Process Engineering Process ∗ Department Centre for Risk Integrity of and SafetyEngineering Engineering (C-RISE) Department of Process Engineering Centre for Risk Integrity and Safety Engineering (C-RISE) Centre for Risk Integrity and Safety (C-RISE) Memorial University of Newfoundland, St. John’s, NL, Canada Centre for Risk Integrity and Safety Engineering Engineering (C-RISE) Memorial University of Newfoundland, Newfoundland, St. John’s, John’s, NL, Canada Memorial University of St. NL, Memorial University of Newfoundland, St. John’s, NL, Canada Canada Abstract: This paper presents design and evaluation of a wavelet-based multi-resolution Abstract: paper presents design and evaluation of a wavelet-based Abstract: This paper presents design evaluation of multi-resolution proportionalThis integral derivative controller (MRPID) for dynamic positioning multi-resolution (DP) of vessels Abstract: This paper presentscontroller design and and evaluation of aa wavelet-based wavelet-based multi-resolution proportional integral derivative (MRPID) for dynamic positioning (DP) of vessels proportional integral derivative controller (MRPID) for dynamic positioning (DP) of under noise and environmental disturbances. In the proposed MRPID controller, the errors proportional integral derivative controller (MRPID) for dynamic positioning (DP) of vessels vessels under noise and environmental disturbances. In the proposed MRPID controller, the errors under noise and environmental disturbances. In the proposed MRPID controller, the errors with position and orientation are decomposed into different frequency components using discrete under noise and disturbances.into In different the proposed MRPID controller, thediscrete errors with position and environmental orientation are decomposed decomposed frequency components using with position and orientation are into different frequency components using discrete wavelet transform. The different scales of wavelet transform represent details such as external with position and orientation are decomposed into different frequency components using discrete wavelet transform. The different scales of wavelet transform such as external wavelet transform. The different scales transform represent details such as external disturbances, process noise and measurement noise. A set of represent sub-PID details controllers are assigned wavelet transform. The different scales of of wavelet wavelet transform represent details suchare as assigned external disturbances, process noise and measurement noise. A set of sub-PID controllers disturbances, process noise and measurement noise. A set of sub-PID controllers are assigned to selected error components and the commands generated by all controllers are added together disturbances, process noise and measurement noise. A set of sub-PID controllers are assigned to selected components and the generated byofall controllers are added together to selected error components and commands generated controllers are produceerror the control command forcommands dynamic positioning the vessel. Performance of the to selected error components and the the commands generated by byofall allthe controllers are added added together together to produce the control command for dynamic positioning vessel. Performance ofwith the to produce the control command for dynamic positioning of the vessel. Performance the controller is the evaluated incommand simulationforand compared with theofconventional PID controllerof to produce control dynamic positioning the vessel. Performance of the controller is evaluated in simulation and compared with the conventional PID controller with controller is evaluated in simulation and compared with the conventional PID controller with acceleration feedback. Nonlinear state estimation is performed by using an unscented Kalman controller is evaluated in simulation and compared with the conventional PID controller with acceleration feedback. Nonlinear state estimation is performed by using an unscented Kalman acceleration feedback. Nonlinear state is filter assuming a typical sensor package for localization. acceleration feedback. Nonlinear state estimation estimation is performed performed by by using using an an unscented unscented Kalman Kalman filter assuming a typical sensor package for localization. filter assuming a typical sensor package for localization. filter assuming a typical sensor package for localization. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Dynamic positioning, wavelet transform, multi resolution control, unscented Keywords: Dynamic Keywords: Dynamic positioning, positioning, wavelet wavelet transform, transform, multi multi resolution resolution control, control, unscented unscented Kalman filter Keywords: Dynamic positioning, wavelet transform, multi resolution control, unscented Kalman filter Kalman filter Kalman filter 1. INTRODUCTION of measurement noise, external disturbances and process 1. of measurement and process 1. INTRODUCTION INTRODUCTION of measurement noise, external disturbances and process dynamics, which noise, are inexternal differentdisturbances scales (Parvez Gao, 1. INTRODUCTION of measurement noise, external disturbances andand process which are in different scales (Parvez and Gao, dynamics, which are in different scales (Parvez and Gao, In Dynamic Positioning (DP), a vessel is kept in a fixed dynamics, 2005). Using this multi-resolution decomposition property dynamics, which are in different scales (Parvez and Gao, In Dynamic Positioning (DP), a vessel is kept in a fixed 2005). Using this multi-resolution decomposition property In Dynamic Positioning (DP), a vessel is kept in a fixed 2005). Using this multi-resolution decomposition property position and heading by providing active thrusts (Fossen, a direct application of wavelet transform to controller In Dynamic Positioning (DP), a vessel is kept in a fixed 2005). Using this multi-resolution decomposition property position and heading by active thrusts (Fossen, of wavelet transform to controller position andvessel’s heading by providing providing active thrusts (Fossen, aa direct direct application of wavelet transform to controller 2011). The propulsion system must withstand the adesign wasapplication proposed by Gao (2005) the position and heading by providing active thrusts (Fossen, direct application of Parvez waveletand transform to where controller 