Robust Output Tracking Control of a Surface Vessel

Transcription

1 8 American Control Conference Westin Seattle Hotel, Seattle, Whington, USA June -3, 8 WeA6. Robust Output Tracking Control of a Surface Vessel DongBinLee,EnverTatlicioglu,Timothy C.Burg,DarrenM.Dawson Abstract Inthispaper,trackingcontrolofathreeegreeof-freeom marine vessel is examine. The primary motivation forthisworkisthe compensationneee forthe ae ms common to surface vessels, resulting in an ymmetric inertia matrix. Two control schemes are consiere: a fullstate feeback controller an an output feeback controller. Numerical simulation results are shown to emonstrate the valiityofthesepropose controllers. I. Introuction Research into the control of marine surface vessels coul be loosely categorize maneuvering [,], ynamic positioning [3,4], tracking (incluing path following or way-point tracking) [6,8], an more recently formation control [5]. From a control perspective, the properties of the ynamic moel of the surface vessel are of great importance. Specifically, the symmetry an the positive efiniteness of the inertia matrix are an important sumption often use in control evelopment. The inertia matrix of a vessel is commonly efine to be equal to the sum of the rigi-boy inertia matrix an the ae ms. The rigi-boy inertia matrix is strictly a symmetric matrix. The ae ms terms result from hyroynamic forces an moments ue to motion of the vessel boy an from the interaction with the fluis an waves. In surface vessel control, the ae ms matrix can eily become ymmetric, especially for higher relative velocity. This ymmetry in the ae ms terms will result in an ymmetric inertia matrix in the ship moel, which may cause system instability or a failure in meeting the control objectives if ignore. As the founation of this work, previous closely relate work is escribe. In [], Skjetne et al. consiere maneuvering control of three egree-of-freeom(3 DOF) marine vessel an presente experiment results for the Cybership II. In [6], Do an Pan presente a global tracking controller of an uneractuate vessel where the system matrices are positive efinite but nonzero o-iagonal terms an Do in [7] propose robust an aaptive output feeback controllers for positioning of a surfacevesselsumethesystemmatricestobepositive efinite at low spee. In [8], Behal et al. utilize a ThisworkhbeensupporteinpartbyaDOEContract,a HonaCorporationGrant. D.Lee,T.C.Burg,D.M.DawsonarewiththeDep tof Electrical an Computer Engineering, Clemson University, Clemson,SC9634,USA. lee@clemson.eu,tburg@clemson.eu ( corresponingauthor),arren.awson@ces.clemson.eu.e.tatliciogluwdep tofelectrical&computerengineering,clemson University.Heisnow with thedep tofelectricalan Electronics Engineering, Izmir Institute of Technology, Urla, Izmir 3543, Turkey. enver@envertatlicioglu.com high-gain observer in the esign to control uneractuate surface vessels with nonintegrable ynamic moels where the inertia matrix w iagonal. However, a few researchers have aresse the ymmetry of the inertia matrix. For example, Skjetne et al. in[] evelope the Coriolis-centripetal an amping moel of the ship for the ymmetric ae ms matrix, but the ms-relate parameters were use in the symmetric form at the experiment, which may be violate at higher operating spees. Inthispaper,wefocusontrackingcontrolofa3DOF surface vessel. The ynamic moel of the ship is sume to be uncertain an the ae ms terms are consiere to be ymmetric which results in an ymmetric inertia matrix. To aress this problem, the ynamic moel of the ship is moifie to have a symmetric an positiveefinite inertia matrix. The novelty of this moification is the multiplication of the system ynamic moel in [] with