指導教授:曾慶耀 博士
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指導教授:曾慶耀 博士 姓名:林柏呈 學號:19967039 PowerPoint PPT Presentation


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指導教授:曾慶耀 博士 姓名:林柏呈 學號:19967039. Fundamental Properties Of Linear Ship Steering Dynamic Models. Index. Abstract Fundamental Properties Of The First Order Nomoto Model Fundamental Properties Of The Second Order Nomoto Model System Overshoot And Zero Location

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指導教授:曾慶耀 博士 姓名:林柏呈 學號:19967039

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指導教授:曾慶耀 博士姓名:林柏呈學號:19967039

Fundamental Properties

Of

Linear Ship Steering Dynamic Models


Index

Index

Abstract

Fundamental Properties Of

The First Order Nomoto Model

Fundamental Properties Of

The Second Order Nomoto Model

System Overshoot And Zero Location

Model Simplification And Bode Plots


1 abstract

1. Abstract

This paper is concerned with the fundamental properties associated with the Nomoto models. Specifically, the state space model associated with the first order Nomoto model is both observable and controllable. The state space model associated with the second order Nomoto model is also observable; however, it is controllable only if the effective sway time constant is different from the effective yaw time constant.

The zero appearing in the transfer function model is found responsible for the overshoot behaviors, which are typical in the yaw rate for large rudder angle steering. This suggests that a second order Nomoto model is more appropriate if the overshoot feature is to be properly modeled.

Both the first and second order Nomoto transfer function models are identifiable, with an ill-conditioning problem associated the latter. This makes the first order Nomoto model very popular in the adaptive autopilot applications.

Model reduction for a fourth order transfer function ship model describing the sway-yaw-roll dynamics is conducted to reach the second order Nomoto model describing the sway-yaw dynamics and the first order Nomoto model describing the yaw dynamics itself, and the Bode plots for these models are given to show the changes in system frequency response caused by model simplification. Thus, appropriate model structures can be selected according to the intended frequency range of application to meet the modeling accuracy requirements.


2 ship steering dynamics model reduction

2. Ship Steering Dynamics Model Reduction

耦合非線性微分方程需要充分代表複雜的船舶運動。

就自航器的設計觀點,多希望能以較簡單數學模式來描述船舶特性,以便於控

制器的設計易於實現。故多將非線性微分方程式針對一平衡狀態,如等速直線

運動予以限性化,達到模式簡化的目的。因此對於高速貨櫃船舶,在此線性化

的結果下,做一方程式將自動解耦合,僅剩下橫移、平擺、橫搖三個自由度的

運動方程式Fig.1:


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對高速船舶進行自航器的設計,在線性化後僅考慮橫移、橫搖、平擺三個自由

度的運動方程式如下:       其中 , …為流體係數

                    為船舶質量(mass of the ship)

                    為對 軸之轉動慣量

                    為對 軸之轉動慣量

                    為橫移速度(sway speed)

                    為縱移速度(surge speed)

                    為平擺角速率(yaw rate)

                    為航向角(heading angle)

     定義為 =

                    為橫搖速率(roll rate)

                    為橫搖角(roll angle)

Eqs.1a-1c採取Laplace轉換可得:       定義為 =

為舵角(rudder angle)

                    為定傾高(metacentric height)


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註解

嚴格而言,流力導數 、 等因船體造波及渦流散播,將與船體之過去運動特

性有關,亦即應以迴旋積分表示所謂記憶效應。等於平靜水域之低頻運動(如

一般商船操縱),可忽略記憶效應而將 、 等視成瞬間特性之函數。

(sway speed)為船舶沿 軸之橫移速度

(surge speed)為船舶沿 軸之縱移速度

(yaw rate)    為平擺角速率、方向角速率

(heading angle) 為方向角、方位角、航向角

(roll angle)   為橫搖角、橫傾角

(metacentric height)定傾高

表明船舶橫搖運動回復能力。

若  為正值,當該船傾斜一小角度時,

則具有一扶正的力矩,且  值愈大時,

扶正力矩愈大。

故定傾高為  值的大小作為船體初穩定

的衡量標準。


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其中


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其中


3 fundamental properties of the first order nomoto model

3. Fundamental Properties Of The First Order Nomoto Model

一階Nomoto模式Eq.7:

以時域方式表示成Eq.9:

平擺角速率為航向角對時間的微分:

因此將Eq.9改寫:

將Eq.11以狀態空間模式描述如下:


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其中

(12c) (12d) (12e)

(12f) (12g) (12h)

系統可控制性 及可觀察性 如下:

Eq.13和Eq.14皆為滿秩數,因此一階Nomoto模式為可控制和可觀測。

至於可鑑定性可由Eq.7決定,因為參數 、 可由輸入舵角 和輸出平擺角速

率 決定出唯一的值,因此一階Nomoto模式也為可鑑定。


4 fundamental properties of the second order nomoto model

4. Fundamental Properties Of The Second Order Nomoto Model

二階Nomoto模式Eq.6:

將船舶之橫搖模式忽略:

因此對Eq.1a和Eq.1c省略橫搖模式:

改寫為:


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Eq.15可以放在狀態空間格式如下:

將Eq.16以狀態空間模式描述如下:

其中

(17c) (17d) (17e) (17f) (17g) (17h)


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矩陣A和B中的元素分別表示如下:


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系統可控制性 如下:

經計算得知,若為Eq.19時,則為滿秩數,二階Nomoto模式為可控制:


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由Eq.15a和Eq.15b也可計算出橫移速度 對舵角 的二階轉移函數

Eq.20與二階Nomoto模式Eq.6做一比較,可得Eq.21:

由Eq.21可得,若 = 時,橫移速度與平擺角速率為依比例關係。而可控制性

仍是藉由輸入控制訊號將系統從初始狀態分別單獨達到所希望的

狀態,故變數間應為獨立不相關。

因此二階Nomoto模式在 ≠ ,即Eq.19的條件下,為可控制系統:


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系統統可觀測性 如下:

Eq.22為滿秩數,因此二階Nomoto模式為可觀測。

至於可鑑定性,則藉由二階Nomoto模式的轉移函數Eq.6進行分析。

由Eq.6發現 =  時,Eq.6將發生對消現象,無法辨識出個別參數,因此對二

階Nomoto模式而言,必須 ≠ 才可鑑定出各個個別參數。

而由一般時傳資料處理中得知Eq.6中一個極點位置與一個零點位置接近,故二

階Nomoto模式為較難鑑定系統。


5 system overshoot and zero location

5. System Overshoot And Zero Location

零點項出現在二階Nomoto模式由Eq.6將進行研究 :

=10

=10

=20

=Fig.2為40

Fig.3為25

Fig.4為15

Fig.5為5


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By examing the position of the zero relative to the pole ,it is found that the closer the zero is located near the imaginary axis, the larger the overshoot is. Moreover, the zero must be located to the right of the poles.

The first order Nomoto model is suitable for describing the small rudder angle yaw dynamics, and needs only two parameters , to characterize the system behavior.

The second order Nomoto model that has two real poles , is indeed appropriate in describing the plane motion-based ship maneuvering dynamics, and has better capability in capturing the overshoot behavior.

一階Nomoto模式Eq.7實際上忽略了橫移對平擺之耦合效應。

二階Nomoto模式Eq.6零點和高頻極點被忽視。

由此推斷,在平擺角速率 的overshoot實際上是橫移耦合所造成的影響。


6 model simplification and bode plots

6. Model Simplification And Bode Plots

所使用船舶數學模式,為Ref.12中所提出的線性船舶狀態空間模式,如下:

其中

(23c) (23d) (23e)

而且

(23f) (23g) (23h)


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利用MATLAB,可計算出輸出平擺角速率 對輸入舵角 的四階轉移函數如下:

若將Eq.24中橫搖模式忽略,可得到簡化之二階Nomoto模式如下:

再將Eq.25簡化,橫移對平擺的耦合效應忽略,可得到一階Nomoto模式如下:


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Fig.6-8分別為依據Eq.24-26所繪出的Bode曲線圖:

(一階) (二階) (四階)

從四階Nomoto模式簡化到二階Nomoto模式其效應是不顯著;即在平擺運動的橫搖模式耦合效應是可忽略。

從二階Nomoto模式簡化到一階Nomoto模式其效應更加顯著;即在平擺運動的橫移模式耦合效應是不可忽略。

Fig.6和Fig.7發現兩者很相近,除了四階模式於頻率  ~rad/sec間有一小凸起。

Fig.6和Fig.8則有稍大差異,在頻率  ~rad/sec間約有5dB振幅和20°的相位差異。


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註解

Fig.9-11分別為依據Eq.24-26所繪出35°舵角的步階響應圖:

(一階) (二階) (四階)

由Fig.9-11中,可以發現三者皆有相同的趨勢,而其中以二階模式較一階模式更接近四階模式;即愈複雜的模式愈能詳細描述系統之真實動態。

船舵迴旋時,使用的最大舵角多只於35°,此因為若舵角在上升,將發生流場

分離(separation)現象。

該現象可視成流場的過度不對稱,反而形成潰散,此將造成舵力不增反減的效

果。該情況類似於過度增加飛機的爬升攻角(angle of attack),最後反而導

致升力驟降,亦即所謂的失速現象。


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