2011). The vessel’s propulsion system must withstand the design was proposed by Parvez and Gao (2005) where the 2011). The vessel’s propulsion system must withstand the design was proposed by Parvez and Gao (2005) where the environmental disturbances such as current, ice load, wind error signal is separated into several components, multi2011). The vessel’s propulsion system must withstand the design was proposed by Parvez and Gao (2005) where the environmental disturbances such as current, ice load, wind error signal is separated into several components, multienvironmental disturbances such as current, ice load, wind error signal is separated into several components, multiand waves and should perform the positioning within an plied with respective gains and then added together to environmental disturbances such as current, ice load, wind error signal is separated into several components, multiand waves and should perform the positioning within an plied with respective gains and then added together to and waves and should perform the positioning within an plied with respective gains and then added together to acceptable tolerance. DP is an important operation in generate the control command. Following this approach, and waves and should DP perform the positioning within an plied withtherespective gains and Following then added together to acceptable tolerance. is an important operation in generate control command. this approach, acceptable tolerance. DP is an important operation in generate the control command. Following this approach, applicationstolerance. such as low speed trajectory following, staseveral applications are reported in the literature such acceptable DP is an important operation in generate the control command. Following this approach, applications such as low speed trajectory following, staapplications are reportedcontroller in the literature such applications such as low low speed speedoftrajectory trajectory following, sta- several several applications are in such tion keeping such and development offshore oil and mineral as development of wavelet-based for permanent applications as following, staseveral applications are reported reportedcontroller in the the literature literature such tion keeping and development of offshore oil and mineral as development of wavelet-based for permanent tion keeping and development of offshore oil and mineral as development of wavelet-based controller for permanent resources. Various control strategies for DP systems have magnetic motor drives (Khan and Rahman, 2008), contion keeping and development of offshore oil and mineral as development of wavelet-based controller for permanent resources. Various control strategies for DP systems have magnetic motor drives (Khan and Rahman, 2008), conresources. Various control strategies for DP systems have magnetic motor drives (Khan and Rahman, 2008), conbeen developed and implemented. A comprehensive study trolling a robotic manipulator (Tolentino et al., 2012) and resources. Various control strategies for DP systemsstudy have magnetic motor drives (Khan (Tolentino and Rahman, 2008), conbeen developed and implemented. A comprehensive trolling a robotic manipulator et al., 2012) and been developed and implemented. A comprehensive study trolling a robotic manipulator (Tolentino et al., 2012) and of previous approaches including PID control, fuzzystudy and recently load frequency control of power systems (Kumar been developed and implemented. A comprehensive trolling aload robotic manipulator (Tolentino et al., 2012) and of previous approaches including PID control, fuzzy and recently frequency control of power systems (Kumar of previous approaches including PID control, fuzzy and recently load frequency control of power systems (Kumar optimal control methods are reported in (Sørensen, 2011). and Ramana, 2016). One disadvantage of this approach of previous approaches including PID control, fuzzy and recently load frequency control of power systems (Kumar optimal control methods are reported in (Sørensen, 2011). and Ramana, 2016). One disadvantage of this approach optimal control methods are control reportedfor in DP (Sørensen, 2011). is and Ramana, 2016). One disadvantage A modelcontrol reference adaptive incorporating that the gain tuning mainly relies of onthis trialapproach and eroptimal methods are reported in (Sørensen, 2011). and Ramana, 2016). Oneis disadvantage of this approach A model reference adaptive control for DP incorporating is that the gain tuning is mainly relies on trial and erA model reference adaptive control for DP incorporating is that the gain tuning is mainly relies on trial and erhybrid force control is presented by Osthus (2014). Reror method. To mitigate this a self-tuning method A modelforce reference adaptive controlbyforOsthus DP incorporating is that the gain tuning is this mainly relies on trial and for erhybrid control is presented (2014). Reror method. To mitigate a self-tuning method for hybrid force control is presented by Osthus (2014). Reror method. To mitigate this a self-tuning method for cently, Du et al. (2015)ispresented a by robust output feedback arorwavelet PID controller is proposed by (Cruz-Tolentino hybrid force control presented Osthus (2014). Remethod. To mitigate this a self-tuning method for cently, Du et al. (2015) presented a robust output feedback PID controller proposed by (Cruz-Tolentino cently, Du by et al. al. (2015) presented presented robust output feedback aa wavelet wavelet PID controller is proposed controller combining a high-gain observer and feedback a neural aet al., 2010) using waveletis cently, Du et (2015) aa robust output wavelet PID controller isnetworks. proposed by by (Cruz-Tolentino (Cruz-Tolentino controller by combining aa high-gain observer and aa neural et al., 2010) using wavelet networks. controller by combining high-gain observer and neural et al., 2010) using wavelet networks. network. Also in (Bidikli et al., 2014), a robust tracking controller by combining a high-gain observer and a neural et al., 2010) using wavelet networks. the above mentioned works are based on pronetwork. in al., 2014), robust tracking network. Also in (Bidikli (Bidikliforet et DP al., vessels 2014), a robust tracking However, controllerAlso is developed with asymmetric However, the above mentioned works are based on pronetwork. Also in (Bidikli et al., 2014), aawith