Research Article Estimation of Observer Parameters for Dynamic Positioning Ships

Transcription

1 Mathematical Problems in Engineering Volume 13, Article ID 17363, 7 ages htt://dx.doi.org/1.1155/13/17363 Research Article Estimation of Observer Parameters for Dynamic Positioning Shis Xiaogong Lin, Yehai Xie, Dawei Zhao, and Shusheng Xu Automation College, Harbin Engineering University, Harbin 151, China Corresondence should be addressed to Dawei Zhao; daweizhao@hrbeu.edu.cn Received 6 November 1; Revised 17 January 13; Acceted 18 January 13 Academic Editor: Valentina E. Balas Coyright 13 Xiaogong Lin et al. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. Considering the roblem of dynamic ositioning systems for the slowly varying disturbances, a arametrically adative observer is resented. The eak frequency of observer is adjusted on-line by autoregressive (AR) sectral estimation; other arameters of observer are otimized using article swarm otimization (PSO). The eak frequency can be calculated by sectral analysis of the itch, roll, and heave measurements. In the sectral estimation, Levinson-Durbin algorithm is used to solve the Yule-Walker equations. Finally, the comuter simulation is given to demonstrate the effectiveness of the roosed method. 1. Introduction Dynamic ositioning systems kee floating structures in fixed osition or redetermined track for marine oeration uroses exclusively by means of active thrusters [1]. Filtering and state estimation are imortant features of a DP system. In most cases, measurements of the vessel velocities are not available. Hence, estimates of the velocities must be comuted from noisy osition and heading measurements through a state observer. Unfortunately, the osition and heading measurements are corruted with colored noise due to wind, waves, and ocean currents as well as sensor noise. However, only the slowly varying disturbances should be counteracted by the roulsion system, whereas the oscillatory motion due to the waves (first-order wave disturbances) should not enter the feedback loo. This is done by using the so-called wave filtering techniques, which searate the osition and heading measurements into a low-frequency and wave frequency osition and heading estimate. The conventional observer filters out the WF motions fromthemeasuredositionandestimatesthelfosition and velocity. In the early studies, Balchen et al. [] and Sørensen et al. [3] used the Kalman filter to filter the WF motion. Later, Fossen and Strand [4] introduced the nonlinear assive observer. It should be noted that the two tyes of observers are based on a riori knowledge of the sea state to filter the WF motions; this means that eak frequency isassumedtobeknown.however,overalongertimeframe, the sea state may change, and therefore eak frequency in general is not known. So, Strand and Fossen [5] imroved the nonlinear assive observer with recursively adative WF filtering; however this design has two main drawbacks. Firstly, only the wave model arameters (not the observer gains) are adated. This means that a riori knowledge of the sea state is required to choose observer gains, such that a notch filtering effect is achieved to remove the firstorder wave comonents. Secondly, the observer tuning can be quite difficult. Guttorm Torsetnes gives a gain scheduled observer. The gain scheduled wave filtering is achieved by measuring the slowly varying wave model arameters on-line and therefore requires no a riori knowledge of the sea state. In this study, article swarm otimization (PSO) is imlemented in conjuncture with an AR sectrum estimation to adajust the arameters of observer. Some numerical simulation results are given to demonstrate the effectiveness oftheroosedmethod.. Shi Model A useful model describing the dynamics of a surface shi sailing in a horizontal lane having 3 degrees of freedom is given in [1], which is common to searate the model into a kinematic model and dynamic model.

