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Rudder roll reduction for small ships 1993

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RUDDER ROLL REDUCTION FOR SMALL SHIPS by Christopher Robert Martin B.A.Sc., The University of British Columbia, 1989 A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming THE UNIVERSITY OF BRITISH COLUMBIA April 1993 © Christopher Robert Martin, 1993 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of  /We^/FYI. The University of British Columbia Vancouver, Canada Date ^?3/607 DE-6 (2/88) Abstract Roll motion on a ship can be very uncomfortable, and in the extreme, dangerous. It is possible to use the rudder to reduce roll motion as has been demonstrated by others on naval and coast guard vessels. For this study, five controllers were chosen and designed to implement a rudder roll reduction system as part of an autopilot on a fishing vessel. A numerical model and a physical model were prepared for testing these controllers. The results from the numerical model indicate rudder roll reduction does not work for the fishing vessels tested. These results are suspect for several reasons. Table of Contents Abstract ^  ii List of Tables ^  vi List of Figures ^  vii Nomenclature ^  ix Part 1: Introduction ^ 1 1.1 Problem 3 1.2 Scope & Assumptions ^ 4 1.3 General Approach 6 1.4 Literature Review ^ 7 Part 2: Design Process ^ 9 2.1 System ^ 9 2.2 Detailed Design ^ 11 2.3 Parameter Identification ^ 15 2.4 PID Controller ^ 17 2.5 Minimum Variance Controller ^ 18 2.6 GPC Controller ^ 20 2.7 Discrete-Time Sliding Mode Control ^ 22 2.8 Pole Placement Control  ^23 iii Part 3: Numerical Simulation ^ 26 3.1 Numerical Model 26 3.2 Controller Parameters ^ 31 3.3 Tests ^ 34 3.4 Results 36 3.4 Evaluation of Roll Reduction ^ 48 Part 4: Physical Model ^ 53 4.1 Yaw Sensor 54 4.2 Roll Sensor ^ 54 4.3 Telemetry System 55 4.4 Data Acquisition 55 4.5 Computer ^ 55 4.6 Radio Control ^ 56 4.7 Towing Tank 57 4.8 Tests ^ 58 Part 5: Conclusions ^ 59 Part 6: Future Work ^ 60 Part 6: Bibliography ^ 61 Appendix I: Approximate Maximum Likelihood Identification .^. 64 Appendix II: Generalized Predictive Control ^ 66 Appendix III: Discrete-Time Variable Structure Control^. •^. 68 iv Appendix IV: Equations of the Transverse Motions ^ 70 Appendix V: Wave Excitation ^  79 Appendix VI: Numerical Simulation Results ^  81 List of Tables Table^ Page 1. Kynoc Particulars ^  28 2. Eastward Ho Particulars. ^  28 3. Dimensionless Quantities  49 4. Model Particulars. ^  53 vi List of Figures Figure Page 1. Ship geometry and motions ^ xi 2. System Diagram ^ 9 3. Sway Excitation Spectrum ^ 30 4. Yaw Excitation Spectrum 30 5. Slope Spectrum ^ 31 6. VSS Impulse Response for Kynoc ^ 37 7. VSS Roll Reduction Impulse^Response for^Kynoc ^ 37 8. Pole Placement Impulse Response for Kynoc 38 9. Pole Placement Rudder Roll Reduction Impulse Response for Kynoc ^ 38 10. PID Impulse Response for Kynoc ^ 39 11. Kynoc 900 Seas ^ 42 12. Kynoc 45° Seas 42 13. Kynoc 60° Seas ^ 43 14. Kynoc 300 Seas 43 15. Eastward Ho 90° Seas ^ 44 16. Eastward Ho 450 Seas 44 17. Eastward Ho 60° Seas 45 18. Eastward Ho 30° Seas ^ 45 19. VSS Roll Reduction Improvement 15°/s Rudder Rate^• • • 46 20. Pole RR Roll Reduction Improvement 150/s Rudder Rate . . 46 21. MV_RR Roll Reduction Improvement 15°/s Rudder Rate . . . 47 22. SS GPC RR Roll Reduction Improvement 15°/s Rudder Rate . 47 23. Example of yaw parameter identification ( random sea vi i test - Eastward Ho) ^ 50 24. Sample Roll Identification (Random Sea CAse - Eastward Ho) ^ 51 25. Yaw Parameter Estimates ^ 52 26. Model Testing Arrangement 57 27. Geomerty of Rudder Moments ^ 71 28. 100 Rudder Step Response 77 29. 100/100 Zig-zag, rudder and yaw^ 78 30. 100/100 Zig-zag, roll^ 78 viii Nomenclature This report covers two engineering areas, each with its own symbol standards. Therefore, two nomenclature listings are provided, one for control and one for naval architecture. There is some cross-over, for example 0 is used for roll angle in both control discussion and naval architecture discussion. Control Nomenclature A — system matrix A(z ▪ ) — system polynomial B — input matrix B(z ) — input polynomial AML S(k) — sliding surface s(k) - ideal set-point TD - derivative time constant T/ — integral time constant T0 - roll natural period C — output matrix^T* - yaw rate time constant G — sliding surface ü — vector of future controls coefficients^x — state vector Emu — predicted output response^ a — EFRA factor J — cost function — EFRA factor K — arbitrary gain^6 — EFRA factor K(t) — gain matrix in AML^e — boundary layer thickness Kr — roll weight^n(t) — a posteriori errors Kry — roll velocity weight^e — estimated parameters Ey yaw velocity weight^X — EFRA exponential forgetting N — prediction horizon factor Nu — control horizon^0 — roll angle P(t) — covariance matrix in^— predicted response ix • - predicted free^KH - hydrodynamic roll moments response^ KR - rudder roll moment - yaw angle K - wave excited roll moment L - ship length m - ship mass Naval Architecture^mx - added mass in the x Nomenclature^ direction my - added mass in the y AR - area of rudder^ direction B - ship's beam NH - hydrodynamic yaw moments b - linear roll damping^NR - rudder yaw moment CH - block coefficient Nw- wave excited yaw moment D - moulded depth^r - yaw rate F, - Froude Number T,d - mean draught GMT - effective metacentric^u - surge velocity height^ V - ship's velocity g - gravitational^v - sway velocity acceleration YH - hydrodynamic sway forces H/ - significant wave^YR - rudder sway force height^ Yw - wave excited sway force Im - roll mass moment of^zR - vertical distance from c.g. inertia^ to rudder. In - yaw mass moment of inertia^ aR - effective rudder in-flow Jm - added roll mass moment^angle of inertia^ A - ship's displacement - added yaw mass moment^6 - rudder command angleLin of inertia^ - roll damping ratio t — wave slope amplitude X — rudder aspect ratio p — density of sea water r — trim x — heading angle relative to waves w — wave frequency we — wave encounter frequency Figure 1. Ship geometry and motions xi Part 1: Introduction Rolling is one of six motions that all ships experience. As rolling becomes more pronounced, it affects both the crew and the vessel. Rolling reduces task proficiency and both directly and indirectly increases the risk of injury. The ship's fittings, such as machinery mounts and cargo holding structures and systems, must resist the resulting forces. Excessive motion can lead to capsize, an endangerment of both life and vessel. By decreasing roll motion, not only is the comfort level increased, but safety is also increased. There are many methods available to decrease roll motion, though all are not currently used. All methods increase roll damping, thereby reducing the dynamic amplification at resonance of the roll motion. They can be divided into two types: passive and active. The most common passive stabilizing devices are bilge keels, which are fitted on most round bilged boats, but rarely on hard chined vessels. Bilge keels work through viscous effects and vortex shedding at very low speeds. As the speed of the boat increases, a hydrodynamic lift begins to act, generating an opposing roll moment. Less common passive devices are passive roll tanks. Roll tanks are designed to dampen the roll motion by having water slosh inside them 90 degrees out of phase of the roll motion. However, they add weight, take up cargo space and reduce overall stability, because they reduce the metacentric height. Roll tanks are rarely used on local fishing vessels. Two passive devices that are used locally in the Pacific Northwest are paravanes and batwings.^Paravanes are triangular sheets of wood or metal that are hung in the water from poles extending laterally outward from the boat. They work by alternately diving downward when the pole to which they are attached rises, producing an opposing roll moment. Paravanes are often found on trawlers because of the existing poles used in the trawling process. The last passive devices are batwings, which work in much the same way as bilge keels. Batwings are sheets of metal perpendicular to and extending laterally outward from the keel. They decrease the roll through vortex shedding and, at higher forward speeds, through dynamic lift. However, they also increase the resistance of the vessel. The second type of stabilizers used to decrease roll motion are active stabilizers. Active systems are characterized by sensors, controllers, and actuators. Active tanks are one type of active stabilizers. Active tanks work on much the same principle as passive tanks; however, a sensor is used to tell a pump when and which way to move the 2 water. An active tank system results in the use of smaller tanks, and therefore little loss in stability and active tanks can be used in a wide range of conditions. The disadvantage is the weight and size of the pump necessary and the energy required to operate the pump. Another type of active damper are fins, which protrude from the hull . A sensor measures the direction and speed of the roll and the fins' angles are adjusted to produce an opposing roll moment. The disadvantages of fins are they are vulnerable to damage, ineffective at low speeds, and in large motions can stall, producing little lift and little opposing roll moment. The rudder, with which almost every ship is fitted, is usually placed below the centre of gravity of the ship. The lift force of the rudder acts below the roll centre of the ship, producing a roll moment. Because the ship's speed of response in roll is much faster than that in yaw, a control scheme has been implemented on some vessels, which reduces the roll without affecting yaw. 1.1 Problem The primary problem is to develop a rudder roll reduction autopilot with applicability to a broad range of small vessels. Because the autopilot is for small vessels, secondary characteristics (problems) that should be considered are: 3 -the autopilot should be cost effective for the small vessel's owner. -the sensors necessary should be inexpensive and few. -there should be no major alteration to the steering gear. 1.2 Scope & Assumptions The control system that was explored in this thesis is a rudder roll reduction system as part of an autopilot. A number of controllers were selected and then evaluated. The system has one input, the rudder, and two outputs, roll and yaw. On all boats the rudder has a maximum angle, in either direction, which it can rotate from the centre line (zero degrees). For the craft considered here, the rudder has a set constant slew rate. It is understood that the constant slew rate is slow, which was factored into the design. It was assumed that the rudder provides sufficient roll moment at forward speed necessary for roll reduction. This one input, two output system to be designed for is a Single Input Multi Output (SIMO) system. Some of the major criteria for the controllers are: - the same controller must be able to be used on different boats easily. - work on one boat under different loading conditions (parameter variations). 