FUNDAMENTAL PROPERTIES OF LINEAR SHIP STEERING DYNAMIC MODELS

Transcription

1 Jounal of Maine Science and Technology, Vol. 7, No. 2, pp (1999) 79 FUNDAMENTAL PROPERTIES OF LINEAR SHIP STEERING DYNAMIC MODELS Ching-Yaw Tzeng* and Ju-Fen Chen** Keywods: Nomoto model, Contollability and obsevability, Identifiability, Model eduction, Oveshoot, Bode plot. ABSTRACT This pape is concened with the fundamental popeties associated with the Nomoto models. Specifically, the state space model associated with the fist ode Nomoto model is both obsevable and contollable. The state space model associated with the second ode Nomoto model is also obsevable; howeve, it is contollable only if the effective sway time constant is diffeent fom the effective yaw time constant. The zeo appeaing in the tansfe function model is found esponsible fo the oveshoot behavios, which ae typical in the yaw ate fo lage udde angle steeing. This suggests that a second ode Nomoto model is moe appopiate if the oveshoot featue is to be popely modeled. Both the fist and second ode Nomoto tansfe function models ae identifiable, with an ill-conditioning poblem associated the latte. This makes the fist ode Nomoto model vey popula in the adaptive autopilot applications. Model eduction fo a fouth ode tansfe function ship model descibing the sway-yaw-oll dynamics is conducted to each the second ode Nomoto model descibing the sway-yaw dynamics and the fist ode Nomoto model descibing the yaw dynamics itself, and the Bode plots fo these models ae given to show the changes in system fequency esponse caused by model simplification. Thus, appopiate model stuctues can be selected accoding to the intended fequency ange of application to meet the modeling accuacy equiements. INTRODUCTION Pape Received August 10, Revised Septembe 17, Accepted Octobe 25, Autho fo Coespondence: Ching-Yaw Tzeng. *Associate Pofesso, Institute of Maitime Technology, National Taiwan Ocean Univesity, Keelung, Taiwan, R.O.C. **Gaduate Student, Institute of Maitime Technology, National Taiwan Ocean Univesity, Keelung, Taiwan, R.O.C. Ship esponse in waves is typically teated as a six degee-of-feedom igid body motion in space. Wheeas, a thee degee-of-feedom plane motion is usually consideed adequate fo ship maneuveing study [1]. Howeve, fo high speed vessels like the containe ships, tuning motion induced oll is not negligible. Hence a fou degee-of-feedom desciption that includes suge, sway, yaw and oll modes is needed [2, 3]. Since the hydodynamics involved in ship steeing is highly nonlinea, coupled nonlinea diffeential equations ae needed to fully descibe the complicated ship maneuveing dynamics. A simple tansfe function model desciption is usually pefeed when a qualitative pediction capability is all we need fom the model. This is the case in a model-based contolle design, since the feedback contolle itself toleates cetain amount of modeling eo and a complicated model might esult in a contolle too complicated to implement. The populaity of the fist ode Nomoto model in ship steeing autopilot design is due to its simplicity and elative accuacy in descibing the couse-keeping yaw dynamics, whee typically, small udde angles ae involved [4, 5]. Extension to lage udde angle yaw dynamics basing upon the Nomoto model has been poposed to bette descibe the nonlinea behavio of yaw dynamics [6]. Fundamental popeties like the contollability and obsevability of linea systems ae known to be impotant in the design fo contolles. The contollability popety ensues that all the system states can be diven to desied values with the contol input. The obsevability popety ensues that all the system states can be etieved fom the measued output [7]. Specifically, the contollability popety itself ensues the state-feedback contolles can be successfully implemented. The contollability and obsevability popeties togethe ensue the output-feedback contolles can be successfully implemented. The identifiability popety ensues that the system model paametes can be uniquely detemined fom measued input output data, which is cucial to the design of adaptive contol systems [8, 9]. Geneally speaking, state space models ae less identifiable than tansfe function models. Ship steeing state space dynamics models ae also known to have identification difficulties due to the cancellation effect [10]. Since most adaptive ship steeing autopilot designs ae based on tansfe function models, ou identifibaility study will be concened with the Nomoto