2 Mathematical Problems in Engineering X E X R ψ X... Wave-Frequency Model. In the controller design synthetic white-noise-driven rocesses consisting of uncouled harmonic oscillators with daming will be used to model the WF motions. The synthetic WF model can be written in statesace form according to ψ d O R (x d, y d ) O (x, y) ξ = A w ξ + E w w w, η w = C w ξ. (4) Y R Y η w =[x w,y w,ψ w ] T is the osition and orientation measurement vector and ξ R 6, w w R 3 1 is a zero-mean Gaussian white noise vector. Consider O E Figure 1: Earth-fixed, reference-arallel, and body-fixed frame. Y E 3 3 I 3 3 A w = [ ], [ A 1 w A w ].1. Kinematic Model. In dynamic ositioning, the motions and state variables of the control system are defined and measured with resect to some reference frames or coordinate systems as shown in Figure 1. The Earth-fixed reference frame is denoted by O E X E Y E Z E, and the body frame is O XYZ (see [1]). The state vector is defined by η = [x, y, ψ] T, (x, y) R is the osition of the shi given in an inertial frame, and ψ [, π) is the heading angle of the shi relative to geograhic North. If only surge, sway, and yaw (3-DOF) are considered, the kinematics and the state vectors are reduced to η = R (ψ) v, (1) wherer(ψ) is defined as cos ψ sin ψ R (ψ) = [ sin ψ cos ψ ]. () [ 1].. Dynamic Model. In the mathematical modeling of shi dynamics, it is common to searate the model into a low-frequency model and wave-frequency model. The WF motion of the shi is due to first-order wave loads. The nonlinear LF equation of motion is driven by second-order mean and slowly varying wave, current and wind loads, and thrust forces...1. Low-Frequency Model. The equations of motion of a large class of surface shis can be described by the following model: Mv + Dv = τ + R T (ψ) e. (3) E w =[ 3 3 Σ ], C w =[ 3 3 I 3 3 ], A 1 w = diag [ ω o1 ω o ω o3 ], A w = diag [ ξ 1w o1 ξ w o ξ 3 w o3 ], Σ = diag [k 1 k k 3 ]. The relative daming ratio ξ i will tyically be in the range [.5.1]; ω oi is the arameter vector of the unknown dominating wave frequencies in surge, sway, and yaw. The bias model may also be modelled as 3. Observer Design (5) e = E b w b. (6) Fossen and Strand have roosed a assive nonlinear observer; see [4]. The observer includes wave filtering roerties, bias state estimation, reconstruction of the LF motion comonents, and noise-free estimates of the nonmeasured vessel velocities. The roosed observer has been roven to be assive and globally exonentially stable (GES). Based on (1), (3), (4), and (6) the observer takes the following form: ξ = A w ξ + K1 y, η = R (ψ y ) v + K y, τ =[τ x,τ y,τ ψ ] T is the control vector consisting of forces and moments roduced by the thruster system; e R 3 1 is the environmental disturbance; M R 3 3 is the system inertia matrix including added mass; D R 3 3 is the daming matrix. ê = T 1 b ê +K 3y, M v = D v + R T (ψ y ) ê + τ + R T (ψ y ) K 4 y, ŷ =[ η + C w ξ, v] T, (7)