4 - the controllers must work on real boats, whose motions are non-linear. - the controllers must work with the presence of wind and waves affecting the vessel (coloured noise). - the controller must take into account the rudder limitations (maximum angle and constant slew rate). The project did not cover all the aspects of autopilot design. The autopilot served only as a regulator, that is, for course keeping only. No mention will be made of turning or manoeuvring. In addition, the possibility of helmsman steering with a rudder roll reduction system also acting was not explored. All controllers have "tuning knobs" for which adjustments can be made to improve the controller's response to different stimuli. This project did not address the tuning of each controller for different, specific tasks. All controllers were tuned using the same test and specified response. The parameters remained fixed through all tests. All design was done in discrete time, that is, using difference equations instead of differential equations and the accompanying analytical tools such as z-domain analysis. The process was linearized for controller design. It was modelled as an Auto-Regressive Moving Average (ARMA) or a state space process. The coloured noise was modelled as an ARMA process driven by white noise. 5 The vessels designed for in this project are in some ways much different than the vessels dealt with in the literature [l7]. Previous controllers were designed for and tested on much larger vessels such as naval vessels. Also, most design information and qualitative discussion of transverse motions in standard literature [20,21] are related to larger vessels. 1.3 General Approach. Controllers were chosen and designed. Their parameters were set using one evaluation criteria for all designs. The first controller designed was of the same form and specifications of autopilots currently on the market. Its responses were used as the baseline against which the other controllers' responses were measured. All controllers were first designed as autopilots alone, then as roll reducing autopilots to see what effect the addition of roll reduction had on the ship response. The controllers were first tested in numerical simulations. Here their responses to specific environments and process parameter variations were examined. The controllers were then to be tested on a physical model. Here their responses to a real physical environment with all its associated non-linearities , noise from different sources, and Numbers refer to entries in the bibliography . 6 other peculiarities, were to be examined. The model tests were to be the more accurate test of whether a rudder roll reduction system would work with fishing and other similar sized boats. At the end of each testing section, the responses of each controller are compared and discussed. Finally, conclusions and recommendations are made. 1.4 Literature Review The first research into rudder roll reduction was undertaken in the mid-seventies, using analog controllers. These controllers were tested using scale model and full scale tests. The types of vessels were fast container ships and naval craft. They achieved limited success. In 1981, Kallstrom [7] conducted a detailed study of a rudder and fin roll control system. Throughout the 1980's van Amerongen [2,3] and various coworkers carried on an extensive project on rudder roll reduction. Their controller was based on Linear Quadratic Gaussian (LQG) design. The weighting factors in the LQ design were adjusted on line; however, there was no on line ship parameter estimation. They designed an automatic gain control system to compensate for the limited slew rate of the rudder. The testing was broad, 7 encompassing mathematical models, analog modelling, scale modelling and, full scale trials. In the 1980's, other researchers, such as Katebi et al.[4], investigated rudder roll reduction using LQ control. The amount of roll reduction reported in these papers ranged between eight and fifty percent. None of these researchers used on-line parameter estimation. Except for van Amerongen et al., there was no consideration of rate saturation. All of the vessels investigated were large container vessels or naval vessels. 8 Part 2: Design Process 2.1 System The system consists of four major components, shown in Figure 2. The components are the ship itself (or the plant), the rudder, the controller, including sensors, and the external environment. All of these components affect the ship motions of interest: yaw and roll. The ship affects itself through the propeller, which can affect yaw through "prop walk," and by the way it is loaded. Figure 2. System Diagram The autopilot is implemented to keep a steady course. It must compensate for external and internal disturbances by using the rudder. The ability to damp roll motions is to be added into the design of the autopilot. That is, the autopilot not only must keep a steady course, but it must also decrease the roll motion. 9 Important characteristics of the system that will affect the design are as follows: - The ship is either neutrally stable or unstable in yaw. Neutrally stable means the ship will hold a course, but the slightest disturbance will send the ship off course, and unstable means the ship cannot hold a course on its own. - The ship is always stable in roll, which means when it is disturbed it will always settle to a steady value. The roll to rudder response is non-minimum phase. In other words when the rudder is moved to one side of neutral, the ship initially rolls in one direction but finally settles in the other. - The yaw virtual moment of inertia of a typical west coast fishing boat is five to six times larger than the roll moment of inertia. - There is cross coupling between the motions, that is, roll affects yaw and yaw affects roll. It was expected that yaw would be influenced by roll much less than it influences roll. - The yaw and roll characteristics will change with speed and loading. Speed is a factor because not only does it change the effectiveness of the rudder, but it also affects the characteristics of the ship. 10 - The rudder is controlled by a hydraulic actuator, which moves at a fixed speed. This can cause difficulties and even instabilities in a roll reduction system, because the rudder angle starts to lag behind the desired angle to such an extent that it starts to increase roll motions. - It is expected that the controllers will be implemented using a digital processor in commercial production. Although the computational speed of computers have increased tremendously over the years, there are still economic factors to be considered. Therefore, some attention will be paid to the computational effort required. - The environmental disturbances are usually caused by wind and waves. The disturbances can be constant such as a steady wind, narrow-band random (usually waves), or wide- band random such as wind gusts. 2.2 Detailed Design The ship will be modelled in a general manner, i.e., using general coefficients rather than specific values. This will be followed by the controller design in the same general manner, i.e., using general coefficients. The ship was modelled as a linear discrete time Single Input Multi Output System (SIMO). Each motion, roll and yaw, was 11 modelled as a second order Auto Regressive Moving Average (ARMA) system using difference equations: A (z1) -,E Bi ( z -1)2 (1) where i=1 is the rudder input i=2 is the cross-coupling of the motions. For yaw ( 1 + al z-1 + a2 z-2) Ip. = JD, z-1 u +b2 z-14•^(2) and for roll ( 1 + al z-1 + a2 z-2) . = b1 z-i. u + b2 (z-i _ z-2) ip^(3) Note that the a; and bi are not the same for the two equations. A standard linear representation in naval architecture for the transverse motions (roll, yaw, and sway) of vessels is a SIMO 12 state space system: I = A0.x+.130u^ (4) with states:  2c= V where tk=yaw angle, v=sway velocity, and 0=roll angle and u is the rudder angle. Sway is often ignored because it is difficult to measure, as are the roll and yaw velocities. They could be estimated by using such methods as Kalman Filtering; however, because the plant parameters are estimated, doing both was felt to be too computationally expensive. Combining the ARMA difference equations into a state space SIMO system, we obtain: x(k+1) = Aa(k)^u(k)^ (5) with states (k) z(k) = (k-1) (k) 4)(k-1) 13 -1The elements of A are the coefficients of A(z-1) and B2(z ) polynomials for each of roll and yaw. The B elements consist of the B1(z-1) polynomial. That is: aly a2y b2y^0 bly A= 1 0 00 B= 0 b2r -b2r air a2r blr 0^0^10 where the y and r subscripts indicate the co-efficient are from the yaw and roll difference equations respectively. Because of the fixed slew rate of the rudder, it is convenient to change the input to Au and augment the states with u(k). That is: 0 x(k) =  The system matrices are now A= aly 1 .1p2r a2y .b2y^0^bly 0^000 -b2r air a2r bir B= bly o kr 0 0 100 o 0 0 001 1 After the plant was modelled, the controllers were chosen. A controller must meet the following criteria: 14 1. be able to compensate non-minimum phase plants. 2. be able to compensate open loop unstable plants. 3. work with or be extendable to multivariable plants. 4. work reasonably well in the presence of "coloured" disturbances. 5. handle large parameter variations. 6. handle a control input (actuator) that has a fixed rate. 7. handle a control input that has amplitude limits. 2.3 Parameter Identification Most controllers' parameters are based on the plant's parameters. To adapt to different plants and to plants under different operating conditions, some controllers use the method of on-line parameter estimation. In other words they estimate the plants' parameters using previous values of the output and the control input, then calculate the next control input. For the controllers in this thesis that require an on-line parameter estimation algorithm, the algorithm used was Approximate Maximum Likelihood, with some modifications. The algorithm and modifications are summarized here. More detail can be found in Appendix I. Approximate Maximum Likelihood was chosen because it has good convergence properties and is easily extendable to the case of a 15 plant with a coloured noise disturbance. It has the form of P(t)x1(t+1)^(t+1)P(t) P(t+1) = P(t)-^ 1+4.