2 80 Jounal of Maine Science and Technology, Vol. 7, No. 2 (1999) tansfe function models. Howeve, the contollability and obsevability issues have to be concened with the state space models. This is because the tansfe function models only epesent the contollable and obsevable pats of the system dynamics. Namely, in tansfoming the state space models into the tansfe function models, if thee is any uncontollable o unobsevable mode, it will be cancelled out and can not be seen in the tansfe function model. Hence, it is the state space countepats of the Nomoto tansfe function models will be used in the study fo the contollability and obsevability popeties. Oveshoot behavios ae typical fo undedamped oscillatoy system. Fo plane motion-based ship steeing dynamics, thee is no estoing foces involved. Hence the yaw ate and sway speed will not exhibit oscillatoy behavios. Howeve, fo lage udde angle steeing, oveshoot behavios in the yaw ate can be obseved. It will be shown that this yaw ate oveshoot phenomenon is due to coupling effect fom the sway mode. A tansfe function point of view will be taken to show that the sway coupling effect intoduces a zeo and a high fequency pole into the system. Specifically, the location of the zeo elative to the imaginay axis on the oveshoot behavio will be exploed and the impotance of selecting the appopiate model stuctue in ode to captue key hydodynamic featues, like the oveshoot behavios can be bette ecognized [11]. Finally, model eduction with espect to a fouth ode tansfe function models descibing the sway-yaw-oll modes of motion will be pesented [12]. Simplifications to a second ode model and to a fist ode model descibing the sway-yaw and yaw modes espectively ae discussed. The Bode plots fo these models will be given to illustate the modeling eo incued due to model simplifications. Based on the fequency esponse infomation of the Bode plots, a tade-off between model complexity and model accuacy can be bette execised accoding to the intended pupose and ange of application of the model. SHIP STEERING DYNAMICS MODEL REDUCTION It is well known that coupled nonlinea diffeential equations ae needed to fully epesent the complicated ship maneuveing dynamics. Howeve fo ship steeing autopilots design, a simple model with aveage pedicting capability is usually pefeed. Based on the lineaized suge-sway-yaw-oll equations of motion, a fouth ode tansfe function elating the yaw ate to the udde angle is deived. Futhe simplifications to the second ode Nomoto model and the fist ode Nomoto model ae also descibed. Upon lineaization with espect to a staight line motion with a constant fowad speed u 0, the suge equation is decoupled and the following linea coupled sway-yaw-oll equations follow immediately. Refeing to Fig. 1, we have m(v + u 0 )=Y V V + Y V V + Y ϕ ϕ + Y P P + Y P P + Y + Y + Y δ δ (1a) I X ϕ = K P P + K P P mggmφ + K V V + K V V + K + K + K δ δ (1b) I Z ψ = N + N + N φ φ + N P P + N P P + N V V + N V V + N δ δ (1c) whee Y V, Y V,, indicate the hydodynamic coefficients; fo instance, Y V indicates the deivative of the sway foce Y to the sway speed V evaluated at the efeence condition; m is the mass of the ship; I X is the moment of inetia about the x-axis; I Z is the moment of inetia about the z-axis; V is the sway speed; u is the suge speed; is the yaw ate; Ψ is the heading angle defined by ψ = ; p is the oll ate; φ is the oll angle defined by φ = p and GM is the metacentic height, which indicates the estoing capability of a ship in olling motion. Taking the Laplace tansfom of Eqs. (1a)-(1c) and eaanging, we have whee Fig. 1. Sway-yaw-oll motion coodinate system. a 1 V = a 2 Φ + a 3 + a 4 δ b 1 Φ = b 2 V + b 3 + b 4 δ c 1 = c 2 V + c 3 Φ + c 4 δ (2a) (2b) (2c)