robust tracking However, the mentioned works are based proportional gain controller and hence may poorly controller is developed for DP vessels asymmetric However, the above above mentioned works areperform based on on procontroller is developed for DP vessels with asymmetric added mass terms, which affect the system dynamics in portional gain controller and hence may perform poorly controller is developed for DP vessels with asymmetric portional gain controller and hence may perform poorly at the presence of disturbances. Furthermore, introducing added mass terms, which affect the system dynamics in portional gain controller and hence may perform poorly added mass terms, which affect the system dynamics in acceleration level. DP control in ice has been reported at the presence disturbances. introducing added mass terms,DP which affect the system dynamics in at the of disturbances. Furthermore, introducing actionof intermediateFurthermore, frequency components in acceleration at the presence presence ofto disturbances. Furthermore, introducing acceleration level. DP control control in in ice ice has has been been reported reported in in derivative Nguyen et al.level. (2009). derivative action to intermediate frequency components in acceleration level. DP control in ice has been reported in derivative action to intermediate frequency components in the decomposed error signal may help to react smoothly Nguyen et al. (2009). derivative action to intermediate frequency components in Nguyen et et al. al. (2009). (2009). decomposed error signal may help to react smoothly Nguyen the decomposed error signal may help to react smoothly The wavelet transform provides multi-scale analysis for the to the fast acting load disturbances in DP. The main the decomposed error signal may help to react smoothly The wavelet transform provides the fast forces acting load disturbances in DP. The The wavelet transform provides multi-scale analysis for to the acting load in DP. main wideband signals and reveals localmulti-scale propertiesanalysis of data.for It to disturbance a vessel are wavemain and The wavelet transform provides multi-scale analysis for to the fast fast forces acting affecting load disturbances disturbances in wind, DP. The The main wideband signals and reveals local properties of data. It disturbance affecting a vessel are wind, wave and wideband signals and reveals local properties of data. It disturbance forces affecting a vessel are wind, wave has been used for signal de-noising over the past. A decurrent effects. While it is easy to measure wind speed and wideband signals and reveals local properties of data. It disturbance forces affecting a vessel are wind, wave and and has been used for signal de-noising over the past. A decurrent effects. While it is easy to measure wind speed has been used for signal de-noising over the past. A decurrent effects. While it is easy to measure wind speed and tailed introduction for signal and image de-noising using direction, the forces due toeasy waves, currents difficult has been used for signal de-noising over de-noising the past. A de- current effects. While it is to and measure windare speed and tailed introduction for signal and image using direction, the forces due to waves, and currents are difficult tailed introduction forpresented signal and and image de-noising using direction, forces due and difficult waveletintroduction transform isfor by image Ergen de-noising (2012). Also in to measurethe and not separable (Fannemel, 2008).are Moreover, tailed signal using direction, the forces due to to waves, waves, and currents currents are difficult wavelet transform is presented by Ergen (2012). Also in to measure and not separable (Fannemel, 2008). Moreover, wavelet transform is presented by Ergen (2012). Also in to measure and not separable (Fannemel, 2008). Moreover, (Chaplais et al., 2004) causal wavelet processing is perforces due to oscillating waves are of high frequency, while wavelet transform is presented by Ergen (2012). Also in to measure and not separable (Fannemel, 2008). Moreover, (Chaplais et al., 2004) causal wavelet processing is perforces due to oscillating waves are of high frequency, while (Chaplais et al., 2004) causal wavelet processing is perforces due to oscillating waves are of high frequency, while formed for de-noising the feedback signal for controlling current introduces a low frequency drift effect. This makes (Chaplais etde-noising al., 2004)the causal wavelet processing is per- forces due to oscillating waves are of high frequency, while formed for feedback signal for controlling current introduces a low frequency drift effect. This makes formed for de-noising the feedback signal for controlling current introduces a low frequency drift effect. This makes reaction wheel system. Applying the wavelet decompodesign ofintroduces conventional PID controller challenging. formed for de-noising the feedback signal for controlling current a low frequency drift effect. This makes reaction wheel system. Applying the wavelet decompodesign of conventional PID controller challenging. reaction wheel system. Applying the wavelet decompoof PID controller challenging. sition to wheel the error signalApplying extracts the the wavelet cumulative effect design reaction system. decompodesign of conventional conventional controllerthe challenging. The wavelet controllerPID decomposes error signal into sition to the error signal extracts the cumulative effect sition to the error signal extracts the cumulative effect 1 The wavelet controller decomposes the error signal into into sition totothe error signal extracts thebecumulative effect The author whom all correspondence should addressed. E-mail: wavelet controller decomposes the error signal different frequency components, scales the components 1 The wavelet controller decomposes the error signal into to all correspondence should 1 author [email protected], different frequency components, scales the components to whom whomTel: all 1-709-864-7652, correspondence Fax:1-709-864-8759 should be be addressed. addressed. E-mail: E-mail: 1 author different frequency components, scales the components author to whom all correspondence should be addressed. E-mail: [email protected], different frequency components, scales the components [email protected], Tel: Tel: 1-709-864-7652, 1-709-864-7652, Fax:1-709-864-8759 Fax:1-709-864-8759