(t+1)P(t)x1(t+1) P(t+1)x1( t+1) 1+4'(ti-up(ox1(t4-3.) e(t+i) = e(t) +K(t+1){37(t+1) -xr(t+1)9(t)] where P is the covariance matrix, x/ is the data vector, and 0 is a vector of the parameters being estimated. The first modification is to add an Exponential Forgetting and Resetting Algorithm (EFRA) (8). This algorithm increases the stability of the AML when excitation is poor (i.e., little or no change in the control input) and tracks time varying parameters. The AML is modified by : aP(t)x.r(t+1)xf(t 1)P(t) P(t+1) -^-  +131-8102(0 A^1+4:(t+1)P(t)x1(t+1) aP( t+1)^ti-1) K( t+1) - 1 +x_r( t+1)Px.r( t+1) The major difference between the EFRA and the usual AML algorithms is in the equation for the covariance matrix. The first two terms correspond to the standard AML with exponential forgetting. The third term provides a lower bound for P, while the last term provides an upper bound. In this multivariable case, the two subsystems will be identified separately. That is, the coefficients in (2) will be identified, then the coefficients in (3) will be identified. K(t+1) - 16 The second modification is a simplification of the coefficients in the yaw subsystem (2). The fact 0 is the integral of 0, with respect to time; that is, a pure integrator, will be taken advantage of. Here al is equal to (-1-a2). Therefore, it is only necessary to identify a2, which can be done easily (see Appendix I). 2.4 PID Controller The Proportional-Integral-Derivative controller would not normally be chosen as a controller because it is used more in tracking applications than for regulation, it is difficult to design for multivariable systems, and it cannot be modified to handle a fixed rate control input. However, since it is the industry standard for autopilots, a PID controller using the industry standard parameters will be implemented. The control algorithm is: u(k) = qoy(k) +q1y(k-1) +q2y(k-2) +u(k-1) where qo = K(1+ -742T ) T qi = -K(1+2-2- - T—2) To Tr TD q2 = Ki--.. The standard values [28,29] for the gain K are 1:0.5 to 1:1, the derivative time constant TD is 0.25 to 0.5 seconds, and the 17 integral time constant T/ is 30 seconds or larger. A sample time To of 0.5 seconds was used for all controllers. 2.5 Minimum Variance Controller The minimum variance controller seeks to minimize the variance of the controlled variable: var [y (k)] =^y2 (k)) where BO is the expected value operator That is, the control is chosen based on the prediction of the output. In this case, with the fixed rudder rate, three possible choices of rudder input exist at any sampling instant: - move the rudder left. - move the rudder right. - keep the rudder at its current position. The input is further restricted when the rudder reaches its maximum amplitude. In this case, the possible choices of rudder input are limited to decrease rudder angle or keep the rudder angle the same. It was found early that minimizing the variance of the output alone did not produce satisfactory results. It is necessary to 18 add two terms into the criterion to be minimized: I (k+1) = Eq y2 (k+1) + K[y(k+1) - y(k) 2 + p u2 (k) } The second term is used to represent the velocity of the output variable and K is weighting on it, and p is a weighting on the control input. After a large number of simulations were run, it was found that the weighting K had a large range of variation for different operating conditions. Therefore a new method was found. In this method an "ideal" set- point, s(k), is added: I(k+1) = E [y(k+1) - s (k+1)] 2 + p u2 (k)} Where s(k+1) is the predicted value of an idealized natural response given the previous values of the output. This method is similar to pole-placement in that a desired response is specified. The controller minimizes the variance between the desired response and the actual response. In implementing this type of control, it is convenient to modify the difference equations (2) and (3) by multiplying them by A ( A=(1-z-1) ). The result is : (1 -z-1) ( 1 +^+ a2 2-2) * =^u +b2 z-1 ( 1 -z-1) ( 1 -z-1) ( 1 +^z-1 + a2 z-2) 4 = /Di z-1A u + b2 (z-1 - z-2) (1-z-1) 4r That is, the output is now dependent on the change in u. 19 The control algorithm then is to: - identify the parameters in the two difference equations using AML. - compute the expected values of lk and 0 for the three possible control inputs (rudder right, rudder left, or no rudder movement). - choose the rudder movement which minimizes : ^I (k+1) = E{ [tIr (ic+1)^(k+1)) 2 + K1[4)(k+1)^(k+1) 2+ pu2 (k) } The minimum variance controller was chosen because it is the basic controller for the regulation of systems with stochastic disturbances. 2.6 GPC Controller The Generalized Predictive Control controller is an N step ahead predictor. It predicts the output N steps ahead, given current values of the inputs, outputs, and plant parameters. In this application, two types of GPC control were evaluated. The first is bang-bang GPC control. The cost function ^J=^+ To] T ImN[HN + To] where HN = Predicted response of the system based on a = Future controls [ Au(k) Au(k+1) ... u (k+1-Nu) )7' 0 = Predicted free response at sample k Nu = Control horizon N = Prediction horizon 20 is calculated explicitly for the three possibilities of Au. If the rudder reaches its amplitude limit, there are only two possibilities for Au. The control horizon used was one. The prediction horizon is based on the slowest rise time of the system, usually the yaw rate. A prediction horizon of 10 was used. The algorithm is then to identify the parameters of the system using AML, predict the output for the three possible inputs, and implement the one which minimizes J. The second type of GPC control was rate constrained GPC. Here the control was calculated to be: a = {{H;UATT' H;:,, To} Because the control horizon is one, the rate constraint is implemented easily. The algorithm is check if Au exceeds the rudder rate, if yes then set Au equal the rudder rate, otherwise implement Au calculated. Setting Au equal to the rudder rate is not arbitrary. It is assumed the constrained optimum lies on the boundary of the cost function and the constraint. A prediction horizon of 10 was also used for the rate constrained GPC. GPC was chosen for rudder roll reduction because it is both a good regulator and tracker. Extensive research done recently on this type of adaptive control has shown it to be robust and stable [9-12]. It can be extended to multivariable systems. 21 2.7 Discrete-Time Sliding Mode Control Sliding Mode control is based on a discontinuous control function that switches when it crosses a boundary in the state space. When the states repeatedly cross this boundary and move toward the origin of the space, the system is said to be sliding. During this time the system is remarkably robust in the presence of disturbances and parameter variation. The resulting control structure is usually non-linear and results in a Variable Structure System (VSS), where the control is different in different parts of the state space. The simplest discontinuous control device is a device with two control states, e.g., a relay. For rudder roll reduction, the two control states are move the rudder right and move the rudder left. The design procedure is to specify a nominal A and B, that is a general or average quantitative expression of the system. Solve the following matrix Ricatti equation for P : P = Q + ATPA - ATPBRRBTP - PBRRBTPA + PBRR13TPBRRBTP where RR = RudderRate Q = Positive Definite Matrix The sliding sub-space coefficients are then: G = BTP 22 and the control input is: A u = -RRsign ( Gx ) The preceding control input can lead to chattering, or the rapid switching of the control between two values. The fact the control input can be zero movement, besides moving right or left, and the idea of a boundary layer will be used to smooth out the chattering. Instead of using a discontinuous surface, a small region will be constructed around the discontinuous surface. If the states are within this region, the control input used will be no change in rudder angle. In other words, If I S(k) I =I Gx(k) I < e then u(k) = 0 where e is the boundary layer width. Sliding Mode control was chosen because it has been used in a situation similar to rudder roll reduction of a ship, the control of transverse motions of aircraft [16]. Sliding mode control can be used with multivariable control and uses a discontinuous control input, as the fixed rudder rate is. 2.8 Pole Placement Control The pole placement controller attempts to place the closed loop poles of the system at some arbitrary location. Given the state 23 space system: x(k+1) = Az(k) +Bu(k) and assuming the input has the form: u(k) = -1Cx(k) then the closed loop response of the system is: .r(k+1) = [A - BIC]x(k) and by setting K, the "natural" response of the system [A-BK] can be set arbitrarily. The order of the "natural" response is the same order of the original system A. For rudder roll reduction, the yaw response was set with two stable exponential decay terms and the roll response was set as having the same natural frequency as, but a much larger damping ratio than, the open-loop roll response. The control algorithm is: - identify the ship parameters in roll and yaw using AML. - calculate the gains K necessary to give the desired closed loop response. - calculate the rudder angle given the current states and the gains calculated. Pole placement was implemented as a rudder roll reduction controller because of the explicit specification of the closed loop response. For example, the roll response is specified to keep the same natural frequency but increase the damping. The 24 possible problem of this controller is if the rudder rate is too slow, the rudder angle may begin to lag the control signal and start exciting the ship in roll. It is thought that if the yaw response is specified slow enough, the rudder lag problem may be avoided. 