3 C.Y. Tzeng & J.F. Chen: Fundamental Popeties of Linea Ship Steeing Dynamic Models 81 a 1 =(m Y V )S Y V a 2 = Y P S 2 + Y P S + Y φ a 3 = Y S + Y + mu 0 a 4 = Y δ b 1 =(I X K P )S 2 K P S + mggm b 2 = K V S + K V b 3 = K S + K b 4 = K δ c 1 =(I Z N )S N c 2 = N V S + N V c 3 = N P S 2 + N P S + N φ c 4 = N δ (3a) (3b) (3c) (3d) (3e) (3f) (3g) (3h) (3i) (3j) (3k) (3l) Afte eliminating the sway speed V and oll angle φ fom Eqs. (2a)-(2c), the following tansfe function elating the yaw ate to the udde angle δ can be obtained: δ = a 1 (b 1 c 4 + b 4 c 3 )+a 2 (b 4 c 2 b 2 c 4 )+a 4 (b 1 c 2 + b 2 c 3 ) a 1 (b 1 c 1 b 3 c 3 ) a 2 (b 2 c 1 + b 3 c 2 ) a 3 (b 1 c 2 + b 2 c 3 ) (4) It can be easily veified that the numeato of Eq. (4) is thid ode in S, while the denominato is fouth ode in S. Hence, Eq. (4) can be expessed in the following fom δ = K(1 + T 3 S)(S 2 +2ηω 0 S + ω 0 (1 + T 1 S)(1 + T 2 S)(S 2 +2ξω n S + ω n2 ) 2 ) (5) whee the quadatic factos ae due to the coupling effect fom the oll mode on the yaw ate. The zeo (1 + T 3 S) and the pole (1 + T 2 S) ae due to the coupling effect fom the sway mode on the yaw dynamics. If the oll mode is neglected, Eq. (5) can be futhe educed to the following fom δ = K(1 + T 3 S) (1 + T 1 S)(1 + T 2 S) (6) Eq. (6) is known as the second ode Nomoto model, whee K is the static yaw ate gain, and T 1, T 2 and T 3 ae time constants. Numeical values of the paametes in Eq. (6) fo a Maine class vessel ae given by T 1 = 118, T 2 = 7.8, T 2 = 18.5 and K = [7]. The zeo tem (1 + T 3 S) and the high fequency pole tem (1 + T 2 S) ae due to the coupling effect fom the sway mode. In pactice, because the pole tem (1 + T 2 S) and the zeo tem (1 + T 3 S) in Eq. (6) nealy cancel each othe, a futhe simplification on Eq. (6) can be done to give the fist ode Nomoto model δ = K (1 + TS) whee (7) T = T 1 + T 2 T 3 (8) in Eq. (8) is called the effective yaw ate time constant. Eq. (8) is obtained by equating the ight handside of Eq. (6) to the ight handside of Eq. (7) equiing the equality elationship to hold up to fist ode in S. If T 2 = T 3 ; namely, a pefect cancellation occus, then the equality elationship is tue up to second ode in S, then T is of couse, equal to T 1. Fo the Maine class vessel, the value of the effective constant T in Eq. (8) is given by T = The fist ode Nomoto model defined by Eq. (7) is widely employed in the ship steeing autopilot design. The yaw dynamics is chaacteized by the paametes K and T, which can be easily identified fom standad maneuveing tests. In pactice, ship steeing autopilots ae design fo heading angle contol. Hence, it is the tansfe function elating the heading angle Ψ to the udde angle δ being needed in the autopilot design. Since the yaw ate is actually the time deivative of Ψ, the equied tansfe function can be eadily obtained by adding an integato 1/S to the tansfe function models defined by Eq. (6) and Eq. (7). FUNDAMENTAL PROPERTIES OF THE FIRST ORDER NOMOTO MODEL The contollability, obsevability and identifiability popeties of the fist ode Nomoto model-based system will be discussed in this section. The identifiability popety will be discussed with espect to the fist ode Nomoto tansfe function model. Howeve, the contollability and obsevability popeties have to be discussed with espect to the state-space model deived fom the Nomoto model. This is because the tansfe function model always epesents the obsevable and contollable pats of the system dynamics. If thee is any unobsevable o uncontollable pats of the dynamics, they ae cancelled out befoe eaching the tansfe function model fom. Hence, it only makes sense to discuss the contollability and obsevability