[email protected], Tel: 1-709-864-7652, Fax:1-709-864-8759 Copyright © 2017, 2017 IFAC 1156Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 1156 Copyright ©under 2017 responsibility IFAC 1156Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 1156 10.1016/j.ifacol.2017.08.396

Proceedings of the 20th IFAC World Congress 1134 Awantha Jayasiri et al. / IFAC PapersOnLine 50-1 (2017) 1133–1138 Toulouse, France, July 9-14, 2017

Disturbances (ice load, current)

Sensor package (GNSS, IMU, magnetometers)

x, y, , PIDH

Thrust allocation algorithm

Force and torque inputs

f x , f y ,

PIDM1

 PIDMN1 PIDL

xˆ, xˆ yˆ, yˆ

Nonlinear State estimator using UKF

ˆ ,ˆ

MRPID controller

Wavelet Decomposition

e

the UKF-based state estimator; Section 3 develops the proposed MRPID controller; section 4 presents simulation results for the proposed control scheme and compares its performance with that for the conventional PID controller; Section 5 concludes the paper.

 

2. SYSTEM MODELING AND STATE ESTIMATION 2.1 Preliminaries and Notations

Set point

xd , yd , d

Fig. 1. Proposed approach for DP of vessels using UKFbased state estimation and MRPID-based control scheme by their respective gains and add together to generate the control signal (Khan and Rahman, 2008; Parvez and Gao, 2005). The DP system, having distinct frequency components, makes the wavelet controller to be an appropriate tool for its control. However, implementation of the wavelet controller to DP systems is yet to be tested. Wave filtering has been reported in the literature, however, for the purpose of filtering disturbances (Hassani et al., 2012, 2013). In this paper we develop a multi-resolution PID controller (MRPID) for DP of vessels introducing sub-PID controllers to selected components of the decomposed error signal. Our approach follows wavelet-based decomposition architecture presented by Parvez and Gao (2005) and neglects the high frequency components of the error signal. However, for low and intermediate frequency terms, it introduces a set of sub-PID controllers where final commands are generated by adding the commands from the sub controllers. As a result, the resolution of the controller will be increased and higher flexibility in controller design can be achieved. The Unscented Kalman Filter (UKF) (Dunik et al., 2012; Wan and van der Merwe, 2001) is used in the proposed method for nonlinear state estimation. The estimated states are used for generating the error signal in X, Y directions and yaw angle based on the set point. The wavelet-based decomposition is performed on these error signals and low and medium frequency terms of the error signal are used for calculating the control command while neglecting the high frequency terms, which mostly represent the noise components. Incorporating both nonlinear state estimation and MRPID control scheme, a complete solution is provided for DP of vessels and it is depicted in Fig. 1. Periodic back and forth motion of ships occurs due to the ocean waves. Correcting these wave induced motion requires high power and often worsens the wear and tear. Therefore, usually this motion is not compensated by the vessel’s DP control system and it is achieved by employing a wave filter, which estimates the vessel position without the wave induced motion (Veksler et al., 2016). In this work, we haven’t consider the wave filtering explicitly. The remainder of paper is organized as follows: Section 2 discusses the 3-DOF model of a ship and presents

Let us consider a simplified 3 Degrees Of Freedom (DOF) vessel model with dynamics and kinematics as follows (Fannemel, 2008): MRB v˙ + MA v˙ r + CRB (v)v + CA (vr )vr + D(vr )vr = τ (1) η˙ = J(ψ)v (2) where MRB ∈ R3×3 and MA ∈ R3×3 are the rigid body mass matrices representing the inertia of the vessel and the hydrodynamic added-mass matrix, CRB ∈ R3×3 and CA ∈ R3×3 are the rigid body Coriolis and centrifugal components as well as added-mass derivatives corresponding to the velocity coupling, the matrix D ∈ R3×3 includes energy dissipative terms due to the relative motion between vehicle and surrounding fluid, τ (= τc + τw + τi ) is the applied force with τc as the control input, τw is the wind force and τi as the ice load. Furthermore, η ∈ R3 is the position and orientation vector expressed in an inertial frame, J ∈ R3×3 is the rotational matrix between the inertial frame and the body frame. Also, η = [x y ψ]T where (x, y) is the horizontal position vector relative to the inertial frame and ψ is the vessel heading. v = [u ω]T are the body-fixed linear and angular velocities with u = [u1 u2 ] having u1 and u2 as the body-fixed velocities in surge and sway directions, respectively, and ω as the yaw rate. The term vr represents velocity relative to the current. The vessel is depicted in figure 2.