25 Part 3: Numerical Simulation 3.1 Numerical Model The first attempt to obtain a numerical model involved the use of data available in the department from full scale tests performed in the summer of 1989. The data consisted of zig-zag trials of an approximately 40' double ender cabin cruiser. A regression analysis was performed to obtain the yaw and roll parameters. The results for yaw were good and made physical sense. The results for roll were not as good and, more importantly, did not make any sense physically. Therefore, a numerical model was implemented. The numerical model used in this section of tests is based on the linear portion of Inoue's [22] manoeuvring model with additional terms for the roll motion taken from Bhattacharya [21]. The 26 equations are (see Appendix IV for more information): ( m + my) V = -22-  pLdV4v - ( mu + mxu - 1pL2d174) r + YR + Yw (I„ + j„ ) I = 1pLdV(L1■71„+Y,Nic]v + -1-pL2dV[ LIVir + Yx5 Jr 1+ - 2 p L2 dVN4) NR Nw (^+ J )4:1 - myz = -b(I) - gAGAI) - pLdV1,„zilv - [ mxu + pL2dV4^+ KR + Kw The hydrodynamic coefficients are estimated from basic ship characteristics alone. The coefficients were calculated from the data of the Kynoc, a 60-foot west coast drum seiner and the Eastward Ho, a 107-foot combination vessel that have been used in other research in the department. The coefficients were calculated for different speeds and loading conditions. From these coefficients a state space model was constructed for each of the different conditions. The particulars and speeds for the different conditions are given in the following tables. 27 Table 1. Kynoc Particulars Particular Condition 1 Half Load Full Speed Condition 2 Half Load Half Speed Condition 3 Full Load Full Speed Condition 4 Full Load Half Speed Lui (in) 17.25 17.25 17.4 17.4 B1(m) 6.1 6.1 6.1 6.1 T(m) 2.53 2.53 2.82 2.82 T 1.24 1.24 1.41 1.41 A(tonnes) 116.5 116.5 141 141 Ca., 0.43 0.43 0.46 .46 L/B 2.83 2.83 2.85 2.85 LCG(m) -0.608 -0.608 -0.851 -0.851 VCG(m) 2.22 2.22 2.13 2.13 GM(m) 0.735 0.735 0.546 0.546 Speed (kts) 11 8 11 8 Table 2. Eastward Ho Particulars. Particular Condition 1 Half Load Full Speed Condition 2 Half Load Half Speed Condition 3 Full Load Full Speed Condition 4 Full Load Half Speed 'Jut (m) 30.5 30.5 30.8 30.8 But (m) 8.74 8.74 8.74 8.74 T.(m) 3.45 3.45 4.04 4.04 r 0.835 0.835 1.073 1.073 A(tonnes) 445 445 572 572 CI., 0.472 0.472 0.514 0.514 L/B 3.49 3.49 3.52 3.52 LCG(m) -1.405 -1.405 -1.893 -1.893 VCG(m) 3.28 3.28 3.38 3.38 GM(m) 1.161 1.161 1.268 1.268 Speed (kts) 13 10 13 9 28 The matrices were discretized using a zero order hold with a sampling period of 0.5s (See for example Ogata [18]). The rudder motion has two restrictions -amplitude and rate. The amplitude is limited to ±40 degrees. The swing rate of the rudder has a maximum of 5 degrees per second. If the demanded rudder angle is within the amplitude limitations, and the difference between the previous rudder angle and the demanded rudder angle is less than or equal to the swing rate multiplied by the control cycle time, the demanded rudder angle is implemented. Otherwise, the rudder angle implemented is the closest value to the demanded angle that does not violate the two restrictions. For the irregular wave disturbances used in some of the tests, digital filter approximations were constructed (see Appendix V). These approximations were based on excitation spectra calculated using the Bretschneider Spectrum. The approximations were matched to the spectra for shape qualitatively and for area under the curve quantitatively. Examples of sway and yaw excitation are contained in Figures 3 and 4. Figure 5 shows an example of wave slope. Roll Excitation was simply AGM x wave slope. 29 Figure 3. Sway Excitation Spectrum Figure 4. Yaw Excitation Spectrum 30 Calculated   Filter Approx. 0.001 0.0009- 0.0008- 0.0007- 0.0006- 0.0004 0.0003 0.0002- 0.0001- 0 0 0.2^0.4^0.6^0.8^1 Frequency ( rad. ) 45 degree heading Sea State 3 Speed 11 kts. 1.2^14 Figure 5. Slope Spectrum In these tests, the state equations were: x(k+1) = Ax(k) + Bu(k) +^(k) where w(k) is a vector of the values from the three excitation filters. 3.2 Controller Parameters The controller parameters that can be adjusted were set for all tests. The test used to specify these parameters consisted of giving the vessel an initial yaw offset of 100 so that the controller had to move the vessel to a zero yaw angle as quickly 31 and smoothly as possible. For the controllers under evaluation, simple autopilots were constructed to compare the amount of roll reduction produced. The parameters for the autopilots were as follows: - PID Controllers. The parameters of the PID controller for the Kynoc were K=-1 ,TD=0.5 , and T1=30. The PID parameters for the Eastward Ho were K=-1 ,TD=1.0 , and T1=45. - Minimum Variance Controllers. The ideal roll response was characterized by the equation 1 - 1.59z-1 + 0.764z-2, corresponding to a natural frequency of 7 seconds and a damping ratio of 0.3. The ideal yaw response for the Kynoc had a characteristic equation of 1 - 1.6z^+ 0.6399z-2. The characteristic equation of the ideal yaw response for the Eastward Ho was 1 - 1.8z^+ 0.8099z -2. Kr was set to 1 for both vessels. The simple autopilots used the same ideal yaw responses for the respective vessels. The Kynoc's autopilot used p=0.0003, the Eastward Ho used p=0.00005. - GPC Controllers. For both vessels, the roll reduction autopilots used N=10. The simple autopilots for both vessels used N=11. - VSS Controllers. The switching surfaces for the roll 32 reduction autopilot, the simple autopilot, and the 15°/s rudder rate roll reduction autopilot were the same for both boats. The nominal system used for rudder roll reduction was: A= 1.8 1 -0.3 -0.8 0 0.3 0 0 1.7624 0 0 -0.9561 0.03 0 0.01 B= 0.03 0 0.01 0 0 1 0 0 0 0 0 0 0 1 1 This system has a yaw rate time constant of 2.24 seconds, a roll natural period of 7 seconds, and a roll damping ratio of 0.05. The roll reduction autopilot's switching surface was G=[43.06 —38.32 16.69 —0.839 4.6215]. The boundary layer used for the Kynoc was 6=0.05; the boundary layer for the Eastward Ho was 6=0.025. The nominal system used for the simple autopilot was: 0 1 0 Ai 0 0 11 0Bi^ ^I 0.8 -2.6 2.8 0.017 The simple autopilot's switching surface was G=[107.8961 —254.926 150.573]. The boundary layers used were 0.09 for the Kynoc and 0.04 for the Eastward Ho. The 15°/s roll reduction autopilot had a switching surface of G=[14.24 —12.67 5.56 —0.3188 1.536]. The boundary layers were 0.06 for the Kynoc and 0.03 for the Eastward Ho. - Pole Placement Controllers. The roll poles, which were the same for both boats, were placed at 0.798±0.3631. These 33 locations correspond to a natural frequency of 7 seconds and a damping ratio of 0.3. The Kynoc's yaw poles were placed at z=0.79 and z=0.81. The Eastward Ho's yaw poles were places at z=0.89 and z=0.91. 3.3 Tests After each controller was chosen, designed, and parameters set, the following tests were simulated using the numerical model: 1. A 15 degree course offset. This offset is 50% larger than the design offset of 100. The half load full speed ship was used for this test. 2. A yaw velocity impulse of 5.7°/s. This test is the base case for the following four tests. 3. Change in speed. The speed was decreased to half speed, which decreases the effectiveness of the rudder and thereby increases the settling time. 4. Change in displacement. The full-load full-speed condition was used for this test. This condition increases the yaw rate decay time and thereby increases the settling time. 5. Change in speed and displacement. This test uses the full- load, half-speed condition. In this test the controller 34 response to both a change in speed and displacement was measured. 6. Rudder servo slows by 20%. The purpose of this test was to see how well the controller works when the actuator's performance has been degraded slightly. This situation could occur due to poor maintenance ( e.g., hydraulic fluid leakage or simple wear). 7. A constant yaw moment. This test simulates effects such as prop walk, steady wind and current effects that would produce a constant yaw moment. The half-load, full-speed condition was used for this test. 8. Irregular beam (90°) seas. This heading should produce the worst roll motions. The full-load, half-speed case was used because this loading condition has the lowest roll stability. The modal frequency of the sea state was set equal to the roll natural frequency and the significant wave height was one hundredth of the length of wave with a period equal the roll period (1-11/2=0.01gT:/(27r) and Tm=T0) . 9. 45 degree quartering seas. Quartering seas produce the most excitation for roll and yaw combined; thus three quartering seas tests were run. The full-load, full-speed condition was used for this test. Sea State 3 for the North Pacific was used for the modal frequency and significant wave height (liti=0.88m and Tra=7•55)• 35 10. 60 degree quartering seas.^The half-load, full-speed condition was used in this test. Sea State 3 for the North Pacific was used for the modal frequency and significant wave height. 11. 30 degree quartering seas. The half-load, half-speed condition was used in this test. Sea State 4 for the North Pacific was used for the modal frequency and significant wave height (H=1.88m and Tm=8•85) . Different load and speed conditions were used in the irregular sea tests to obtain a better idea of the range of behaviour. The irregular sea cases (tests 8-11) were then rerun for both vessels using a rudder rate of 15°/s. Only the four better roll reduction autopilots, the VSS, the pole placement, the minimum variance, and the rate constrained GPC, as well as the PID and the pole placement autopilots were run. 3.4 Results The results are tabulated in Appendix VI. The impulse responses from the Rynoc of the PID autopilot, the VSS roll reduction and simple autopilots, and the pole placement roll reduction and simple autopilots are shown in Figures 6 to 10. These figures are examples of the deterministic tests and show the effect of adding rudder roll reduction . 