4 82 Jounal of Maine Science and Technology, Vol. 7, No. 2 (1999) popeties of the state space model, which contains the obsevable, unobsevable, contollable and uncontollable modes [11]. Eq. (7) can be expessed in time domain as T + = Kδ (9) With the notation ψ = (10) whee Ψ is the heading of the ship. Eq. (9) can be witten as Tψ + ψ = Kδ (11) Eq. (10) and Eq. (11) can be aanged in the standad state space fom whee and x = Ax + Bu y = Cx x = ψ u = δ y = Ψ A = B = K 0 T T C = [1 0] (12a) (12b) (12c) (12d) (12e) (12f) (12g) (12h) Accoding to linea system theoy, the system defined by Eqs. (12) is contollable if the following matix U is of full ank U = [B AB] of full ank V = C CA = (14) 1 Staightfowad computation indicates that the matix U and the matix V defined by Eq. (13) and Eq. (14) espectively ae full ank. Hence, the fist ode Nomoto model-based system is both contollable and obsevable. Eq. (12b) descibes the measuement infomation. Accoding to Eq. (12h), the measued signal is the heading angle Ψ, which is eadily available onboad almost all the vessels. In the above discussion, contollability indicates the system states (Ψ, ) can be contolled to abitay value via application of the udde δ. Obsevability indicates the system states (Ψ, ) can be obtained via the measued data Ψ. Moeove, contollability implies the state-feedback contolle will be successful. With the addition of obsevability implies that the output-feedback contolle will be successful. Identifiability of the fist ode Nomoto model defined by Eq. (7) implies that the paametes K and T can be uniquely detemined fom the input udde angle δ and the output yaw ate. Since, this is equivalent to fitting a fist ode model to the measued input-output data, and the gain K and the time constant T ae uniquely detemined. Consequently, the fist ode Nomoto model satisfies the identifiability popety. Hence, on-line estimation of the model paametes based on the measued udde and yaw ate infomation will be possible and adaptive contol stategy can be successfully implemented. FUNDAMENTAL PROPERTIES OF THE SECOND ORDER NOMOTO MODEL Simila to pevious discussions the identifiability condition will be discussed basing upon the tansfe function model defined by Eq. (7), and the contollability and obsevability conditions will be discussed basing upon the coesponding state space model. Recall that the second ode model Nomoto model defined by Eq. (6) is obtained fom Eq. (1a) and Eq. (1c), while neglecting the oll mode. Namely, = 0 K T K T K T 2 (13) m(v + u 0 )=Y v v + Y v v + Y + Y + Y δ δ I z = N v v + N v v + N + N + N δ δ (15a) (15b) and the system is obsevable if the following matix V is Eqs. (15) can be put in state space fomat as

5 C.Y. Tzeng & J.F. Chen: Fundamental Popeties of Linea Ship Steeing Dynamic Models 83 m Y v Y 0 N v I z N v ψ + Y v mu 0 Y 0 N v N v ψ b 11 = b 21 = (I z N )Y δ + Y N δ (m Y v )(I z N ) N v Y (17m) (m Y v )N δ + N v N δ (m Y v )(I z N ) N v Y (17n) = Y δ N δ δ (16) 0 Eq. (16) can be futhe eaanged in the standad state space fomat as Contollability of the second ode Nomoto modelbased system descibed Eqs. (17) is satisfied if the following matix U is of full ank U = [B AB A 2 B] x = Ax + Bu (17a) a 2 11 b 11 + a 11 a 12 b 21 y = Cx whee (17b) = b 11 a 11 b 11 + a 12 b 21 + a 12 a 21 b 11 + a 12 a 22 b 21 a 21 a 11 b 11 + a 21 a 12 b 21 b 21 a 21 b 11 + a 22 b 21 + a 22 a 21 b 11 + a 22 b 21 0 b 21 a 21 b 11 + a 22 b 21 x = v ψ (17c) (18) u = δ y = Ψ (17d) (17e) Afte some computation, it can be veified that the matix U defined by Eq. (18) is of full ank if a 12 b 21 a 22 b 11 b 11 (a 21 b 11 a 11 b 21 ) (19) and A = B = a 11 a 12 0 a 21 a b 11 b 21 0 C = [0 0 1] The elements in matix A and B ae given by a 11 = (17f) (17g) (17h) (I z N )Y v + Y N v (m Y v )(I z N ) N v Y (17i) a 12 = (I z N )(Y v mu 0 )+Y N (m Y v )(I z N ) N v Y (17j) a 21 = (m Y v )N v + N v Y v (m Y v )(I z N ) N v Y (17k) a 22 = (m Y v )N + N v (Y v mu 0 ) (m Y v )(I z N ) N v Y (17l) Altenatively, the condition defined by Eq. (19) can be veified by equiing the columns in Eq. (18) not popotional to each othe and the ows in Eq. (18) not popotional each othe. Contollability condition of the system defined by Eqs. (17) implies that the states Ψ, and v can be diven independently to abitay values via the udde angle δ. Implication of