3 Mathematical Problems in Engineering 3 where K 1 R 6 3, K, K 3 and K 4 R 3 3 are observer gain matrices; y = y ŷ is the estimation error. Consider K 1 =[ diag {k 1,k,k 3 } diag {k 4,k 5,k 6 } ], K = diag {k 7,k 8,k 9 }, K 3 = diag {k 31,k 3,k 33 }, K 4 = diag {k 41,k 4,k 43 }. k i = (ζ ni ζ i ) ω ci ω i, i = 1,,3, k i =ω i (ζ ni ζ i ) ω ci ω i, i = 4,5,6, ω ci = 1.ω i, k i =ω ci i = 7, 8, 9, where ζ i is the desired daming ratio (tyically 1) and ω i is oftensetequaltothewaveeakfrequency.ω ci = 1.ω i is the cutoff frequency of the notch filter. ζ ni >ζ i is a tuning arameter to be set between.1 and 1.. K 3 and K 4 should be sufficient high to ensure roer bias estimation. K 3 and K 4 are simly chosen so that 1/T bi < K 3i /K 4i <ω i <ω ci is fulfilled. We can see that ω i, K 3,andK 4 are the unknown arameters in the wave model. 4. Measuring ω i by AR Estimation The dominating wave frequency ω i isassumedtobetheeak frequency of the unbiased Power Density Sectrum (PDS) based on itch, roll, and heave resonse of the vessel. At this oint, it is worth noting that the itch, roll, and heave motions are usually not controlled in DP systems for shis, and they will therefore not be affected by the controller in the same degree as the three horizontal motions; see [6]. The ower sectrum estimation method used here is the arametric method of Autoregressive (AR) Sectral Estimation, which has several advantages comared to nonarametric methods; see [7]. Parametric methods are recommended since they give better estimates for shorter finite-length data records. An autoregressive (AR) model is a ossible method of modeling a stationary stochastic rocess. Given the signal x(n), x(n 1), x(n ),...,x(n ), the autoregressive model canbewrittenas x (n) =w(n) k=1 (8) (9) a k x (n k), (1) where w(n) is an indeendent identically distributed stochasticsequencewithzeromeanandvariance.thecoefficients a k are called the autoregressive arameters of the model, and is the model order. Once such a model of the rocess is available, its sectrum can be easily comuted: σ w P xx (e jw )= 1+ k=1 a ke jw. (11) The values of the arameters a k and σ w are unknown. The roblemistoestimatetheunknownarameters.a k and σ w canbecalculatedbysolvingthefollowing: φ xx (m) =E[x (n) x (n+m)] = E[x(n) ( = k=1 k=1 a k x (n k+m) +w(n+m))] a k φ xx (m k) +E[x (n) w (n+m)]. Since E[x(n)w(n + m)] = { m> σ w m=,wehave (1) a { k φ xx (m k), m >, k=1 φ xx (m) = (13) { a k φ xx ( k) +σ w {, m =. k=1 From (1) and(13), we obtain φ xx () φ xx ( 1) φ xx ( ) 1 φ xx (1) φ xx () φ xx ( ( 1)) a 1. [... ] [. ] [ φ xx () φ xx ( 1) φ xx () ] [ a ] (14) σ w = [.. ] [ ] Equation (14) is called the Yule-Walker equations and describes the unknown AR coefficients in terms of the +1estimated autocorrelation coefficients. Solving the Yule- Walker equations for a k is termed the autocorrelation method of AR arameter estimation and can be accomlished using a recursive technique such as the Levinson-Durbin algorithm. The Levinson-Durbin algorithm is given by k m = [φ xx (m) + m 1 k=1 a m 1 (k) φ xx (m k)], σ m 1 a m (k) =a m 1 (k) +k m a m 1 (m k), σ m =σ m 1 (1 k m ). (15) The coefficients a m (k) are generated recursively from their (m 1)-order values. σ m is the minimum backward rediction errors of m-order model, and k m is the reflection coefficient. 5. Otimizing K 3 and K 4 The tuning of the arameters is a necessary art of design observer, and the aroriate observer can guarantee the system obtaining fast dynamic resonse and robust. In this section, the matrics K 3 and K 4 of observer are otimised by PSO.