 

 

 

   

Fig. 2. A vessel navigation system The rotational matrix J(ψ) is computed as follows: [ ] cos(ψ) −sin(ψ) 0 J(ψ) = sin(ψ) cos(ψ) 0 (3) 0 0 1 Moreover, τw is assumed measured and τi is modeled as follows (Kjerstad and Skjetne, 2014): τi = J(ψ)T bi + wi b˙ i = wb

(4)

where bi is a Wiener process (Kjerstad and Skjetne, 2014) and wi and wb are zero mean Gaussian noise vectors.

1157

Proceedings of the 20th IFAC World Congress Awantha Jayasiri et al. / IFAC PapersOnLine 50-1 (2017) 1133–1138 Toulouse, France, July 9-14, 2017

Measurand

Frequency

Noise, σ

Accelerometer Gyroscope Attitude sensor DGPS sensor DVL

Specific force Angular velocity Orientation Position Linear velocity

100 Hz 100 Hz 10 Hz 10 Hz 10 Hz

0.01 m/s2 0.0005 rad/s 0.05 rad 0.1 m 0.01 m/s

2.2 Sensor modeling IMU measurements: The Inertial Measurement Unit (IMU) sensor consists of accelerometers and gyroscopes (gyros), which measure specific forces and angular rates in the body-fixed coordinate system. The accelerometer reading Za , can be modeled as (Miller et al., 2010): → (7) Z = u˙ + [ω×] u + J T − g +b +n a

a

a

where [ω×] is the skew symmetric matrix cross product form of the vector ω, which is given as: [ ] 0 −ω3 ω2 0 −ω1 , [ω×] = ω3 (8) −ω2 ω1 0 → where − g = [0 0 9.81 m/s2 ]T is the gravity vector, and ba = 0.01 m/s2 is the bias of the reading with b˙ a = 0. The measurement noise is distributed as: na ∼ N (0, Qa ), with Qa = σa2 as the variance for each direction. The gyro reading Zg , is modeled as (Miller et al., 2010): Zg = ω + bg + ng (9) where bg = 0.02 rad/s is the bias of the reading with b˙ g = 0. The noise of the reading is distributed as: ng ∼ N (0, Qg ), with Qg = σg2 as the variance on each axis. Attitude measurements: Attitude sensor composed of magnetometer (and compass) measures yaw angle. This sensor is modeled as: (10) Z ψ = ψ + nψ where the sensor noise is distributed as nψ ∼ N (0, Qψ ) having Qψ = σψ2 as the variance of ψ measurement. DVL sensor reading: The DVL update (in processed form) is modeled as: Zdvl = u + ndvl (11) where Zdvl is the DVL reading and the sensor noise is distributed as ndvl ∼ N (0, Qdvl ) with Qdvl = σdvl having 2 σdvl as the variance in surge and sway directions.

2.3 System modeling Incorporating the kinematics and measurements, the state-space equations can be written as follows:  ψk+1 = ψk + ∆t(Zg(k+1) − bg(k) )      bg(k+1) = bg(k)    2 ∆t J(ψk )uk (Za(k+1) − ba(k) ) pk+1 = pk + ∆tJ(ψk )uk +  2     ba(k+1) = ba(k)   ( )  uk+1 = uk + ∆t (Za(k+1) − ba(k) ) − [ωk ×] uk (13) The state vector consists of 9 states as follows: [ ]T y = η T v T bTg bTa (14) The process noise covariance matrix is given as:   0 0 0 J(σv2 )J T dt2 2  0 Qa dt 0 0  (15) R=  0 0 Qg 0 0 0 0 0 The measurement model is written as Z = h + Q where, as defined next, Z is the measurement vector, h is the measurement function, and Q is the measurement noise covariance. T (16) Z = [Zxy Zψ Zdvl ] T

(17) h = [p ψ u] ] [ 0 Qxy 0 0 Qψ 0 (18) Q= 0 0 Qdvl Figure 3 shows the bias estimations in gyroscope and accelerometers, which were set to 0.02rads−1 and 0.01ms−2 , respectively. The bias estimation shows quick convergence to the set values. The error plots shown in Fig. 4 and 5 show that the accuracy of state estimation is within an acceptable level. 0.04 0.02 0 −0.02 0 0.05

1000

2000

3000

4000

5000

6000

0 0 0.05

1000

2000

3000

4000

5000

6000

0 0

1000

2000

3000 4000 Time (× 0.1s)

5000

6000

a1

Sensor type

Position estimation using DGPS: The position estimation in processed from is represented as: Zxy = p + nxy (12) where Zxy is the DGPS reading about the ship’s horizontal position p and nxy ∼ N (0, Qxy ) with Qxy as the variance in (x, y) directions.