the contollability condition will be futhe exploed. Recall that by eliminating the sway velocity v in Eq. (15a) and Eq. (15b) leads to the second ode Nomoto model defined by Eq. (6). Similaly by eliminating the yaw ate in Eq. (15a) and Eq. (15b) leads to the sway to udde tansfe function v δ = K v (1 + T v S) (20) (1 + T 1 S)(1 + T 2 S) whee K v is the static sway gain coefficient and T v is the sway time constant. It is to be noted that the poles of the sway-udde model defined by Eq. (20) ae exactly the same as those in the yaw-udde model defined by Eq. (6). Fom Eq. (6) and Eq. (20), it follows that v = K v (1 + T v S) K(1 + TS) (21) Eq. (21) shows that, if T v = T, then v is popotional to. Namely, v is dependent on, since K v /K is a

6 84 Jounal of Maine Science and Technology, Vol. 7, No. 2 (1999) Fig. 2. Unit step esponse, 10(1 + 40S) (1+10S)(1 + 20S). Fig. 4. Unit step esponse, 10(1 + 15S) (1+10S)(1 + 20S). constant. Howeve, we ecall that fo the second ode Nomoto model-based system defi]öd by Eqs. (17). If the system is contollable, then the state vaiables Ψ,, v have to be able to move independently via application of the udde δ. It is thus infeed that fo the system to be contollable, T v must be diffeent fom T. Namely, T v T is equivalent to the condition given in Eq. (19) that ensues the contollability of the system. Obsevability condition of the system defined by Eqs. (17) is satisfied if the following matix V is of full ank V = = Fig. 3. Unit step esponse, C CA CA a 21 a (1 + 25S) (1+10S)(1 + 20S). (22) It can be easily veified that Eq. (22) is of full ank fo any ship. Hence, the second ode Nomoto modelbased state space system is obsevable. This implies that all the states (Ψ,, v) can be econstucted fom the measued heading angle Ψ. Identifiability of the second ode Nomoto model defined by Eq. (6) implies that the paametes K, T 1, T 2 and T 3 appeaing in Eq. (6) can be uniquely detemined fom the input δ and output measuements. It is clea that if T 2 = T 3, then it is impossible to identify all the fou paametes K, T 1, T 2 and T 3. Since the zeo tem (1 + T 3 S) will cancel out the pole tem (1 + T 2 S). It is then infeed that the system defined by Eq. (6) is identifiable if T 2 T 3. In pactice, the value of T 2 is nealy equal to the value of T 3 and we have a nea cancellation situation. This will esult in an ill-conditioning poblem when tying to identify the values of T 2 and T 3 appeaing in the second ode Nomoto model defined by Eq. (6). Due to this ill-conditioning popety, the fist ode Nomoto model defined by Eq. (7) is pefeed to Eq. (6) in the design fo an adaptive autopilot, whee eliable on-line estimation of the model paametes is needed. SYSTEM OVERSHOOT AND ZERO LOCATION In this section, the effect of the zeo tem appeaing in the second ode Nomoto model defined by Eq. (6) will be studied. Unit step esponse will be given to illustate the elation between the zeo location and the existence of oveshoot behavios. Unit step esponses fo the second ode Nomoto model defined by Eq. (6) with the following numeical data will be pesented. Specifically, K = 10, T 1 = 10, T 2 = 20. Fou values will be assigned to T 3 to epesent diffeent locations of the zeo. Specifically, Fig. 2 coesponds to the unit step esponse fo T 3 = 40, Fig. 3 coesponds to T 3 = 25, Fig. 4 coesponds to T 3 = 15 and Fig. 5 coesponds to T 3 = 5 espectively. The oveshoots ae obseved in Fig. 2 and Fig. 3 but do not appea in Fig. 4