an upper triangular matrix which results in amoelwithymmetricinertiamatrix.afterthis moification, the resulting ynamic moel becomes a special ce of the multi-input multi-output system that wconsierein[](seeequationin[] ).Next,the robust full-state feeback (FSFB) an output feeback (OFB) control strategies in [], which were esigne for general cls of multi-input multi-output nonlinear systems, are tailore to fit this ynamic surface vessel moel. The paper is organize follows: Section II presents aynamicanakinematicmoelofthe3dofsurface vessel. The error system evelopment an the control strategies are provie in Section III. The numerical simulation results are shown in Section IV followe by conclusions in Section V. II. SystemMoel In this section, the system moel an relevant properties are iscusse. The ynamic an kinematic moels of a 3 DOF surface vessel expresse in the boy-fixe frame,b,aregiven[, ] M +C+D = () ẋ = R () where the vector ẋ(t)r 3 represents the position an orientation rate in which x =[x p,y p,] > enotes the The robust control evelopment in [9] is similar to [] with aminormoification in the matrix ecomposition (see Lemma in both [9]an []).In thatsense,throughoutthispaperwe will referto [9],however,thereaerisalso referre to [] /8/$5. 8 AACC. 544

2 linear position (x p,y p )alongthex-anthey-axes an the yaw angle (). The vectors v r (t), (t)=[u, v, ] >,an (t)r 3 enotetherelativevelocitybetween the fluis an vessel an the velocity an acceleration of the rigi-boy ship, respectively. M(),C(, r ), D(, r )R 3x3 representtheinertiamatrix,centripetal an Coriolis force, an hyroynamic amping terms, respectively. In the subsequent control evelopment, M( ),C( ),and( ) are sume to be uncertain an continuously ierentiable up to their secon time erivatives.in(),(t)r 3 representsthecontrolinput vector which h the following form =[,, 3 ] > (3) where (t) an (t)r are the translational forces inthex-any-irections,respectivelyan 3 (t)r is the moment about the Z-axis. The matrix, R() SO(3),enotes the rotation matrix, containing the yaw angle, about the Z-axis.Thecoorinateframeofthe surface vessel is presente in Figure, where B is the boy-fixe reference frame of the vessel an a fixe inertial frame, approximate by the earth-fixe frame (North-Et-Down convention), is enote by I.The To facilitate the subsequent control evelopment, the ynamic moel in () will now be moifie to obtain a symmetric inertia matrix. There will exist an upper iagonalmatrix,t( )R 3x3,suchthatthemultiplication oft( )anm( )resultsinymmetric,positiveefinite matrix, enote by M s ( )R 3x3. After multiplying () with T( ), the following expression is obtaine M s =T(C+D)+T. (5) To further moify the ynamic moel, the time erivativeof()isobtaineinthefollowing form =R > ẍr > ṘR > ẋ (6) wherethepropertyoftherotationmatrixthatr - ()= R > ()wutilize.substituting(6)into(5)yiels M s R > ẍ= h i M s S 3 ( )T(C+D) R > ẋ+t (7) where the time erivative of the orientation matrix, enotebyṙ(),ankew-symmetricmatrixs 3( ) R 3x3 canbecalculatefollows Ṙ()=RS( ),S 3 ( )=. (8) Afterpremultiplying(7)withR(),thefollowingmoel is obtaine M()ẍ= C(x,ẋ,, r )ẋ+rt (9) where M()an C(x,ẋ,, r )R 3x3 areefine M,RM s R > () h i C=R M s S 3 ( )T(C+D) R >. () Fig.. Diagram ofasurfacevessel states are meure from the center point (CP) of the ship frame expresse in B an x g enotes the istance between the center point an the center of gravity(cg) oftheship. Theinertiamatrix,M( ),oftheshipisefine[] M,M RB +M A (4) wherem RB ( )R 3x3 representstherigi-boysymmetricinertiaanm A ( )R 3x3 accountsfortheymmetric ae ms. Since M A ( ) is ymmetric the inertia matrixofthesurfacevessel,m( ),isalsoanymmetric matrix. It shoul be note that the form of (9) w motivate bytheynamicmoelin[9],[]anhence,solvingthe same control problem. Property Thematrix : M()ispositiveefinite, symmetric, an satisfies the following inequalities kk T M kk,r 3 () where, R are positive bouning constants. III. Control Development A. Full-State Feeback Control The subsequent evelopment is be on the sumptionthatallthestatesofthevesselaremeurable. 545