36 Figure 6. VSS Impulse Response for Kynoc Figure 7. VSS Roll Reduction Impulse Response for Kynoc 37 Figure 8. Pole Placement Impulse Response for Kynoc Figure 9. Pole Placement Rudder Roll Reduction ImpulseResponse for Kynoc 38 Figure 10. PID Impulse Response for Kynoc The more relevant tests are the irregular sea tests because in these tests both roll and yaw are excited and because these tests typify the conditions where a ship's operator desires roll reduction. For convenience, the results from the irregular sea tests for controllers with a 5°/s rudder rate are presented in Figures 11 to 18. The figures show the RMS roll and yaw angles, smaller values being better for each. The increase in roll reduction for the controllers tested with a 15°/s rudder rate are also provided in Figures 19 to 22. All tests were run for a time of 10 minutes. An angular value of 20 degrees for any of yaw, roll, or rudder angle was considered to exceed the valid range of the numerical model. In general, the simple autopilots had good results, often better 39 than their roll reduction counterparts. The adaptive controllers on the Eastward Ho tests did not do well, often exceeding the valid yaw range of the numerical model. The constant yaw moment test on both vessels caused problems with many of the controllers. One unusual result was that better course keeping resulted in better roll response. It was expected there would be a trade-off of worse yaw response for better roll response. The other unusual result was that the increase in the rudder rate to fifteen degrees per second did not improve the roll results for any of the controllers. The controller helped most by the increased rudder rate was the pole placement roll reduction controller; however, the increase in performance was not consistent over all the tests. The results are summarized for each controller as follows: - The PID autopilot did surprisingly well, as it was expected to have problems with rudder phase lag in the more extreme irregular seas. The actuated rudder angle always matched the command rudder angle from controller. - The minimum variance autopilot had excellent course keeping in almost every test. 40 - The VSS autopilot did extremely well, its course keeping was phenomenal. - The pole placement autopilot had excellent results; however, the actuated rudder angle did not always match the command rudder angle from the controller. - Both GPC autopilots had fair results, but better performance was expected. - All the roll reduction autopilots did poorly. The best roll reduction controller is the VSS roll reduction controller which outperforms the PID controller in six out of eight cases. The next best controller is the pole placement roll reduction autopilot, which is better than the PID controller four out of eight times. 41 _IFS^fil_GPC_PAS^VSS_RRS^Poi RAS^SSAPCJIRS 013_CPC^VSS Poi Pi^55_0PCController Figure 11. Kynoc 900 Seas Figure 12. Kynoc 45° Seas 42 I. Figure 13. Kynoc 600 Seas PD^IN_115^IR_CPC-RAS VSS-1115^Poi RAS^SS_CPC_DIS IN^RLCPC^VSS^Pala Ph^SS_CPC Controller Mg Yaw MI Roll 2.00 1.60 1.60 0.40 O.20 O.00 Figure 14. Kynoc 30° Seas 43 Figure 15. Eastward Ho 900 Seas Figure 16. Eastward Ho 450 Seas 44 Figure 17. Eastward Ho 600 Seas Figure 18. Eastward Ho 30° Seas 45 90 45 60 30 90 45 60 30 Kynoc^Eastward Ho Heading Figure 19. VSS Roll Reduction Improvement 15°/s RudderRate Figure 20. Pole_ RR Roll Reduction Improvement 15°/sRudder Rate 46 Figure 21. MV_RR Roll Reduction Improvement 15°/s Rudder Rate Figure 22. SS_GPC_RR Roll Reduction Improvement 150/s Rudder Rate 47 3.4 Evaluation of Roll Reduction Based on the numerical simulations, rudder roll reduction does not work well enough for commercial implementation on fishing boats. None of the roll reduction controllers reduced the roll motion below that of the PID controller in every irregular sea case. The result from the numerical simulations that rudder roll reduction does not work is suspect for several reasons. The first reason is that some of the controllers exceeded the valid yaw range of the numerical model in the Eastward Ho tests. The reason for these results was found by comparing heeling moment produced by the rudder to the wave induced roll moment. For a wave induced roll moment produced by a wave with a period equal to the Eastward Ho's natural roll period and a wave height of one hundredth the wave length, the rudder angle necessary for the heeling moment to counteract the wave moment is 51°. The resulting heading angle is 60°. Both these values are well outside the numerical model's range. In comparison, the Kynoc's rudder angle is 15° and the yaw angle is 19° for a wave with the Kynoc's natural roll period. The reason for the large difference between the two boats is the Kynoc has a more effective rudder. The second reason comes from comparing the results in this study to previous results in literature [3,5,8], where roll reduction was reported successful. To compare the same qualities on 48 Table 3. Dimensionless Quantities Dimension less quantity GM't/L zR/L AR/L2 Fo To/To 1/(6^x To) Kynoc 0.037 0.126 0.0046 0.32 - 0.21 0.0280.435 Eastward 0.040 0.077 0.0027 0.26 - 0.23 0.029 Ho 0.39 USCGC 0.0096 0.037 0.0015 0.21 - 1.37 0.02 Jarvis 6 0.24 Amerongen2 0.17 0.014 Kallstrome 0.0026 0.031 0.0012 0.17 1.52 0.0054 different ships, the technique of dimensional analysis was used. The dimensionless quantities are presented in Table 3. The three quantities vertical distance to rudder centre of effort to length, total rudder area to length squared, and Froude Number (zR/L, AR/L2, and Fn) are measures of how effective the rudder is in roll and the two boats in this study would appear to have more effective rudders. The ratio of yaw-rate decay-time to roll natural period (VT.) for the Kynoc and Eastward Ho are not any better or worse than the vessels used in previous studies. The rudder rate to roll period fraction indicates that the boats in this study have slower rudder rates for their given roll periods; however, the rudder rates are not much slower than the USCGC Jarvis's rudder rate and the Eastward Ho, which has a slightly faster rudder, had poorer results than the Kynoc. The third reason is the result that better course keeping gives better roll response. Intuitively, this result does not seem correct because there should be some trade-off between course keeping and roll reduction. Reasons for this result are unknown 49 0.9^ 2y •^b2y 400^500 Time (s) boy 600 0.6- 200 at this time. The last reason to suspect the result from the numerical model tests is that increasing the rudder rate did not improve the roll response. Previous results [2,3,5] indicate that increasing the rudder rate always increases the roll response and intuitively, improving the actuator should improve the system response when system improvement is possible. Reasons for the lack of improvement are unknown at this time. The identification algorithm, used in the adaptive controllers, worked well (see Figures 23 and 24 ). The yaw parameter a2y was typically slightly higher than expected, possibly due to the presence of sway motion. Figure 23. Example of yaw parameter identification ( random sea test - Eastward Ho) 50 1i" bir a2r I b2r- _ - a ir _ 160 260 360 460 500 . 60 Time (s) -1 - o 0.! -0.5 Figure 24 Sample Roll Identification (Random SeaCAse - Eastward Ho) The poor results with the adaptive controllers were first thought to be from the adaptive algorithm because at the same time into the simulation when the yaw results started to exceed the model's range, the yaw parameter a2y would rapidly change to a new value (see Figure 25). A sensitivity experiment was performed using the minimum variance roll reduction autopilot. The controller parameter Kr was reduced, in effect reducing the importance of roll reduction, and the results compared. When Kr was small enough the numerical model did not exceed its range of validity and the a2y parameter did not have a rapid change (as in Figure 23). Therefore, from this experiment and the comparison of heeling moment to wave-induced roll moment, the problems with the 51 b 1 ,, 0 b 2 y 160^260^360^460^500 Time (s) -01 ^ 0 600 1 0.9- 0.8- 0.7- 0.6- 0.5- 0.4- 0.3- 0.2- 0.1 - adaptive controllers are either from the controller algorithms or the numerical model and not from the adaptive algorithm or the process model used in the adaptive algorithm. Figure 25 Yaw Parameter Estimates The results for the VSS, the minimum variance, and the pole placement autopilots are very good, not just because of their good yaw response, but because of the form of the algorithms. These controllers have just one "tuning knob", compared to the PID controller which has three, making it easier for the ship's operator to adjust the autopilot. The minimum variance and pole placement would appear to have more than one parameter to adjust; however, the pole locations could be chosen by the controller based on the identification results, most likely the Joh, parameter because it shows how effective the rudder is and therefore how quickly the boat can respond. 52 Part 4: Physical Model A physical model that had been used for other self-propelled model tests was prepared for autopilot testing. The model is a 15:1 scale model of the Eastward Ho, a 107' steel combination vessel. The principle particulars of the model are given in the following table. Table 4. Model Particulars. Particular Length (m) 1.805 Beam (m) 0.54 Depth (m) 0.35 Draft (m) 0.22 Ch 0.5 A(kg) 102 Preparations for the tests included re-surfacing the outer hull of the model with a fabric composite, splining the outer hull and installing a new power bus. Two 12V 24Ah batteries and one 12V 7Ah battery were installed for power. One of the 24Ah batteries was replaces with a 31Ah battery after the old battery became damaged. A choke and capacitor were installed as power conditioners for the existing motor control unit and a capacitor was installed as a power conditioner for the rudder servomotor, respectively. 53 4.1 Yaw Sensor A yaw sensor is necessary for measuring the model's direction. The first sensor installed for measuring yaw angle was a International Navigation INI-100 Digital Heading Sensor. This device is a fluxgate magnetometer, or a solid state compass. After many trials with poor results, it was discovered that there was too much distortion of the earth's magnetic field within the towing tank building. A directional gyroscope, an Aviation Instrument Manufacturing Co. 205-2A, was then installed. This instrument is connected to a modified Wagner SEAutopilot Generation 2 autopilot to resolve the synchronous output of the gyroscope to a dc signal for the telemetry system. 4.2 Roll Sensor The other sensor necessary in a roll reduction system is one to measure roll angle; therefore, a Humphrey VG24-0825-1 vertical gyroscope was installed. This gyroscope has a potentiometric output, i.e., dc signal output; however, the signal to noise ratio was too large. Therefore, an amplifier between the gyroscope and telemetry system was built. 