7 C.Y. Tzeng & J.F. Chen: Fundamental Popeties of Linea Ship Steeing Dynamic Models 85 and Fig. 5. Moeove, the oveshoot is lage in Fig. 2 than that in Fig. 3. By examing the position of the zeo elative to the pole, it is found that the close the zeo is located nea the imaginay axis, the lage the oveshoot is. Moeove, fo the oveshoot to exist, the zeo must be located to the ight of the poles. In pactice the oveshoot in the yaw ate is obseved in lage udde angle tuning maneuve. Thus, it is infeed that the second ode Nomoto model has to be employed, if the oveshoot phenomenon is to be captued. Since the fist ode Nomoto model can neve exhibit an oveshoot phenomenon, it is indeed only suitable fo the desciption of small udde angle yaw dynamics, whee oveshoot in the yaw ate will not appea. Moeove, fo the steeing dynamics, thee is no hydodynamic estoing foce involved. Hence the yaw ate will not have an oscillatoy behavio and the ship steeing dynamics will always act like an ove-damped system. Consequently, the poles ae always of eal values and this futhe justifies that the second ode Nomoto model that has two eal poles is indeed appopiate in descibing the plane motion-based ship maneuveing dynamics. Recall that when educing the second ode Nomoto model defined by Eq. (6) to the fist ode Nomoto model defined by Eq. (7) actually neglects the sway coupling effect on the yaw mode. The zeo and the high fequency pole ae then neglected. Thus it can be infeed that the oveshoot in the yaw ate is actually caused by the sway coupling effect, and the tighte the maneuve is, the lage the sway coupling effect is. Fom the above discussions it is concluded that the fist ode Nomoto model is suitable fo descibing the small udde angle yaw dynamics, and needs only two paametes to chaacteize the system behavio. This makes it elatively easy to identify the model fom expeiment data. The second ode Nomoto model has bette capability in captuing the oveshoot behavio. Howeve, nea cancellation of one of the poles and zeo makes it an ill-conditioning poblem in identifying the model paametes fom expeiment data makes it less attactive fo an adaptive autopilot application. In this pape, an altenative appoach is suggested fo an adaptive autopilot implementation basing upon the second ode Nomoto model. Since the zeo of the tansfe function helps bette descibing the yaw dynamic oveshoot behavio, its stuctue is etained and the paamete is fixed at a value detemined off-line fom input-output expeiment data. The model descibed by Eq. (6) is thus unchanged, except that the paamete T 3 is fixed. Hence, only the paamete K, T 1 and T 2 need to be identified on-line. This stategy peseves the featue of the second ode Nomoto model without intoducing the ill-conditioning poblem in identifying the model paametes makes it moe attactive than the conventional implementation of an adaptive autopilot based on the fist ode Nomoto model defined by Eq. (7). MODEL SIMPLIFICATION AND BODE PLOTS A fouth ode linea state space model epesenting the sway-yaw-oll modes of motion given in Ref. [12] will be used as the nominal model in constucting the coesponding yaw to udde tansfe function. Futhe simplification to the second ode Nomoto model and the fist ode Nomoto model will also be pesented. The nominal state space model is given by whee and x = Ax + Bu y = Cx x = φ p v u = δ y = A = Fig. 5. Unit step esponse, 10(1 + 5S) (1+10S)(1 + 20S) (23a) (23b) (23c) (23d) (23e) (23f)

8 86 Jounal of Maine Science and Technology, Vol. 7, No. 2 (1999) Fig. 6. Bode diagams, fouth ode model. Fig. 8. Bode diagams, fist ode model. B = Fig. 7. Bode diagams, second ode model C = [ ] (23g) (23h) Using the MATLAB, the tansfe function fom input udde δ to the output yaw ate is obtained as δ = S S S S S S S (24) The numeato and denominato of Eq. (24) ae of thid ode and fouth ode in S espectively, and this agees with that of Eq. (4). It is to be noted that Eq. (24) epesents the yaw dynamics with inclusion of the oll and sway coupling effects. By neglecting the oll mode in Eq. (23), the following yaw ate to udde tansfe function that includes the sway coupling effect can be obtained S = (25) δ S S Eq. (25) coesponds to the second ode Nomoto model defined by Eq. (6) Futhe neglecting the sway coupling effect esults in the model that has yaw mode itself only δ = (26) S Eq. (26) coesponds to the fist ode Nomoto model defined by Eq. (7) Bode plots that epesent the fequency domain yaw esponses of the models defined by Eqs. (24)-(26) ae given in Figs. 6-8 espectively. Each figue has a magnitude plot and a phase plot. Specifically, Fig.6 coesponds to the fouth ode model, Fig. 7 coesponds