A NONLILEAR CONTROLLER FOR SHIP AUTOPILOTS

Transcription

1 Vietnam Journal of Mehanis, VAST, Vol. 4, No. (), pp. A NONLILEAR CONTROLLER FOR SHIP AUTOPILOTS Le Thanh Tung Hanoi University of Siene and Tehnology, Vietnam Abstrat. Conventional ship autopilots are designed based on a linear ship model using pole - plaement tehnique or linear optimal theory. However, in operation, the ship kinematial parameters an go out the linear limits. In this paper, a nonlinear optimal ontrol law based on aggregated variables is presented. The riterion is hosen so that the dynami harateristis of objet are inluded. The stability of the losed-loop system is global aording to the Lyapunov stability theory. The ontrol law depends expliitly on ship model parameters, so that it is an be easily to tune when the parameters hange. ey words: Ship autopilot, nonlinear ontrol, aggregated regulator, aggregated variable.. INTRODUCTION Conventional ship autopilots are designed based on a linear ship model using pole - plaement tehnique or linear optimal theory without onsidering steering dynamis. However, in operation, the ship kinematial parameters an go out the linear limits; the steering mehanism has limit speed of defletion. In suh a ase the stability of the losed - loop system may not be guaranteed. More ever, negleting the nonlinearity and dynamis of steering mahine may effet on system performane, usually redues it. For a nonlinear system the bak stepping or state feedbak linearization tehniques an be used [, ], but no optimal riterion is onsidered in these approahes. In this paper, a nonlinear optimal ontrol law based on so - alled aggregated variables is presented. The riterion is hosen so that the dynami harateristis of objet are inluded. The stability of the losed - loop system is global aording to the Lyapunov stability theory. The ontrol law depends expliitly on ship model parameters, so that it is an be easily to tune when the parameters hange. The struture of the paper is organized as follows. A brief desription of ontroller design method and ship motion model is presented in the next setion. Controller designing and simulation are presented in the Setions and 4 respetively. Conlusion is presented in the last setion.

2 4 Le Thanh Tung. A BRIEF DESCRIPTION ON CONTROLLER DESIGN METHOD, SHIP STEERING EQUATIONS OF MOTION AND WAVE MODEL.. Controller design method based on aggregated variables A framework of design method based on so - alled aggregated variables is proposed by olesnikov []. In this setion, the theoretial basis is taken from [4]. Consider a dynami system ẋ i = f i (x), i = n m, ẋ n m+j = f n m+j (x) + b j u j, j = m, x = (x, x,..., x n ) T, where x = (x, x,..., x n ) T - vetor of state variables, u j - ontrol ations. Let Ψ j (x) =, j = m - aggregated maro variables. The motion of synthesis system has to satisfy minimum of ost funtions defined as J j = t t () ( Ψ j + φ j(ψ j )) dτ, () where φ j (Ψ j ) - a funtion of Ψ j (x), whih must be hosen so that, the differential equation Ψ j + φ j (Ψ j ) = has stable solution Ψ j (x). Stable minimum of the ost funtion is solution of the following differential equations Define Ψ j as Ψ j = n i= We have the ontrol law ( Ψj u j = b j x n m+j ) n i= Ψ j + φ j (Ψ j ) =. () Ψ j x i f i (x) + n i=n m+ Ψ j x i f i (x) + φ j (Ψ j ) +.. Ship steering equations of motion Ψ j x i b n i u n i. (4) n Ψ j b n i u n i x i i = n m + i n m + j For heading problem, the most used ship mathematial models are following: The models of Nomoto et al [, 5] For small rudder angles, the transfer funtion between the rudder angle δ and the yawing rate ω of a surfae ship an be desribed by the linear models of Nomoto et al. Nomoto s nd order model is written as ω(s) δ(s) = ( + T s) ( + T s)( + T s), (6) (5)

3 A nonlilear ontroller for ship autopilots 5 where s is used to denote the variable of Laplae operator, is the gain onstant and T i (i =,, ) are three time onstants. A first - order approximation is obtained by defining the effetive time onstant as: T = T + T T. Hene, ω(s) δ(s) = The yaw angle (ourse) ϕ(t) is related to yaw rate ω(t) as + T s. (7) ϕ(t) = ω(t). (8) The model of Beh and Wagner Smith [, 6] The linear ship steering equations of motion an be modified to desribe large rudder angles and ourse - unstable ships by simply adding a nonlinear maneuvering harateristi to Nomoto s nd order model. Beh and Wagner proposed the model T T ϕ () + (T + T ) ϕ + H B ( ϕ) = (δ + T δ), (9) where the funtion H B ( ϕ) desribes the nonlinear maneuvering harateristi produed by Beh s reverse spiral maneuver, that is H B ( ϕ) = b ϕ + b ϕ + b ϕ + b. () For a ourse - stable ship, the b >. A single srew propeller or ship with asymmetry in hull has b of non - zero value. Hene, b = for a symmetrial hull. The model of Norrbin [, 7] An extension of Nomoto s st order model an be made by defining where the maneuvering harateristi H N ( ϕ) is defined as T ϕ () + H N ( ϕ) = δ, () H N ( ϕ) = n ϕ + n ϕ + n ϕ + n. () The Norrbin oeffiients: n i (i = ) are related to those of Beh s model by: n i = b i / b. n = ± for stable and unstable - ourse ship respetively. Sine a onstant rudder angle is required to ompensate for onstant or slowly - varying wind and urrent disturbanes, the bias term n ould be treated as an additional rudder off - set. That is, a large number of ships an be desribed by.. The wave model H N ( ϕ) = n ϕ + n ϕ. () The following wave model is adopted from Paulsen et al [8] ẋ w = A w x w + b w η, w = C T x w, (4) where A w = [ ] [ ] [ ωn, b ξω w =, C n w =, (5) w ]