54 4.3 Telemetry System A Sigma Data Systems telemetry system was used to send the sensor signals to the shore-based computer. The system consists of a transmitter installed on the model, a radio receiver and a reader unit on shore. The time delay from the sensor reading on the model through to the output from the reader was approximately 40ms. 4.4 Data Acquisition A Strawberry Tree ACJr-12 was used for data acquisition. The card has a maximum of eight channels of which two were used. With two channels used, the card has a sampling rate of 1 kHz at a resolution of 11 bits. 4.5 Computer A Toshiba T3200 portable computer was used as a host for the data acquisition card and to implement controllers in software. The computer is based on the 16-bit Intel 80286 microprocessor. The operating system used was MS-DOS 4.0. The sampling period used for the controllers was 0.11s. This value, which is constrained by the operating system, when added to the delay time of the telemetry system, results in the closest value to the desired sampling period of 0.13s (Froude scaling of the 0.5s full scale 55 sampling period). 4.6 Radio Control The radio control system is based on Futaba R/C components. The transmitter is a Futaba FP-5FG/K Digital Proportional Radio Control. It is interfaced with the computer by means of a custom circuit board between the serial port of the computer and the trainer port of the R/C transmitter. The rudder servo on the model has a maximum rudder rate of 91 degrees per second. The constrained rudder rate of 19 degrees per s (Froude scaling of the 5°/s full scale rudder rate) was therefore implemented in software. 56 A diagram of the complete system is provided in Figure 26. Figure 26. Model Testing Arrangement 4.7 Towing Tank The location of model testing is the towing tank at the Ocean Engineering Centre at B.C. Research. The tank is 220 feet long, 12 feet wide, and 8 feet deep. Physical model testing was disrupted prior to completion by B.C.Research going into receivership. 57 4.8 Tests Only some preliminary tests were completed, due to the loss of the facility. These preliminary tests consisted of nine zig-zag trials, in which only yaw motion was recorded, and one zero-speed roll-decay test. From the model tests, the yaw rate time constant was found to be around two seconds. The model time constant corresponds to a full scale time constant of around seven seconds. The model tests were made at speeds estimated at three to five knots (full scale). No comparison was made to the numerical model because the numerical model did not produce good responses at these low speeds. From the roll decay test, the roll natural period was 1.49 seconds and the damping ratio was 0.075. These values were found two ways: first, using logarithmic decrement for the damping ratio and visual measurement of the period, then from regression analysis. The natural period of the model corresponds to a full scale roll period of 5.77 seconds. A comparison to the numerical model roll period was not possible because the GM of the model was not measured. The damping ratio of the physical model is approximately twice that of the numerical model. This discrepancy is most likely due to the bilge keels fitted on the physical model, as it was assumed there were none with the numerical model. 58 Part 5: Conclusions 1. Based on the results from the numerical simulation, rudder roll reduction does not work well enough for commercial implementation on fishing boats. 2. The yaw motion resulting from the rudder action necessary to reduce roll motion on fishing boats can be too large for rudder roll reduction to be effective. 3. The results from the numerical simulation are suspect because: - the results do not indicate better roll response with a faster rudder rate. - the results indicate a straighter course gives better roll response. - comparison to previous results indicates little difference in the relative capabilities of ships used in these tests and ships used in previous, successful implementations. 4. The VSS, minimum variance, and pole placement controllers had excellent results as autopilots with no roll reduction. 59 Part 6: Future Work The research that remains is: 1. Confirm that rudder roll reduction does not work using model tests and the two better rudder roll reduction controllers, i.e., the VSS rudder roll reduction and the pole placement rudder roll reduction controllers. 2. Confirm the autopilot results using model tests. 3. Explore the idea of combining the pole placement and minimum variance controllers into one controller. The resulting controller should have smooth, exact control (like that of the pole placement controller but which the minimum variance tends to lack) without any rudder saturation (which the pole placement has but the minimum variance does not). 60 Part 6: Bibliography On Rudder Roll Reduction 1. Roberts, G.N., and S.W. Braham."A Design Study on the Control Of Warship Rolling Motion Using Rudder and Stabilizing Fins." International Conference on Control (1991): 838-843. 2. Amerongen, J. van, et al. "Rudder Roll Stabilization for Ships." Automatica 26 (1990): 679-690. 3. Amerongen, J. van, et al. "Rudder Roll Stabilization - Controller design and experimental results." 8th Ship Control Systems Symposium Proceedings 1 (1987): 1.120-1.142. 4. Katebi, M.R., et al. "LQG Autopilot and Rudder Roll Stabilization Control System Design." 8th Ship Control Symposium Proceedings 3 (1987): 3.69-3.84. 5. Baitis, E., et al. "Rudder Roll Stabilization for Coast Guard Cutters and Frigates." Naval Engineers Journal (May 1983): 267-282. 6. Kallstrom, C.G., et al. "Roll Reduction by Rudder Control." Proceedings 13th STAR Symposium (1988): 67-76. 7. Kallstrom, C.G. "Control of Yaw and Roll by a Rudder/Fin- Stabilization System." Proceedings 6th Ship Control Systems Symposium (1981): F2 3-1-F2 3-23. On Identification 8. Salgado, M.E., et al. "Modified Least Squares Alogrithm Incorporating Exponential Resetting and Forgetting." International Journal of Control 47, No.2 (1988): 477-491. On Generalized Predictive Control 9.^Clarke, D.W., et al. "Generalized Predictive Control - Parts I and II." Automatica 23, No.2 (1987): 137-160. 61 10. Tsang, T.T.C., and D.W. Clarke. "Generalized Predictive Control with Input Constraints." IEE Proceedings 135, Pt.D, No. 6 (November 1988): 451-460. 11. Clarke, D.W., and C. Mohtadi. "Properties of Generalized Predictive Control." IFAC 10th World Congress on Automatic Controls 10: 63-74. 12. Kinnaert, M. "Adaptive Generalized Predictive Controller for MIMO systems." International Journal of Control 50, No.1 (1989): 161-172. On Sliding Mode Control 13. Willer, P.C. "Stability Analysis of Regular and Sliding Motions of Linear Dynamical Systems with Ideal Relay." American Control Conference 3 (1985): 1675-1699. 14. Sarturk, S.Z., et al. "On the Stability of Discrete-Time Sliding Mode Control Systems." IEEE Transactions on Automatic Control AC-32, No.10 (October 1987): 930-932. 15. Kotta, U. "Comments on 'On the Stability of Discrete-Time Sliding Mode Control Systems'." IEEE Transactions on Automatic Control AC-34, No.9 (September 1989): pp1021-1022. 16. Spurgeon, S.K. "On the Development of Discrete-Time Sliding Mode Control Systems." International Conference on Control (1991): 505-510. 17. Spurgeon, S.K. "Hyperplane Design Techniques for Discrete- Time Variable Structure Control Systems." International Journal of Control 55, No.2 (1992): 445-456. On Discrete-Time Control 18. Ogata. K. Discrete-Time Control Systems. Englewood Cliffs: Prentice-Hall Inc., 1987. 19. Isermann, R. Digital Control Systems. Heidelberg: Springer- Verlag, 1981. 62 For the Numerical Simulation 20. Lewis, E.V. (Editor). Priciples of Naval Architecture Volumes I-III (Second Revision). Jersey City: The Society of Naval Architects and Marine Engineers., 1988. 21. Bhattacharyya, R. Dynamics of Marine Vehicles. New York: Wiley Interscience, 1978. 22. Inoue, S., et al. "A Practical Calculation Method of Ship Maneuvering Motion." International Shipbuilding Progress 28, No.325 (September 1981): 207-222. 23. Inoue, S., et al. "Hydrodynamic Derivatives on Ship Manoeuvring." International Shipbuilding Progress 28, No.321 (May 1981): 112-125. 24. Clarke, D., et al. "The Application of Manoeuvring Criteria in Hull Design Using Linear Theory." Transactions of the Royal Institute of Naval Architects (1983): 45-68. 25. Bass, D.W., and M.R. Haddara. "Roll and Sway-Roll Damping for Three Small Fishing Vessels." International Shipbuilding Progress 38, No.413 (1991): 51-71. 26. Rydill, L.J. "A Linear Theory for the Steered Motion of Ships in Waves." Transactions of the Royal Institute of Naval Architects (1959): 81-112. 27. Reid, R.E., et al. "The Use of Wave Filter Design in Kalman Filter State Estimation of the Automatic Steering Problem of a Tanker in a Seaway." IEEE Transactions on Automatic Control AC-29, No.7 (July 1984): 577-584. On PID Autopilots 28. Dickson, Bill, R&D Director, CompuNav Systems. Telephone Conversation with Author, 8 April 1992. 29. Wagner, Paul, Wagner Marine Systems Inc. Telephone Conversation with Author, 8 April 1992. 63 Appendix I: Approximate Maximum Likelihood Identification The following is a more detailed description of the identification method used and of the simplification of theidentification of the yaw parameters. Given i A (z-1) y =E Bi (z-1) ui i=o where j is the number of inputs and consider the cross-couplingterms as another form of input. A (z-1) =1 + aiz-1 + a2z-2 +... +anz-n B(z1) = boi + bliz-1+...+bmiz-m AML identification has the form : P(t+1) = P(t)- P(t)x.r(t+1)xr(t+1)P(t)1+xf(t+1)P(t)x1(t+1) P( t+1) xi ( t+1) 1+4 (t+1) P(t)x.r(t+1) e(t+1) = y(t+1) - xf (t+1)0(t) 0(t+1) = 0( t) +1C(t+1) e(t+1) to include the Exponential Forgetting and Resetting algorithm [8] the co-variance and gain equations are modified: = 1 aP(t)x1(t+1)x.r(t+1)P(t) A^1 +XI( t+1) P( t) Xi( t+1) -^ccP(t+1)x1(t+1) 1+x. r(t+1)Px.r(t+1) x/ is the vector of inputs and outputs.e is the vector of the parameters being estimated. x . [-y(k-1) ,..., -y(k-n) , Ell (k-1) , ..., ul(k-m1) , ....1- , u2 (k-1) ,..., ui (K-mi) ,ri (k-1) ,...,11 (k-n)] T 0 = [al, a2, --, an, bii, -, blm,,b12, •••, bimi, ci, •••, cn} T IC( t+1) - +111-8p2(t) 64 n are the a posteriori errors and n ( t) = y(t) - xr(t)0(t) For the yaw sub-system, it is assumed n=2, j=2, m1=1, and m2= . that is ( 1 + al z-1 + a2 z-2) lir = bil z-1 u +b12 Z-1 04) A simplification for the identification of the yaw subsystem is made by assuming the natural response of yaw is a decay term and an integrator : A(z1) = (1-z-1) (1-a2z') or al= -(1+a2) therefore it is only necessary to identify a2. The following changes must be made to x1 and 8: xi = Dir (k-1) - Ip. (k-2) ,u(k-1) ,4)(k-1)]T 9 = [a21b111b12] T e(k) = lir (k) - Ili (k-1) -49 65 Appendix II: Generalized Predictive Control The following is a more detailed description of the state spacegeneralized predictive controller. Given a state space system x(k+i) = Ax(k) +Bu(k) y(k) = Ca(k) where A is (nxn), B is (nxm), C is (pxn), x is a vector of lengthn representing the states, u is a vector of length m representingthe incremental inputs, and y is a vector of length prepresenting the outputs. Assuming a prediction horizon of length N, and a control horizon of length Nu, Nu N, the global predictive model is: IF= Hira yo where CB^0^0 CAB^CB %.