to the second ode model and Fig. 8 coesponds to the fist ode model. The magnitude plot and phase plot of the second ode Nomoto model ae about the same as those of the fouth ode model, expect that thee ae humps associated with the fouth ode model. The diffeence between the second ode Nomoto model and the fist ode Nomoto model is somewhat lage. Specifically, thee is a 5db magnitude diffeence and a 20 deg phase diffeence fo the fequency ange between 10 2 ad/sec to 10 1 ad/sec. Based on the esults of Figs. 6-8, it is clea that the effect of model simplification fom the fouth ode model to the second ode Nomoto model is not significant; namely, the coupling effect of the oll mode on the yaw motion is negligible. Howeve, the effect of model eduction fom the second ode Nomoto model to the fist ode Nomoto model is moe significant; namely, the coupling effect of the sway mode on the

9 C.Y. Tzeng & J.F. Chen: Fundamental Popeties of Linea Ship Steeing Dynamic Models 87 Fig. 9. Yaw ate esponse due to udde, fouth ode model. Fig. 11. Yaw ate esponse due to udde, fist ode model. CONCLUSIONS Fig. 10. Yaw ate esponse due to udde, second ode model. yaw motion is not negligible. Hence fo the ship model epoted in Ref. [12], it seems easonable to use the second ode Nomoto model defined by Eq. (25) to epesent the behavio of the fouth ode model epesented by Eq. (24) in an autopilot design. Step input esponse of the ship models descibed by Eqs. (24)-(26) is also given to exhibit thei time domain behavios. Specifically, Figs coespond to the yaw ate esponse, due to 35 deg udde input, of a fouth ode, second ode and fist ode model espectively. With the fequency esponse infomation of the Bode plots fo models with diffeent level of complexity, modeling eos caused by model eduction can be bette assessed. The yaw ate esponse in the time domain povides a convenient way of assessing the behavio of each model. It is well known that a moe complicated model tends to bette descibe the system behavio in a wide fequency ange. Howeve, the intended fequency ange of a dynamic system is usually limited a finite inteval, which is usually oughly known in advance. This infomation is helpful in choosing the appopiate model that achieves easonable accuacy in the fequency ange of inteest. The fist ode Nomoto model being vey popula in the design fo ship steeing autopilots is not without eason. Its simplicity and easonable accuacy in descibing small udde angle yaw dynamics make it attactive. Moeove, the elative easiness in identifying the model paametes makes it suitable fo adaptive autopilot application, whee on-line estimation of the model paametes is impotant. The second ode Nomoto model includes the coupling effect fom the sway to the yaw mode. This intoduces a zeo and a high fequency pole into the tansfe function. The zeo stuctue contibutes to the oveshoot phenomenon, which can be seen in the yaw ate fo lage udde angle maneuves. Howeve, the ill-conditioning poblem associated with the second ode Nomoto model in the identification of the model paametes fom input-output data seems to outweigh the impovement gained in its modeling capability. An appoach that etains the zeo stuctue while avoiding the ill-conditioning poblem duing on-line estimation has been poposed. The fist ode Nomoto model s state space countepat is found to be both contollable and obsevable; hence, both the state feedback and output feedback contolles can be successfully implemented. The second ode Nomoto model s state space countepat is also obsevable; howeve, the system is not contollable, if the effective sway time constant T v is equal to the effective yaw ate time constant T. Bode plots of models with diffeent level of complexity ae useful in chaacteizing the elative modeling eos incued due to model eduction on the fequency domain. With the help of Bode plots, justification of using a simple model can be done within the intended fequency ange of application to ensue that the fequency esponse of the simple model easonably appoximates that of the complicated ones.