4 4 Mathematical Problems in Engineering 5.1. Overview of the PSO. The article swarm otimization (PSO) is an evolutionary comutation technique develoed by Kennedy and Eberhart in 1995; see [8 11]. Similar to the genetic algorithm (GA), the PSO algorithm is an otimization toolbasedonoulation,andthesystemisinitializedwitha oulation of random solutions and can search for otima by the udating of generations. It can be described as follows. At each iteration, each article can adjust its velocity vector, based on its momentum and the influence of its best osition as well as the best osition of its neighbors, and then comute a new osition where the article is to fly to. Suosing that the dimension for a searching sace is J, thetotalnumber of articles is N, the osition of the ith article can be exressed as vector l i = (l i 1, li,...,li J )T ;thebestosition of the ith article being searching until now is denoted by l i lbest = (l i 1, li,...,li J )T, and the best osition of the total article swarm being searching until now is denoted by vector l g gbest = (l g 1, lg,...,lg J )T ;thevelocityoftheith article is reresented by vector u i =(u i 1, ui,...,ui J )T.So,thevelocity and osition of the ith article can be udated as follows: u i (t+1) =Wu i (t) +h 1 r 1 (l i lbest (t) li (t)) +h r (l gbest l i (t)), l i (t+1) = l i (t) + u i (t+1), (16) where r 1 and r are random numbers between and 1; h 1 and h are the acceleration constants with ositive values; W is the weight function. The following weighting function is usually used in (16): W=W start W start W end t, (17) t max where W start is the initial weight, W end is the final weight, t max is the maximum iteration number, and t is the current iteration number. 5.. Fitness Function. Adoting the absolute error (ITAE) as the minimum, we can gain fitness function as J= β z 1 (t) dt, (18) with β being ositive constant Otimization Procedure. The PSO algorithm comrises the following stes. Ste 1. Set t = (t is the iterative number). Initialization of article swarm is generated randomly as follows: l i () = (l i 1 (), li (),...,li J ())T = (k 31,k 3,k 33,k 41,k 4,k 43 ) T ; the swarm has 6-dimension articles which can guarantee K 3 and K 4 to be ositive matrices. K 3 and K 4 are used in the observer mentioned in Section 3. Thearticlevelocitiesare generated randomly in the range 1 and as follows: u i = (u i 1 (), ui (),...,ui J ())T. Table 1: Suly vessel main articulars. Over length 175. m Breadth 5.4 m Design draught 9.5 m M 1 7 [.64 ; ; ] D 1 5 [. ; ; ] Ste. Define objective function values of the articles that areevaluatedgivenby(18). To each article in the articles swarm, set its best osition l j lbest () (the fitness function is the least). Finding the least fitness function values in the initializing articles swarm, the best osition is l gbest (). Ste 3. The osition and the velocity of each article are udated using (16). The fitness function values are calculated for the udated ositions of the articles. If J(l j (t)) J(l j (t 1)), j = 1,...,J,thenl j lbest = lj (t); else l j lbest = lj (t 1). Setting the least fitness function values in the current articles swarm is the gtharticle. If J(l g (t)) J(l gbest (t 1)), g = 1,,..., J, then l gbest = l g (t); else l gbest = l gbest (t 1). Ste 4. If we attain the maximum iteration, exit, and otherwise go to iterate. 6. Simulation Research In this section, the erformance of the observer tuning by PSO and AR sectrum estimation is verified. A suly shi named as S-175 is used as a case study, and the vessel main articularsareshownintable 1. For the urose of comarison, all the simulations deloy the same arameter settings as follows. The shi controlled by PID controller [1] is maintained in the desired osition and heading η d = [ ] T. The oulation size of PSO is 5, the maximum velocity is set as 1, the iteration is, and the weights are W start =.9 and W end = Examle 1. Verify the PSO. In order to demonstrate the efficiency of the roosed PSO, otimization observer by PSO is comared with an observer using genetic algorithmic (GA) [1]. The vessel was exosed to the JONSWAP distributed waves with significant wave height. m and eak frequency ω i =.65rad/s, ς 1 =ς =ς 3 =.1. The arameters are shown in Table, and the simulation curve is shown in Figure. From Figure, we can see that the roosed PSO otimization observer can give better results, with a small osition and heading bias. 6.. Examle. Verify the Parametrically Adative Observer. To demonstrate the erformance of arametrically adative observer in this aer, the roosed arametrically adative

5 Mathematical Problems in Engineering 5 Table : Parameters of observer. Matrix Emirical value Parameter results by PSO K 3.1 (45 4 ) diag([1.4, 1.33,.74]).1 (45 4 ) diag([1.4, 1.94, 1.]) K 4 (45 4 ) diag([1.35, 1.8,.57]) (45 4 ) diag([1.41, 1.89,.16]) Measured osition using GA Estimated osition using GA Measured osition using PSO Estimated osition using PSO (a) Simulation curve of observer using GA (b) Simulation curve of observer using PSO Measured osition using GA Measured osition using PSO (c) ThemeasuredositionofshiofobserverusingGAandPSO Figure : Simulation curve of observer using GA and using PSO. observer is comared with the observer (7). The aforementioned two observer gains are otimized using PSO. The JONSWAPwavesectrumwithsignificantwaveheighttobe set at m and eak frequency to be set at.8.65 rad/s is used to imitate sea conditions. The simulation curve is shown in Figure 3. FromFigures3(a) and 3(b),the observer withar sectral estimation can avoid the transients caused by the instability of verying sea states. From Figure 3(c), we can clearly see that the shi using observer with AR estimation has better osition and heading. It can be seen from Figure 3(d) that the ω i estimated by AR method is very close to redefined eak frequency. 7. Conclusions This aer roosed a arametrically adative observer of shi for the slowly varying seas. The eak frequency of observer can be calculated by AR sectral estimation of the itch, roll, and heave measurements, some arameters of observer are tuned on-line based on eak frequency, and other arameters of observer are otimized off-line using