3 ) Error System Development : The tracking error for position an orientation, enote by e (t)r 3,isefine e,x x (3) where x (t) R 3 is the esire trajectory. For the subsequent stability analysis, the esire trajectory an its first an secon time erivatives are sume to be boune (i.e., x (t), ẋ (t), an ẍ (t) L ). To facilitate the subsequent error system evelopment, a filtereerror,enoteby e (t)r 3,isefine e,ė +e. (4) Inorertosimplifytheerrorsystemantofacilitatethe stability analysis, a filtere tracking error is introuce r,e +e. (5) Theynamicsofr(t)canbeobtainefollows ṙ=ẍ ẍ+ė (6) where the secon time erivatives of(3) an(4) were utilize. After premultiplying (6) with M(),wecan obtain the following expression Mṙ= Mẍ CẋRT+Mė (7) where (9) w utilize. The expression in (7) can be rearrangeafteraingansubtractingtheterms. Mr (t),e (t),anr(t)totheright-han sie Mṙ=NR. Mre R(TI 3 ) (8) wheretheauxiliarysignaln( )R 3 isefineby N, Mẍ Cẋ+Mė +. Mr+e. (9) To facilitate the control evelopment, the open-loop error ynamics can be obtaine from(8) Mṙ=Ñ+N. Mre R () wherethesignals(t), (t)r areefine +,R(TI 3 ), () wherethethircomponentof(ti 3 )iszero(anexample of T( ) isshowninthesimulationsection)ann ( ), Ñ( )R 3 areefinefollows N,N x=x, ẋ=ẋ, ẍ=ẍ, Ñ,NN () where the esire trajectory an its first two time erivativesaresumetobebouneanhencen ( ) an Ṅ( )areboune signals. Remark: The term Ñ( ) in () is upper boune Ñ N (kzk)kzk (3) where N ( )isagloballyinvertible,non-ecreingfunction an the auxiliary error vector z(t)r 9 is efine z(t)=[e >,e >,r > ] >. ) Control Input : Beontheopen-looperrorynamicsin()anthe resultin[9], thecontrol input(t)isesigne,r > (K+I 3 )r+r >ˆf (4) where KR 3x3 is a constant positive efinite iagonal gain matrix an ˆf(t)R 3 is a feeforwar component that is introuce to compensate for N (t) an (t). After substituting (4) into (), the following closeloop error system can be obtaine Mṙ=+. Mre (K+I 3 )r (5) where (t) an (t) R 3 are auxiliary functions efine follows,ñ,,n ˆf. (6) Notethatsince ˆf( )isusetocompensateforfunctions of the esire trajectory it is known that ˆf( ) L apriori. Remark: The controller propose in (4) is an application of the previous theoretical evelopment in [9] where the propose control yiels a semi-global, uniformly, an ultimately boune (sguub) tracking result. Thus, the reaer is referre to [9] for a etaile stability analysis. Remark3: The term ˆf( ) in (4) an (35) is not irectly specifie here but in practice it can be implemente in other ways incluing a neural network. B. Output Feeback Control The following evelopment is be on the sumption that the position an the orientation of the ship, x(t), is the only state that is meurable. ) Observer Design : To facilitate the subsequent control evelopment, the auxiliaryerrorvector,enotebyz(t)r 6,isreefine z(t)=[e >,r > ] > (7) where e (t) an r(t) are error signals efine in (3) an (5), respectively. The following expression can be obtaine for the ynamics of z(t) ż=[(re ) >, ṙ > ] > (8) where(3) an(4) were utilize. An estimate of the unmeurable, z(t) in (7), is introuce follows ẑ, ê >,ˆr > > (9) where ê (t),ˆr(t) R 3 are high-gain observers that are introuce to estimate the error signals e (t) an r(t), respectively []. The time erivative of (9) can be obtaine. ẑ, ". ê.ˆr # ˆrê + = (e ê ) (e ê ). (3) 546