^0 CB CAR-1B CAN-2B CA"uB CA ={^ix(k) CAN umr.(10 = [Au (10 Au (k+1) - Au(k+ Nu - 1) 12" The control law is to choose ü which minimizes: Jr = 1";IX + arn where P is a diagonal weighting matrix for the input. For the case of the rudder with the fixed swing rate, the incremental input has only three possible choices, swing left, swing right, or do not swing. In this case, and assuming acontrol horizon ( N4 ) of 1, it is easier to calculate J for each of the three possibilities of input and choose the one which HR. = and 66 minimizes J. When the rudder reaches its amplitude constraint, the number of choices decreases to two, decrease the rudder angle or let it remain. For the case of the rudder with a maximum swing rate, the incremental input which minimizes J is calculated from: a = {[H,TruHA H: Tu o The calculated input is then checked against the maximum input and if the calculated value exceeds the maximum, the maximum is implemented. This setting the input to the maximum results from assuming the constrained optimum lies on the boundary of the cost function and the constraint. 67 Appendix III: Discrete-Time Variable Structure Control A design equation will be derived for finding the discrete sliding plane co-efficients for a state space system with bounded relay control. The design equation has the form of a matrix Riccati equation. Given a state space system: x(k+1) = Ax(k)+Bu(k)^(AIII.1) where A is (nxn), B is (nxm) and each ui has the form: u1(k) = -Kisigm(Si(k)) or u(k) = -Ksign(S(k))^(Anrr.2) where 17= diag(Ki) sign(Si(k))" sign (3(k)) = sign(Sm(k))_ and Si (k) Gix^ (AIII.3) where Gi are row vector of size n. The Ricatti equation will be derived using the second method of Lyaponov. First, consider the stability of the state trajectories off the sliding subspace using a matrix Lyapanov function : V(x) =.KT(.1c)FX(k) then A.17(20 = v(x(k+1)) - v(x(k)) = xr(k+i)px(k+1) -xT(k)Px(k) using (AIII.1): AV(x) = [ Ax(k) + Bu(k)]TP[Ax(k) + Bu(k) - xr(k)Px(k) = xT(k) (ATPA-P)x(k) + xr(k)ATPBu(k) + uT(k)BYPAx(k) + uT(k)BTPBu(k) using (AIII.2): AV(x) = XT(k) (ATPA-P)x(k) - xr(k)ATPBKsign(Gx(k)) - [Ksign(Gx(k)) ]TBTPAx(k) + f sign(G(x(k)) ]TBTPB[Ksign(Gx(k)) 68 Assume G = B TP AV (x) = XT (k) (ATPA - P) x (k) - xT (k) ATPBKsign (BTPx (k) ) - [ Ksign (BTPx(k) ) ] TB TPAx (k) + [ Ks i gn (B TPx (k) ) ] TB TPB [ Ks i gn (B TPx (k) ) ] Letting sign(BTPx(k)) = 12-1BTPx(k) where R is an arbitraryweighting matrix of form diag( ri ) A V (x) = XT (k) TPA- P) x(k) - XT(k) (ATPRKR-IBTP) x(k) - XT (k) (PB (R-1) TKBTPA) x(k) + xT (k) (PB (1?-1)TKBTPBKR-1BTP) x (k) Let = _[ A TPA _ A TpBKR-lBTp _ pBK- ( R-1) TB TpA pB T (R-1) TKB TpBKR-1B Tpl The system is stable if AV(x) is negative definite, Q must bepositive definite. This leads to the following Ricatti equation: P = + A TPA _ A T pBR'KB Tp pB(R-1)7703 TPA pB (R-1) TKB TpBR-1KBTp (AIII.4) The design procedure is specify a nominal system A,B with controlconstraints K, and specify the arbitrary weighting matrices Q andR. Then find P from (AIII.4), then G=BP and implementu=—Ksign(Gx(k)). 69 Appendix IV: Equations of the Transverse Motions This appendix presents the derivation of the state space equations for the transverse motions of a vessel. The equations will be based on the linear equations of motion. A sample calculation for one speed and loading condition will be performed. From Inoue et al [22]. ( see also v.3, Chapter 9, section 16.7 of [20]) the transverse equations of motion are: Sway^m(V + ur) =^+ YR + Yw^(AIV.1) Yaw Izzt = NH ÷ NR NW (A/V. 2 ) Roll^,x4) = KH KR + Kw^(A/V. 3) The force and moments due to wave action (.) have been added. The free response of the three motions will be developed first, starting with sway (AIV.1) 1m(li+ur) = -m P--mxur + -2- pLdV2{117 + rL1 + YR + Yw vV^r Iri M M 1.7 —1 pLdITYfy + (mu + m^1u - —pL2 dVYIr) .r =YR+ Yw 2^ 2 Yaw (AIV.2): = -J„t+4pLdV2[Nlvvi+leirl+4pL2dV2[44)]+NR+Nw ( Izz + Jzz) - 4 pLdVILNIv+ YlvKs I V — 4 pL2dvtizer+Y/gcsli- - l_pL2dv244) = 1\TR + Ivw 2 Roll (AIV.3): J - N(4) - gAGZ(4)) - YHZH +KR +Kw (^+^)4) + N(44) + gAGZ (4)) + YyzH = KR + Kw Using linear damping and stiffness, as in Bhattacharya [21] ( /xx + LT) + NI) + gAIGMA + YHzH = KR + Kw Expand YH (^+J ) + bd) + gAMT24) + [ -myx>. mxur + ipLdVYI,v+ pL2dVYIrr]zH = KR + Kw ( Im,r+ J )(T) + b4) gAGMT4- myzHV + 4 p LdVY/vzilv + [ -mxu + p L2 dVilzHr = KR + Kw 70 Geometries xR - The rudder centre of pressure with respect to the ship's centre of gravity. It is measured longitudinally, and is negative aft (see Figure 27). zH - Location of the ship's centre of pressure with respect to the ship's centre of gravity. This is the point at which the lateral hydrodynamic forces are assumed to act. Is measured vertically, negative up. For all ships in this study, it was assumed to be on-half the mean draft. zR - Location of the rudder's centre of pressure with respect to the ship's centre of gravity. It is measure vertically, and is negative up (see Figure 27). Figure 27. Geomerty of Rudder Moments Masses and Moments of Inertia Added mass in the y-direction. ny = k2 A Added mass in the x-direction mx = ki A Added yaw mass moment of inertia J = k'Inzz The co-efficients, k1, k2, and k' are found in Vol.3, Chapter 9, Section 9.4 of [20] . They are based on an ellipsoid of revolution with a major axis of one half the waterline length, and a minor axis of the mean draft. Yaw mass moment of inertia. 71 The yaw radius of gyration is assumed to be approximately one quarter of the waterline length, i.e., kzz = 0.25 4a then the yaw mass moment of inertia is Izz = Alc!z Roll mass moment of inertia. The roll radius of gyration is assumed to be: = CVB2 + .D2 where C varies between 0.33 and 0.39. The added roll mass moment of inertia was assumed to be 15% of the mass moment of inertia. Hydrodynamic Derivatives The following are the linear hydrodynamic derivatives from Inoue et al. [23]: Y. = -1-17tk + 1.4C 411.0 + —\2T ^ ‘2^BLA^31"/ = 17k[1 + 0.81]^r ^4 _ 0.271 1 T = -[0 . 54k - k2][1.0 + 0.3] where^k- 2T L pp 1 - ^ 13 ^[k + 1.4C 2-312 B L T = T. - TFwD The roll damping coefficient depended on the hull form. For hard chine hulls a damping ratio of 0.09 [25] was used. For round bilge 72 hulls the following formulae, from [20] were used: 0.55(.0024LBZ0d554)1 Co - ^AB2 )21 2 Fn 1+ Fn +2( Fn CB Cf = 0.00085H(-L m CB^CBB G C = CO + Cf where Co - zero speed damping ratio Cf - forward speed correction Fn - Froude Number - V sIgT, d - distance the intersection of the waterline and centreline to the turn of the bilge The critical roll damping was found from: = 2 VgAGMT(Ixx + Jxx) Rudder Forces The rudder forces and moments are : YR -(1 + aH)FNCOS8 NR = -(1 + aB) XRFNCOS8 KR = (1 +^zHFNcosO where - 1 p  6.131  ADVsina R-^2 1+2.25 - - aH = 0.627CB- 0.153 The method for finding VR in Inoue et al. is quite complicated, so 73 instead equation (169) from Vol.3, Chapter 9 of [20] was used TIR=k(1 + Sa) V where^0.8<k<1.0 depending how much of the rudder - is in the propeller slipstream.Pn VSa - For aR, assume aRz6. Also, it was assumed cos(5)=1. Sample Calculations The following are the necessary parameters needed to calculated the transverse motion co-efficients. Vessel name & Condition:^Eastward Ho Ship Data Full load 13kts Lwl 30.7848 Cb 0.513686Bwl 8.738616 L/B 3.522846Depth 4.2672 B/T 2.163774Taft 4.54025 B/D 2.047857Tfwd 3.4671Tmean 4.0386 tau 1.07315 Displ (Vol.) 558.094 Displ (Mass) 572.0463 Engine^ PropellerType Dia.^2.1336Bhp 850^Pitch 1.524RPM^1225 k 0.99Gear Red^4 RPM^300 knots^m/s Speed^13 6.68772 Fn 0.384836 Stability x - positive fwd; z - positive downLCG^-1.8928^VCG^3.377184GM 1.267968 zh 1.357884 below c.g. Round bilge or Hard Chine ( b or c ) distance from C.L. @ W.L. to turn of bilge 4.60 Pn where P - propeller pitch n - rev/ unittime V - Ship velocity 74 Rudder height^2.286^xr^-12.509 Span 2.286 zr 2.36584 Chord 1.2192 Area^2.787091 Aspect Ratio^1.875 The added masses, mass moments of inertia and hydrodynamic derivatives were calculated using the above formulae: Added Mass x-direction^45.04215 Added Mass y-direction^494.4331 Yaw Radius of Gyration^7.6962 Yaw Mass Moment of Inertia 33883.16 Added yaw m.m. of i.^19587.96 Roll Radius of Gyration^3.50094 Roll Mass Moment of Inertia7011.333 Added Roll M.M. of I. 1402.267 Zero Speed Roll Damping Coefficient 0.011913 Roll Damping 675.5547 0.262376 L-beta 0.425741 Yv prime -0.72545 Yr prime 0.249876 NV prime -0.21816 Nr prime -0.07865 Nphi prime -0.0076 For the rudder forces: Sa 0.122346 Rudder Velocity 7.430879 Rudder Normal Force 219.7674 ah 0.169081 The co-efficients for the equations of motion are then found to be: Sway Equation Sway acceleration 1066.479 Sway Velocity 309.136 Yaw rate 848.9842 75 Yaw Equation Yaw Acceleration^53471.11 Yaw Rate^ 25557.17 Sway Velocity 3447.02 Roll Angle 666.7563 Roll Equation Roll Acceleration^8413.6 Roll Velocity 675.5547 Roll Angle^ 7115.55 Sway Acceleration^-671.383 Sway Velocity -419.771 Yaw Rate 4042.015 Rudder forces Sway^ -256.926 Yaw 3213.883 Roll 607.8455 The co-efficents are then arranged into proper matrices A and B: where the state vector is: Ac 0 1 0 0 0 0 -0.47796 -0.06447 -0.01247 0 0 -0.79606 -0.28987 0 0 0 0 0 0 1 0 -0.54394 0.026761 -0.84572 -0.08029 Bc 0 0.060105 -0.24091 0 0.053022 Kc o o o 1 0 0 0 1 0 0 0 0 0 0.197267 1 These matrices are discretized using a zero order hold to obtain the discrete time matrices: A 1 0.445675 -0.00711 -0.00142 0 0 0.792773 -0.02667 -0.00535 -0.0014 0 -0.32935 0.870468 0.001073 0 0 -0.06138 0.003745 0.897635 0.473025 0 -0.23094 0.01553 -0.39928 0.859656 76 0.007248 0.028484 -0.11763 0.005723 0.020789 0.11573 -0.00122^0 0.445675 -0.00713 0 -0.08775 0.466396^0 -0.01052 0.010064 0.121193 -0.06138 0.040641 0.473025 A comparison between the continuous time model (using a Runge-Kutta differential equation solver) and the discrete time model is given in the following figures. As can be seen in the figures, not only do both models seem to be reasonable representations of the transverse motions, for the purposes of this study the difference between them is unmeasureable. Figure 28. 