10 88 Jounal of Maine Science and Technology, Vol. 7, No. 2 (1999) REFERENCES 1. Lewis, E.V., Edito, Pinciples of Naval Achitectue, Vol. III Motion in Waves and Contollability, Society of Naval Achitects and Maine Enginees, New Jesey (1989). 2. Son, K.H. and Nomoto K., On the Coupled Motion of Steeing and Rolling of High Speed Containe Ship, Jounal of Society of Naval Achitects, Japan, Vol. 150, pp (1981). 3. Oltmann, P., Roll-An often Neglected Element of Maneuveing, Intenational Confeence on maine Simulation and Maneuveability, New Foundland, Canada (1993). 4. Nomoto, K., Taguchi, K., Honda, K. and Hiano, S., On the Steeing Quality of Ships, Intenational Shipbuilding Pogess, Vol. 4, pp (1957). 5. van Ameongen, J., Adaptive Steeing of Ships A Model Refeence Appoach to Impoved Maneuveing and Economical Couse-Keeping, Ph.D. Thesis, Delft Univesity of Technology, The Nethelands (1982). 6. Nobin, N.H., On the Design and Analysis of the Zig-Zag Test on Base of Quasilinea Fequency Response, Technical Repot No. B140-3, The Sweden State Shipbuilding Expeimental Tank (SSPA), Gothenbug, Sweden (1963). 7. Fossen, T.I., Guidance and Contol of Ocean Vehicles, John Wiley and Sons, NY (1994). 8. Astom, K.J. and Kallstom, C.G., Identification of Ship Steeing Dynamics, Automatica, Vol. 12, pp (1976). 9. Holzhute, T. and Stauch, H., A Commecial Adaptive Autopilot fo Ships : Design and Opeational Expeiences, IFAC 10th Tiennial Wold Confeences, pp , Munich, Gemany (1987). 10. Hwang, W.Y., Cancellation Effect and Paamete Identifiability of Ship Steeing Dynamics, Intenational Shipbuilding Pogess, Vol. 26, No. 332, pp (1982). 11. Goodwin, G.C., Gaebe S.E. and Salgado, M.E., Pinciples of Contol System Design, Depatment of Electical and Compute Engineeing, Univesity of Newcastle, Callaghan, Austalia (1997). 12. Zhou, W.W., Chechas, D.B. and Calisal, S., Identification of Rudde-Yaw and Rudde-Roll Steeing Models by Using Recusive Pediction Eo Techniques, Optimal Contol Application and Methods, Vol. 15, pp (1994).!"#$%&'(!!"#$%&'()*+!!"#$%&'(&')!"#$%&'()*+,-./01!"#$%&'()*+,*-./0123!"#$%&'()!*+,-./012!"#Nomoto!"#$%& '(!"#$%&'()*+,-./01234!"#$%&'()*+Nomoto!"!"#$%&'Nomoto!"#$%!"#$%#&'()*+,-./012!"#$%&'()*+,-Nomoto!"#$%&'()*+,-.*/0-12 Nomoto!"#$%& '()*+!"#$%&'()*+,$%&'$-./!"#Nomoto!"#$%&'()$!"#$%&'()*+,-./012!"#$%&'()*+,-./01234!"#$%&'()*+,-#Nomoto!"#$%&'()*+,-./0+1#2!"#$%&'()*+,-./$0123!"#$%&'(!)*+, #$%-'!"#$Bode!"#$%&'()*!"#$%&'()*+,-./01234!"#$!%&'()*+,-./012!"#$%