4 To facilitate the subsequent analysis, the following observer errors, enote by (t) an (t) R 3, are introuce = (e ê ), =rˆr. (3) Theynamicsfortheobservererrorscanbeobtaine follows = ( + ), = +ṙ (3) where(3) an(3) were utilize. After combining(3), the following simplifie expression can be obtaine.=a o+g (33) where (t),[ >, > ] > R 6 an the signals g(t)r 6 an A o R 6x6 areefinefollows I A o = 3 I 3,g=. (34) I 3 O 3 ṙ ) Control Input : Similarto(4)antheresultin[9],theoutputfeeback controller for ṙ(t) in(34) is esigne follows,r > sat{(k+i 3 )ˆr}+R >ˆf (35) where sat{ } R 3 represents the vector saturation function an ˆf(t) R 3 isthefeeforwarterm(see Remark 3). Substituting this control input into () yiels the following close-loop error system Mṙ=+. Mre sat{(k+i 3 )ˆr} (36) where( )an ( )wereintroucein(6). Remark 4: Since the output feeback controller esigne in (35) is a special ce of the evelopment in [9], a semi-global, uniformly, an ultimately boune (sguub) tracking result can be inferre. IV. Numerical Simulation Results Two numerical simulations were performe to show the valiity of the propose controllers. The rigi-boy inertia matrix incluing the ae ms terms are of the following form[] M = m+x u n a n n c n b (37) where n a,n b,n c, an n are auxiliary terms that are efine follows n a = my v,n b =I z Nṙ n c = mx g N v,n =mx g Yṙ. From(37),itcanbeseenthattheaemsmatrix,enotebyM A ( ),hnonzeroo-iagonalhyroynamic amping terms (usually refere to hyroynamic erivatives). In [], Yṙ an N v were set equal to zero which resulte in a symmetric inertia matrix. It is clear that if the values of Yṙ an N v are ierent, then the resulting matrix is ymmetric. For the simulation, the following values were chosen for these parameters Yṙ=.,N v =., (38) which prouce an ymmetric inertia matrix. Be on the inertia matrix in (37), the following T( ) matrix is efine to moify the system ynamic moel T = n a n b (39) + n c n a n + n an c n c n + n an b whereisanon-zeroauxiliarytermefine = mi z +mnṙ+y v I z Y v Nṙ+(mx g ) mx g N v mx g Yṙ+YṙN v. The Coriolis-centripetal term C( ) in () isefine by combining the rigi-boy matrix C RB () R 3x3 an corresponingaemsc A (, r )R 3x3 C(, r )= c c (4) c c wherec ( r )=mu+(x u u r )anc (, r )=m(x g + v)+(y v v r +.5(Yṙ+N v ) ). The amping matrix D( ) in () is efine by combining the linear matrix term D L ()R 3x3 an nonlinear matrix D NL (, r )R 3x3 D(, r )= (4) where ( r )=X u +(X u u u r X uuu u r), (, r )=Y v +(Y v v u r Y rv ), 33 (, r )= N r +(Y v v u r Y r v ), 3 (, r )=Y r + (Y v r u r Y r v ), an 3 (, r )=N v + (N v v v r N rv ). The saturator which limts the signals to upper an lower w use by ± for the control input both FSFB an OFB controllers. The other system parameters were obtaine from the experimental resultsin[].theesirepositionanyawtrajectory are specifie below sin(.t) x (t)= (m), cos(.t) (t)=.t(ra), an the vessel w consiere to be initially at rest in the following configuration x() = [.,, - 8 ]>. The surface vessel w specifie to have the relative velocity, v r =[3,,] > where the relative velocity is sume to have the constant surge spee, 3[m/s], about the X-axis. A. Full-State Feeback(FSFB) Control For the FSFB controller simulation, the constant control parameter w chosen K = iag 5 5 ª. In Figure, tracking the esire trajectory of the surface vessel in XY-plane is emonstrate. In Figure 3, the position an yaw angle 547