4 6 Le Thanh Tung η is a zero mean Gaussian white noise sequene, ω n - the dominating wave frequeny, ξ - the relative damping ratio of the wave, w is the gain that is dependt on the wave energy. In the transfer funtion form the model is w = C T (si A w ) w s B = s + ξω n s + ωn. (6). CONTROLLER DESIGN For designing of ontrol law, the model of Norrbin is hosen. By ombining the steering mehanism dynamis, desribed by equation δ = u, the ship steering equation of motion, wave model and denoting x = ϕ, x = ω = ϕ, x = δ, we have the augmented system dynamis on the wave as ẋ = x, ẋ = n T x n T x + T (x + w), ẋ = u. To design the ontrol law we use the system dynamis on the still water w =. Then, the autopilot performane is tested in ases: still water and in wave. Define the aggregated variable as ψ = x + x + x + x. (8) Choose the ost funtion as J = t t (7) ( T Ψ ) + Ψ ) dτ, T >. (9) Then, the ontrol signal u is defined from the equation T ψ + ψ = as u = [ n x + ( T T + )x + ( n T T T n T )x n T x5 + ( T + T )x + T x x Under the ation of this ontrol signal, the point defining system position in state spae is moved into neighborhood of the surfae Ψ =. In this ase, system motion along it is desribed by a set of differential equations as following ẋ = x, ẋ = T x ( n T + T )x ( n T + T )x. The design parameters,,, must be hosen so that the motion is stable. Consider a Lyapunov funtion andidate as ]. () () V = p x + p x, p i >, i =,. ()

5 A nonlilear ontroller for ship autopilots 7 Time differentiation gives V = p ẋ x + p ẋ x = x x (p p T we have By the hoie of Hene V ) x ( n T + T )p x 4 ( n T + T )p. () = p p T, > n, > n. (4) n T + T V = x ( n T + T >, n T + T )p x 4 ( n T + T Obvioualy, when t, x, from the nd equation of () we get Sine T T >. (5) )p. (6) x =. (7), then x =. Then the global stability is guaranteed. 4. SIMULATION STUDY The ship model used in the simulation study is adopted from Fossen []. The ship parameters are following: =.5(/s), T = (s), n =, n =.4(s ). w t(s) Fig.. Wave disturbane w versus time

6 - - 8 Le Thanh Tung Clearly it is a ourse - stable ship. The parameters of the wave model are hosen as: ω n =.7, ξ =, w =. [8], the wave harateristi in the time domain is presented in Fig.. The domain of design parameters variation and the phase portrait of the losed - loop system are shown in Figs., respetively. n pt p n Fig.. Domain of design parameters x 5 x x x Fig.. Phase portrait of losed - loop system The desired yaw angle is ϕ r =. The design parameters are hosen to be: p =.6, p =, =.7, =.8, =.99, =., T = 7.5.

7 A nonlilear ontroller for ship autopilots 9 At first we assume no wave disturbane, that is w =. The transition harateristis of losed - loop system are shown in Fig. 4a. We see that the yaw angle and yaw rate onverge to the desired value in finite time t(s) (a) without wave disturbane w - t(s) (b) with wave disturbane w Fig. 4. Closed - loop system harateristis: - yaw angle ϕ (deg), ω (deg/s), - rudder angle δ(deg), 4 - ontrol signal u(deg/s) - yaw rate In the Fig. 4b, the hange of state variables and ontrol signal in presene of the disturbane is presented. We see that, the yaw angle is bounded approximately by the value of deg after about 7 s, the rudder angle - 5 deg after the same time. 5. CONCLUSIONS A nonlinear autopilot based on aggregated maro variables has been derived. The domain of design parameters has been defined. The ontrol law depends expliitly on the ship parameters so that it is flexible to hange. The performane of the ontroller was illustrated through a simulation study. Global stability of losed - loop system was proven by applying Lyapunov stability theory. ACNOWLEDGEMENTS This work has been partially ompleted by supports of the sientifi researh program "Researh and development tehnology applied for mini submarine" (oded: B-.) at Hanoi University of Siene and Tehnology. REFERENCES [] T. Fossen, M. Paulsen, Adaptive Feedbak Linearization Applied to Steering of Ships, st IEEE Conferene on Control Appliations, (99). []. Husa, T. Fossen, Baksteeping design for nonlinear way - point traking of ships, 4 th IFAC Conferene on Manoeuvring and Control of Matine Craft, (997).

8 Le Thanh Tung [] olesniov A. A, Analytial onstrution of nonlinear aggregated regulators for a given set of invariants manifolds, Eletromehania, Izv. VTU, Novoherkask, Part I: - 9, Part II: (987) (in Russian). [4] Terekhov V. A. et al., Neural network ontrol systems, M.: IPRGR () (in Russian). [5]. Nomoto et al., On the steering Qualities of Ships, Tehnial report, International Shibuilding Progress, 4 (957). [6] M. I. Beh and L. Wagner Smith, Analogue Simulation of Ship Manoeuvres, Tehnial Report Hy-4, Hydro- and Aerodynamis Laboratory, Lyngby, Denmark (969). [7] N. H. Norrbin, On the Design and Analyses of Zig - Zag Test on Base of Quasi Linear Freqqueny Response, Tehnial Report B 4 -, The Swedish State Shipbuilding Experimental Tank (SSPA), Gothenburg, Sweden, (96). [8] M. Paulsen et al., An output Feedbak Controller with Wave Filter for Marine Vehiles, (994). Reeived August 6,