6 6 Mathematical Problems in Engineering Measured osition without AR estimation Estimated osition without AR estimation (a) Simulation curve of observer without AR estimation Measured osition without AR estimation Measured osition using AR estimation (c) The measured osition of shi of observer without and with AR estimation ω i / (rad/s) Measured osition using AR estimation Estimated osition using AR estimation (b) Simulation curve of observer using AR estimation (d) The estimated eak frequency Figure 3: Simulation curve of observer without and with AR estimation. PSO. The simulation results show that the arametrically adative observer can avoid the transients caused by the instability of varying sea states and can imrove the accuracy of dynamic ositioning systems. Acknowledgment The aer was suorted by the National Nature Science Foundation of China, no References [1] T. I. Fossen, MarineControlSystems:Guidance,Navigation and Control of Shis, Rigsand Underwater Vecicles, Trondheim, Norway, 1st edition,. [] J. G. Balchen, N. A. Jenssen, and S. Sælid, Dynamic ositioning using kalman filtering and otimal control theory, in Proceedings of the IFAC/IFIP Symosium on Automation in Offshore Oil Field Oeration, , Amsterdam, The Netherlands, [3] A. J. Sørensen, S. I. Sagatun, and T. I. Fossen, Design of a dynamic ositioning system using model-based control, Control Engineering Practice,vol.4,no.3, ,1996. [4] T. I. Fossen and J. P. Strand, Passive nonlinear observer design for shis using Lyaunov methods: full-scale exeriments with a suly vessel, Automatica, vol. 35, no. 1,. 3 16, [5] J. P. Strand and T. I. Fossen, Nonlinear assive observer design for shis with adative wave filtering, in New Directions in Nonlinear Observer Design, vol.44,.89 94,Sringer, London, UK, [6]E.A.Tannuri,J.V.Sarano,A.N.Simos,andJ.J.DaCruz, Estimating directional wave sectrum based on stationary shi motion measurements, Alied Ocean Research, vol. 5, no. 5, , 3. [7] J. Pardey, S. Roberts, and L. Tarassenko, A review of arametric modelling techniques for EEG analysis, Medical Engineering and Physics,vol.18,no.1,. 11,1996.

7 Mathematical Problems in Engineering 7 [8] J. Kennedy and R. Eberhart, Particle swarm otimization, in Proceedings of the IEEE International Conference on Neural Networks, , Piscataway, NJ, USA, December [9] Z. L. Gaing, A article swarm otimization aroach for otimum design of PID controller in AVR system, IEEE Transactions on Energy Conversion, vol.19,no., , 4. [1] R. J. Wai and K. L. Chuang, Design of backsteing articleswarm-otimisation control for maglev transortation system, IET Control Theory & Alications, vol.4,no.4, , 1. [11] M. Yang, X. Wang, and K. Zheng, Nonlinear controller design for ermanent magnet synchronous motor using adative weighted PSO, in Proceedings of the American Control Conference (ACC 1), , Baltimore, Md, USA, July 1. [1] O. I. Hassanein, A. A. Aly, and A. A. Abo-Ismail, Parameter tuning via genetic algorithm of fuzzy controller for fire tube boiler, International Intelligent Systems and Alications,vol.4,no.4,.9 18,1.

8 Advances in Oerations Research htt:// Volume 14 Advances in Decision Sciences htt:// Volume 14 Alied Mathematics Algebra htt:// htt:// Volume 14 Probability and Statistics Volume 14 The Scientific World Journal htt:// htt:// Volume 14 International Differential Equations htt:// Volume 14 Volume 14 Submit your manuscrits at htt:// International Advances in Combinatorics htt:// Mathematical Physics htt:// Volume 14 Comlex Analysis htt:// Volume 14 International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics htt:// Volume 14 htt:// Volume 14 Volume 14 htt:// Volume 14 Discrete Mathematics Volume 14 htt:// Discrete Dynamics in Nature and Society Function Saces htt:// Abstract and Alied Analysis Volume 14 htt:// Volume 14 htt:// Volume 14 International Stochastic Analysis Otimization htt:// htt:// Volume 14 Volume 14