5 Desire Actual Position x Error in Boy fixe Frame Y Position y Error in Boy fixe Frame Yaw (ψ) Angle Error 5 X Fig.. Trackingemonstration inthexy-plane(fsfb) of the ship are presente along with their corresponing esire trajectories. The error signal e (t) is presente in Figure 4. From Figure 4, it is clear that the surface vessel tracke the esire trajectory. The control input (t)isshowninfigure5. Position Tracking along x axis actual esire Position Tracking along y axis Orientation Tracking along yaw irection Fig. 4. Tracking Errors in Position (x p,y p)anyawangle() (FSFB) [N] Translational Force Input u Translational Force Input u Rotational Torque Input u Fig.5. ForcesanTorqueInput(FSFB) Fig.3. Trackingemonstration (FSFB) B. Output Feeback(OFB) Control For the OFB controller, the constant control parameter w chosen K =iag 4 4 ª an the following control parameters were use =, =, an =.. In Figure 6, the position an the esire trajectory of the surface vessel in XY-plane is emonstrate. Figure 7 shows the position an yaw angle of the ship along with their corresponing esire trajectories. Figure 8 isplays the small, boune trackingerrorsignals.infigure9,thecontrolinput(t) is presente. V. Conclusion The control problem of surface vessels with ymmetric inertia matrices w aresse. The significance of this work w the moification of the inertia matrix by pre-multiplying with an upper triangular matrix to obtain a symmetric form. Then, a full-state feeback an an output feeback controller were esigne. Numerical simulation results were shown to emonstrate the propose control strategies. References 548

6 Desire Actual Position x Error Position y Error Y Yaw (ψ) Angle Error 5 X Fig.6. Trackingemonstration in thexy-plane(ofb) Position Tracking along x axis actual esire Position Tracking along y axis Orientation Tracking along yaw irection Fig.8. Tracking Errorsin Position an Yaw Angle (OFB) [N] 5 Translational Force Input u Translational Force Input u Rotational Torque Input u Fig.7. Trackingemonstration (OFB) Fig.9. ForcesanTorqueInput(OFB) []T.I.Fossen,MarineControlSystems:Guiance,Navigation, an Control of Ships, Rigs an Unerwater Vehicles, Tronheim:MarineCyberneticsAS,Norway,ste.. [] R.Skjetne,O.N.Smogeli,anT.I.Fossen, ANonlinearShip ManoeuveringMoel:Ientificationanaaptivecontrolwith experiments for a moel ship, Moeling, Ientification an Control,Vol5,No.,pp.3-7,4. [3] T. D. Nguyen, A. J. Sørensen, an S. T. Quek, Design of Hybri Controller for Dynamic Positioning from Calm to ExtremeSeaConitions,Automatica,Vol.43,No.5,pp ,May7. [4] E. A. Tannuri an H.M. Morishita, Experimental an numerical evaluation of a typical ynamic positioning system, Applie Ocean Research Vol. 8, No., pp , April 6. [5] F.Fahimi, NonlinearMoelPreictiveFormationControlfor GroupsofAutonomousSurfaceVessels,InternationalJournal ofcontrol,vol.8,issue8,pp.48-59,august7. [6] K. D. Do an J. Pan, Global tracking control of uneractuate ships with nonzero o-iagonal terms in their system matrices,automatica,vol.4,pp.87-95,5. [7] K. D. Do, Global Robust an Aaptive Output Feeback Dynamic Positioning of Surface Ships, IEEE International Conference on Robotics an Automation, Rome, Italy, pp ,April7. [8]A.Behal,D.M.Dawson,W.E.Dixon,anY.Fang, Tracking an Regulation Control of an Uneractuate Surface Vessel with NonintegrableDynamics,IEEE TransactionsonAutomaticControl,Vol.47,No.3,pp.495-5,March. [9]X.Zhang,A.Behal,D.M.Dawson,anB.Xian, Output Feeback Control for a Cls of Uncertain MIMO Nonlinear SystemswithNon-SymmetricInputGainMatrix,Proc.44th IEEEConferenceonDecisionanControl,antheEuropean ControlConference,Seville,Spain,pp ,Dec.5. []J.Chen,A.Behal,D.M.Dawson,anX.Zhang, Robust an Aaptive Output Feeback Control Strategies for a Cls of Uncertain MIMO Nonlinear Systems, Technical Report, CRB, Clemson University, April 6, Available: [] A. N. Atsi an H. K. Khalil, A Separtion Principle for the Stabilization of a Cls of Nonlinear Systems, IEEE TransactionsonAutomaticControl,Vol.44,No.9,pp ,Sept