2 ba2(m/s )b (m/s2)bg(rad/s)

The 2D non-rotational current velocity is modeled as (Fannemel, 2008): ] [ Vc cos(βc ) (5) η˙ c = Vc sin(βc ) 0 where V c and βc are slowly-varying 1st-order GaussMarkov processes given as: V˙ c + µ1 (Vc − Vc0 ) = w1 (6) β˙ c + µ2 (βc − βc0 ) = w2 with µ1 , µ2 > 0, Vc0 , βc0 are the mean values and w1 , w2 are zero mean Gaussian variables. The vessel under consideration has a typical set of sensors as mentioned in Table 1. Table 1. Sensor characteristics

1135

Fig. 3. Gyro (bg ) and accelerometer (ba ) bias estimation 3. WAVELET-BASED MRPID CONTROL The Discrete Wavelet Transform (DWT) is calculated using the sub-band coding scheme (Parvez and Gao, 2005) using filters h(k) and g(k). The complete decomposition process is depicted in the figure 6.

1158

Proceedings of the 20th IFAC World Congress 1136 Awantha Jayasiri et al. / IFAC PapersOnLine 50-1 (2017) 1133–1138 Toulouse, France, July 9-14, 2017

−3

5

0

−5 0 0.02

err

% y

−13

x 10

Reconstruction error in xerr,yerr,ψerr signals

% xerr

5

1000

2000

3000

4000

5000

6000

0

% ψerr

−0.02 0 0.05

1000

2000

1000

2000

3000

4000

5000

6000

3000

4000

5000

6000

0

−0.05 0

Time (× 0.1s)

% u2err % u1err

0

% ωerr

−0.1 0 0.1

1000

2000

3000

4000

5000

6000

1000

2000

3000

4000

5000

6000

−0.1 0 0.01

1000

2000

3000

4000

5000

6000

0

−0.01 0

Time (× 0.1s)

Fig. 5. Error in body-fixed velocity estimation

h

c j 1,k

2

h

2

c j  2,k

f (k)  c j ,k

d j  2,k

Error in ψ

err

0

1000

2000

3000 4000 Time (× 0.01s)

5000

6000

in the presence of disturbances. Furthermore, the complete removal of the derivative action may not be effective as the derivative terms for some intermediate frequency components of the error signal may yield better control in a reactive environment. Hence, in this paper control commands, which represent the forces and torque input of the vessel (Fx , Fy , Tpsi ) are calculated as in equation (19).

0

d j  2,k

err

Error in yerr

Fig. 7. Reconstruction error in wavelet-based decomposition for three error signals:Xerr , Yerr and ψerr

0.1

d j 3,k

Error in x

−5 0

Fig. 4. Error in X, Y, ψ estimation

c j 3,k

x 10

2

h

2

g

cj 3,k d j 3,k

2

h

2

h

2

g

cj 3,k

h

2

g

2

g

2

g

2

2

d j 3,k

2

d j  2,k

2

2

h h h

g

c j 3,k d j 3,k d j  2,k

d j 1,k

fH

f2 f1 fL

Fig. 6. Decomposition analysis (above) and decomposition synthesis (below) steps using sub-band coding scheme showing three-level decomposition (adopted from (Parvez and Gao, 2005) and (Cruz-Tolentino et al., 2010)). Here, ↓ 2 and ↑ 2 represent undersampling and oversampling steps respectively. Generally in a PID control scheme, the roportional (P) and integral (I) terms capture low-frequency information of the error signal where as the derivative (D) term captures high-frequency information. The MRPID decomposes the signal into its high, low, and intermediate scale components. In the approach presented in (Parvez and Gao, 2005), the final control action is computed by multiplying each component with their respective gains and adding together the resulting commands. However, the high frequency components are mainly due to the noise and they are neglected in control computation. A main drawback of the conventional wavelet-based controller is the lack of integral action and consequently it may perform poorly

uA = P IDA,H eH + P IDA,M1 eM1 + ... + P IDA,MN −1 eMN −1 + P IDL eL such that A ∈ (X, Y, ψ) (19) The MRPID controller will have more tuning parameters and hence better resolution can be achieved. The proposed approach begins with decomposing the 3dimensional error signal (in X, Y and ψ) to a set of low frequency and high frequency components. To perform multilevel decomposition the error signal is generated by mirroring and appending the past data as described by Parvez and Gao (2005). The 3D decomposition is implemented on the resultant error vector. Number of decomposition levels are calculated based on equation (20). ) ( 2L − 1 N ≤ log2 +1 (20) F −1 where N is the number of levels that the signal is decomposed, L is the number of past observations and F is the size of the filter (Parvez and Gao, 2005). There are several wavelets to chose and based on the literature we selected “Daubechies” of order 4. Also L = 8 and F = 4 resulted in N = 2. Hence, two-level decomposition is performed. Table 2 shows the Daubechies fourth order filter coefficients used for decomposition analysis and synthesis steps, respectively. After the decomposition synthesis, the Table 2. Filter coefficients for decomposition (Daubechies 4) ¯ h g¯ h g

-0.1294 -0.4829 -0.4829 -0.1294

0.2241 0.8365 0.8365 -0.2241

0.8365 -0.2241 0.2241 0.8365

0.4829 -0.1294 -0.1294 -0.4829

decomposed high and low frequency components of the error signal must add up to the original signal. To ensure that the reconstruction is preserved to a higher degree, the above is verified for the implementation and negligible reconstruction error is observed as shown in figure 7.

1159

Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Awantha Jayasiri et al. / IFAC PapersOnLine 50-1 (2017) 1133–1138

1137

4. IMPLEMENTATION

5. CONCLUDING REMARKS

The proposed wavelet-based MRPID controller is implemented in a simulation setting for vessel DP and its performance is compared against that of the conventional PID + acceleration feedback controller. The 3-DOF nonlinear vessel parameters used for the simulations are reported in (Fannemel, 2008). In these simulations the sample time is set to 0.01s and the set point is given as (0, 0, π/3) in X,Y and ψ values r,espectively. For applications such as DP require to withstand environmental disturbances present in the form of waves, currents, wind and ice loads. To represent those a periodic sinusoidal force (having a period of 20s) superimposed by a noise (with σ = 20) is applied to the vessel dynamical system in X, Y and yaw directions. The simulations are performed for approximately 30 seconds.

In this work we have implemented a UKF-based nonlinear state estimator and a wavelet-based MRPID controller for dynamic positioning of vessels. Simulation results using are presented and performance of the proposed control scheme is compared with that of the conventional PID controller. The proposed approach has more parameters to tune than the conventional method. Adding derivative and integral terms with intermediate and low frequency decomposed components of the error signal yields better disturbance rejection capability in the system. The MRPID controller inherits high resolution and offers more flexibility in controller design. However, the number of tuning parameters increases with the level of decomposition and hence tuning can be tedious. In future work, we plan to perform a comparison of different control approaches employed for dynamic positioning and rate them. Also, using a more realistic model to include wave and wind loads as well as implementation of a feedforward controller to compensate for the wind load will be considered.

Figure 8 shows the simulation results using the conventional PID controller and Figure 9 shows the results using the proposed MRPID controller. In figures 8(a) and 9(a) the vessel trajectories in set point stabilization are given for both approaches. Figures 8(b) and 9(b) show the yaw angles of both of these approaches. Figures 8(c) and 9(c) depict the evolution of applied forces and torques. Finally, Figures 8(d) and 9(d) show the disturbance forces acting on the vessel. 4.1 Discussion Disturbances forces acting on a vessel may have very different frequency components. For example, oscillating waves are of high frequency, while current introduces a low frequency drift effect. Similarly the frequency characteristics of wind and ice-load are different. This makes design of conventional PID controller for DP systems challenging; however, the same problem makes the wavelet controller a relevant tool for the DP system. As the wavelet controller decomposes the error signal into different frequency components and scales the components by different gains, effects of different disturbances can be better controlled using the approach. In the proposed approach the nonlinear state estimation is performed by using a UKF and the error in state estimation is within an acceptable range. In the controller implementation we stared from decomposing the error signal to a set of low and high frequency components. The de-noising is performed by removing the highest frequency component of the signal. Although, the low frequency components are least affected by the derivative term, it may not be the same for intermediate frequency terms. Hence, we have used sub-PID controllers for each of the error components. It can be observed that the DP performance using the proposed approach (shown in Figure 9) outperforms that using the conventional PID method (shown in Figure 8) when the same level of disturbances are present. Both approaches are capable of keeping the vessel within 2m radius of the set point. The evolution of yaw angle in the proposed approach (shown in Figure 9(b)) shows better response close to the set value (π/3), compared to the yaw angle obtained using the conventional PID approach (shown in Figure 8(b)).

REFERENCES Bidikli, B., Tatlicioglu, E., and Zergeroglu, E. (2014). A robust tracking controller for dynamically positioned surface vessels with added mass. In Decision and Control (CDC), 2014 IEEE 53rd Annual Conference on, 4385–4390. Chaplais, F., Tsiotras, P., , and Jung, D. (2004). On-line wavelet denoising with application to the control of a reaction wheel system. In AIAA Guidance, Navigation, and Control Conference. Cruz-Tolentino, J.A., Ramos-Velasco, L.E., and EspejelRivera, M.A. (2010). A self-tuning of a wavelet pid controller. In Electronics, Communications and Computer (CONIELECOMP), 2010 20th International Conference on, 73–78. Du, J., Hu, X., Liu, H., and Chen, C. (2015). Adaptive robust output feedback control for a marine dynamic positioning system based on a high-gain observer. Neural Networks and Learning Systems, IEEE Transactions on, 26(11), 2775–2786. Dunik, J., Simandl, M., and Straka, O. (2012). Unscented kalman filter: Aspects and adaptive setting of scaling parameter. IEEE Transactions on Automatic Control, 57(9), 2411–2416. Ergen, B. (2012). Signal and image denoising using wavelet transform. In Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology. InTech. Fannemel, ˚ A.V. (2008). Dynamic Positioning by Nonlinear Model Predictive Control,. Master’s thesis, Department of Engineering Cybernetics, Norwegian University of Science and Technology. Fossen, T.I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley & Sons. Hassani, V., Sørensen, A.J., Pascoal, A.M., and Aguiar, A.P. (2012). Multiple model adaptive wave filtering for dynamic positioning of marine vessels. In 2012 American Control Conference (ACC), 6222–6228. doi: 10.1109/ACC.2012.6315094. Hassani, V., Sørensen, A.J., and Pascoal, A.M. (2013). Adaptive wave filtering for dynamic positioning of ma-

1160

Proceedings of the 20th IFAC World Congress 1138 Awantha Jayasiri et al. / IFAC PapersOnLine 50-1 (2017) 1133–1138 Toulouse, France, July 9-14, 2017

1

1

2 1.5

0

2 1.5

0

1

1

−1

0

−0.5

−3

0.5 Y (m)

−2

Y (m)

0.5 Y (m)

Y (m)

−1

−2 −3

−1

−1 −4

−4

−1.5

−5

−4

−2 X (m)

0

−2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 X (m)

−1.5

−5

−4

1 1000

1500 2000 Time (× 0.01s)

2500

0 0

3000

500

1000

F

F

X

1500

2000

y

2500

500

1000

1500 2000 Time (× 0.01s)

Fy −50 0 1000

2500

3000

−1000 0

1000

1500 2000 Time (× 0.01s)

Dy

D

2500

ψ

3000

Disturbance (N,Nm)

Disturbance (N,Nm)

0

500

1000

1500

2000

2500

3000

500

1000

1500 2000 Time (× 0.01s)

2500

3000

(c) Applied forces in X, Y and Yaw

50

Dx

500

Fx

0

(c) Applied forces in X, Y and Yaw

−50 0

3000

0

3000

0 −1000 0

2500

50

Force (N)

0

1000

1500 2000 Time (× 0.01s)

(b) Heading (ψ)

Torque (Nm)

Torque (Nm)

Force (N)

(b) Heading (ψ)

500

−2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 X (m)

1

50

−50 0 1000

0

2

ψ (rad)

ψ (rad)

2

500

−2 X (m)

(a) X,Y plot of DP with proposed MRPID control

(a) X,Y plot of DP with PID+acceleration feedback control

0 0

0 −0.5

50

0 D

X

−50 0

(d) Disturbances in X, Y and Yaw (in N, Nm)

500

1000

1500 2000 Time (× 0.01s)

D

Y

2500

D

ψ

3000

(d) Disturbances in X, Y and Yaw (in N, Nm)

Fig. 8. DP using PID+acceleration feedback control

Fig. 9. DP using proposed MRPID control

rine vessels using maximum likelihood identification: Theory and experiments. IFAC Proceedings Volumes, 46(33), 203 – 208. Khan, M.A.S.K. and Rahman, M.A. (2008). Implementation of a new wavelet controller for interior permanentmagnet motor drives. IEEE Transactions on Industry Applications, 44(6), 1957–1965. Kjerstad, O. and Skjetne, R. (2014). Modeling and control for dynamic positioned marine vessels in drifting managed sea ice. Modeling, Identification and Control, 35(4), 249–262. Kumar, A.G.D. and Ramana, N.V. (2016). A wavelet based multi resolution controller for load frequency control of multi area deregulated power system. In Proceedings of the 3rd International Conference on Electrical, Electronics, Engineering Trends, Communication, Optimization and Sciences (EEECOS). Miller, P., Farrell, J., Zhao, Y., and Djapic, V. (2010). Autonomous underwater vehicle navigation. Oceanic Engineering, IEEE Journal of, 35(3), 663–678. Nguyen, D.T., Sørbø, A.H., and Sørensen, A.J. (2009). Modelling and control for dynamic positioned vessels in level ice. IFAC Proceedings Volumes, 42(18), 229 – 236.

Osthus, V. (2014). Robust Adaptive Control of a Surface Vessel in Managed Ice Using Hybrid Position- and Force Control. Master’s thesis, Department of Engineering Cybernetics, Norwegian University of Science and Technology. Parvez, S. and Gao, Z. (2005). A wavelet-based multiresolution pid controller. IEEE Transactions on Industry Applications, 41(2), 537–543. Sørensen, A.J. (2011). A survey of dynamic positioning control systems. Annual Reviews in Control, 35, 123– 136. Tolentino, J.A.C., Silva, A.J., Velasco, L.E.R., and Ramrez, O.A.D. (2012). Wavelet pid and wavenet pid theory and applications. In PID Controller Design Approaches Theory, Tuning and Application to Frontier Areas. InTech. Veksler, A., Johansen, T.A., Borrelli, F., and Realfsen, B. (2016). Dynamic positioning with model predictive control. IEEE Transactions on Control Systems Technology, PP(99), 1–14. Wan, E.A. and van der Merwe, R. (2001). The unscented kalman filter. In S. Haykin (ed.), Kalman Filtering and Neural Networks. Wiley - Technology & Engineering.

1161