10° Rudder Step Response. 77 Figure 29. 100/100 Zig-zag, rudder and yaw. Figure 30. 100/100 Zig-zag, roll. 78 Appendix V: Wave Excitation The wave action on a ship is highly non-linear, and is not easily simplified. This appendix describes the steps used to obtain a numerical model of wave excitation for implementation in a discrete simulation. The excitation is usually represented as a force or moment equation that is dependent on frequency. From these equations, digital filters must be obtained that have as their inputs white noise. From ref. 26, the sway excitation force is Yw = p ge-kZSirlX NCOS (ki dx trnsin ( wet) and the yaw excitation moment is: Arw = -pge-kzsinxfAxxsin (1c/x) dx fl,cos (co et) where Z = distance of the M-3" below the waterline kl= kcosx k = the wave number x = the wave encounter angle ( 0° following seas, 1800 head seas) we = wave encounter frequency A = 1,4 Ta _ 47ue )2-^L2 In this simulation, the cross-sectional area Ax is that of anellipsoid of revolution, with the minor axis equal to the draughtand the major axis equal to one half the length. From [21] , the roll excitation moment is: = gAGMTsinx Esin(wet) The wave slope spectrum used in this simulation is based on theBretschneider Spectrum (wave height): _BA^4S(W) = w5 e " where A and B depend on the modal frequency and variance. Forwave slope spectrum, the Bretscneider Spectrum is modified: S(w) = W4 S(w) - Acog2eg2 79 The spectrum must be further modified so it is based on the frequency of encounter we: St (() e ) — S (c) - 2 6) uco sd where: () 2 U (4)e = 16)^COS)] All of the forces and moments are an amplitude multiplied by thewave slope Esin(wet + e); however, both the sway and yawamplitudes are functions of the wave number and hence the wave frequency. Therefore the spectra of these amplitudes must be found. They have the form: Se(we) = Ihre(wle) ps, (6)4) This complicates the calculation of the spectrum. Because each wehas three corresponding values of w, the amplitudes and wave slopes must be calculated for each of the three frequencies,multiplied, then summed ( see ref ). These spectra were then approximated by continuous time transfer functions whose power spectra were matched for shape visually andarea under their curves quantitatively. The transfer functions were discretized using the bi-lineartransform with the appropriate sampling period. For beam seas ( x=90 ), the excitation was further simplified. The sway excitation force used in this case was: Yw-  mg  sin(wet)m + m The yaw excitation was taken to be zero because the hull shapeapproximation of an ellipsoid of revolution means the hull hasfore-aft symmetry resulting in zero net excitation. The roll excitation remained the same. The only necessary spectrum calculation was then for wave slope,and in beam seas there is no complication with different wavefrequencies for encounter frequency, i.e. w=coe. 80 Appendix VI: Numerical Simulation Results K noc 15° Offset Controller Final yawangle Maximum yaw angles Settlingtime (s) Rise Time(s) PID 8.50e-08 -2e-06 9e-06 45 5.5 MV 0.125072 -5e-05 5e-05 20.5 20.5 MV RRS -0.38812 -2e-04 2e-04 27.5 11.5 BB GPC -0.90094 -2e-04 2e-04 22 6 BB GPC RRS -0.38774 -5e-05 4e-05 42.5 7.5 VSS 0.638019 -3e-05 3e-05 20.5 18.5 VSS RRS 0.125071 -5e-05 5e-05 33.5 7 Pole Plc -4.0e-09 -2e-06 2e-06 16 13.5 Pole RRS 0 0 0 23.5 12 SS GPC 1.00e-09 -2e-06 2e-06 21 7 SS GPC RRS -2.4e-07 -3e-04 3e-04 43.5 7.5 K noc 5.7° s Impulse Controller Final yaw angle _ Maximum Yaw angles Settling time (s) Rise Time(s) PID 2.10e-08 -2e-06 5e-06 44 5.5 MV -0.08354 -5e-06 4e-06 27.5 13.5 MV RRS 0.429435 -6e-05 4e-05 39 7 BB GPC 0.942373 -6e-05 5e-05 30 4 BB GPC RRS -0.08349 -4e-05 4e-05 27 5 VSS 0.429384 -5e-06 3e-06 27.5 18.5 VSS RRS -0.0836 -4e-05 5e-05 47.5 10.5 Pole Plc -1.2e-08 -6e-06 4e-06 27 7 Pole RRS 0 0 0 33.5 6.5 SS GPC -9.0e-09 -3e-06 2e-06 25 5 SS GPC RRS 0 0 0 23 5 81. Kynoc Change in Speed Controller Final yaw angle Maximum yaw angles Settling time (s) Rise Time (s) PID 0.000968 -0.025 0.06 112.5 118 MV 0.137847 -3e-05 3e-05 18.5 8.5 MV RRS -0.09171 -8e-05 9e-05 41.5 11.5 BB GPC -0.01489 -0.393 0.389 599 6.5 BB GPC RRS -0.09163 -2e-05 2e-05 32 8.5 VSS -0.55074 -le-04 le-04 22.5 13.5 VSS RRS -0.09163 -2e-05 2e-05 20 18.5 Pole Plc -3.0e-09 -2e-06 2e-06 19 8.5 Pole RRS 0 0 0 31 10 SS GPC -6.0e-09 -3e-06 2e-06 30 7.5 SS GPC RRS -2.6e-08 -le-05 8e-06 33 8.5 Kynoc Change in Displacement Controller final yaw angle Maximum Yaw Angles Settling time (s) Rise Time (s) PID -8.2e-08 -2e-05 3e-05 49 5.5 MV -0.3404 -4e-05 7e-05 35.5 13.5 MV RRS 0.199443 -5e-05 9e-05 44 10 BB GPC -0.88027 -le-04 2e-04 48 4.5 BB GPC RRS 0.199578 -6e-05 5e-05 38 4.5 VSS 0.199515 -2e-05 3e-05 35.5 22 VSS RRS 0.199452 -3e-04 6e-04 78 12 Pole Plc 7.80e-08 -3e-05 4e-05 35.5 6.5 Pole RRS 0 0 0 31.5 6.5 SS GPC 2.50e-08 -le-05 2e-05 33 5 SS GPC RRS -1.0e-09 -le-06 le-09 34.5 4.5 82 Kynoc Change in Speed and Dis lacemnt Controller Final yaw angle Maximum Yaw angles Settling time^(s) Rise Time (s) PID 0.000507 -0.048 0.108 118 9 MV -0.72037 -3e-05 3e-05 22.5 8 MV RRS -0.20882 -3e-05 8e-05 45.5 11.5 BB GPC -0.28261 -10.03 11.2 186 7 BB GPC RRS 0.047001 -6e-05 3e-05 55.5 8.5 VSS 1.069895 -le-04 le-04 22 16.5 VSS RRS -0.20894 -le-04 le-04 31.5 14.5 Pole Plc -9.0e-09 -le-05 le-05 24.5 7.5 Pole RRS 0 0 0 36 8 SS GPC -2.9e-08 -2e-05 2e-05 37.5 7.5 SS GPC RRS 3.05e-07 -0.002 0.002 53 8 K noc Rudder Servo Slows Controller Final yaw angle Maximum Yaw Angles Settling time (s) Rise Time (s) PID 2.30e-08 -2e-06 5e-06 44.5 599.5 MV -0.18619 -5e-05 5e-05 27.5 13.5 MV RRS -0.18596 -le-04 le-04 28.5 7.5 BB GPC -1.00688 -5e-05 5e-05 21 4.5 BB GPC RRS -0.18634 -2e-04 2e-04 27 5.5 VSS 0.634489 -5e-05 5e-05 27.5 15 VSS RRS -0.18619 -5e-05 5e-05 35.5 11.5 Pole Plc -1.0e-08 -5e-06 4e-06 27 11.5 Pole RRS 0 0 0 31.5 6.5 $S GPC -9.0e-09 -3e-06 2e-06 23 5.5 SS GPC RRS 0 0 0 24.5 5.5 83 Kynoc Constant Yaw Moment Controller Final yaw angle Maximum Yaw Angle Settling time^(s) Rise Time (s) PID -1.9e-06 -le-06 3e-06 40.5 94 MV 3.042305 -3.162 10.99 595.5 5.5 MV RRS 148.3223 -126.2 179 520 76.5 BB GPC 0.760904 -3.342 3.932 530.5 5.5 BB GPC RRS 25.81668 -162.8 180.3 593 7 VSS -0.1463 -0.132 0.129 597.5 16 VSS RRS -3.4502 -0.143 0.133 599.5 6.5 Pole Plc 0.613604 -1.095 1.169 583.5 32 Pole RRS 0.824794 -11.78 5.514 584 10.5 SS GPC 0.562783 -0.326 0.331 599 35 SS GPC RRS 14.26879 -118.4 135.1 599.5 7.5 The column %PID, in the tables of the irregular sea test results, is the amount of roll reduction using a controller as compared to using the PID controller. K noc 900 Seas Controller RMS rudder angle RMS yaw angle Maximum yaw angles RMS roll angle Maximum roll angle %PID PID 1.68 1.53 -5.61 3.39 2.78 -8.28 8.45 0.0% MV 6.42 1.56 -5.12 5.71 3.37 -9.49 10.4 -21.3% MV RRS 5.68 1.64 -4.67 6.28 3.45 -9.07 9.27 -24.0% BB GPC 7.44 2.41 -9.08 10.9 3.3 -10.5 8.97 -18.7% BB GPC RRS 7.32 3.35 -13.4 11.9 2.82 -8.08 10.5 -1.3% VSS 4.98 1.07 -3.58 4.94 3.23 -9.74 8.51 -16.1% VSS RRS 4.68 1.72 -5.22 6.15 1.76 -5.27 5.11 36.8% Pole Plc 4.5 1.14 -5.54 4.37 3.28 -9.53 10.4 -17.9% Pole RRS 5.52 3.14 -10.3 12.6 3.03 -9.93 11.1 -8.9% SS GPC 2.47 1.17 -3.46 3.86 2.82 -7.86 8.37 -1.5% SS GPC RRS 5.09 1.91 -6.68 4.88 2.22 -5.74 8.05 20.1% 84 Rynoc 450 Seas Controller RMSrudder angle RMS yaw angle Maximum yawangle RMSroll angle Maximum roll angle %PID PID 2.36 2.19 -5.71 5.36 2.59 -6.84 6.99 0.0% MV 2.08 0.27 -0.87 0.91 1.34 -3.57 3.95 48.3% MV RRS 2.3 1.37 -5.22 4.45 2.43 -7.11 8.05 6.2% BB GPC 2.19 0.42 -1.36 1.75 1.57 -3.48 4.48 39.3% BB GPC RRS 3.35 2.73 -9.85 8.93 2.37 -7.31 8.35 8.5% VSS 1.93 0.27 -0.85 0.67 1.3 -3.52 4.5 49.9% VSS RRS 5.27 5.91 -16.3 15.9 3.46 -10.5 9.84 -33.6 Pole Plc 1.87 0.48 -2.11 2.08 1.61 -4.4 5.25 37.9% Pole RRS 2.42 1.13 -4.14 4.37 2.09 -9.85 8.48 19.3% SS GPC 2.41 0.88 -3.57 3.34 2.1 -7.65 7.38 18.9% SS GPC RRS 3.13 1.24 -3.47 7.35 3.12 -10.4 9.68 -20.6 Kynoc 600 Seas Controller RMS rudder angle RMS yaw angle Maximum yaw angle RMS roll angle Maximum roll angle %PID PID 2.17 1.94 -6.67 5.43 2.56 -8.98 8.81 0.0% MV 2.78 0.4 -1.43 1.91 2.03 -6.53 6.46 20.5% MV RRS 3.15 1.46 -6.21 4.75 3.03 -9.02 8.66 -18.4 BB GPC 2.64 0.51 -1.92 1.8 2.54 -8.33 7.8 0.6% BB GPC RRS 3.64 2 -9.41 7.12 3.14 -9.4 9.91 -22.9 VSS 2.78 0.59 -2.27 1.91 2.29 -6.89 8.33 10.6% VSS RRS 5.18 5.64 -12.7 19.4 3.03 -8.4 8.38 -18.3 Pole Plc 2.18 0.61 -3.06 1.5 2.16 -6.92 6.05 15.6% Pole RRS 2.46 1.09 -2.87 3.21 2.09 -6.99 6.28 18.2% 1SS GPC 2.34 0.85 -2.87 2.43 2.37 -6.43 7.18 7.2% SS GPC RRS 4.08 1.42 -4.15 8.5 4.24 -14.4 11 -65.9, 85 Kynoc 30° Seas Controller RMSrudder angle RMS yaw angle Maximum yawangle RMS roll angle Maximum roll angle %PID PID 1.63 1.44 -4.98 4.66 1.34 -3.9 4.81 0.0% MV 5.71 1.03 -5.08 4.37 1.9 -5.57 5.2 -41.6% MV RRS 4.81 0.85 -2.67 3.35 1.85 -5.89 4.88 -38.2% BB GPC 6.89 1.85 -6.01 9.23 1.85 -4.9 6 -38.0% BB GPC RRS 4.37 1.72 -5.23 6.74 1.45 -4.58 4.61 -7.8% VSS 4.55 0.53 -2.68 2.41 1.65 -4.46 4.1 -22.8% VSS RRS 3.43 1.84 -5.06 4.7 1.21 -2.95 3.51 9.8% Pole Plc 3.78 1.02 -8.71 3.72 1.61 -4.07 6.23 -19.7% Pole RRS 3.01 0.86 -3.42 2.53 1.35 -3.95 5.33 -0.6% SS GPC 2.1 0.92 -2.59 3.43 1.38 -4.23 3.93 -3.1% SS GPC RRS 3.93 1.05 -3.27 3.26 1.5 -4.62 4.6 -11.7% Eastward Ho 150 Offset Controller Final yaw angle Maximum yaw angles Settlingtime (s) Rise Time(s) PID -7e-06 -0.0006 0.0002 61.5 9.5 MV -0.5592 -5e-05 0.0002 57.5 29.5 MV RRS -0.7745 -141.09 142.86 599.5 9.5 BB GPC -0.0004 -0.7497 0.7434 599.5 7 BB GPC RRS 0.00436 -4.1305 3.1941 173 7.5 VSS -0.5593 -0.0002 0.0005 45 20.5 VSS RRS -0.0035 -7e-05 5e-05 54.5 10 Pole Plc -0.0006 -0.0283 0.0088 112.5 13 Pole RRS 3.0e-06 -le-05 0.0001 61 18 SS GPC 1.6e-07 -0.0001 0.0002 46.5 7.5 SS GPC RRS -7e-05 -0.3572 0.3867 125 7.5 86 Eastward Ho 5.7°/s Impulse Controller Final yawangle Maximum yaw angles Settlingtime (s) Rise Time(s) PID -6e-06 -0.0007 0.0002 53 11.5 MV -0.6879 -0.0003 0.0004 52.5 26.5 MV RRS -3.5587 -141.76 150.63 588.5 23 BB GPC 0.7869 -62.223 61.5 599.5 13 BB GPC RRS -0.176 -5.9902 4.4323 215.5 20 VSS -0.1325 -0.0003 0.0007 37.5 27.5 VSS RRS -0.1321 -0.0002 0.0066 86.5 13.5 Pole Plc 0.00013 -0.0162 0.0488 105.5 19 Pole RRS 8.5e-06 -4e-05 0.0005 64.5 27 SS GPC -0.4564 -53.359 56.018 599.5 13 SS GPC RRS -0.0215 -1.4335 1.4286 188 18 Eastward Ho Chan e in S eed Controller Final yaw angle _ Maximum yaw angles Settling time (s) Rise Time (s) PID 0.00002 -0.0072 0.0213 82 18.5 MV 0.02614 -0.0002 0.0019 59.5 94 MV RRS 0.39059 -61.347 62.129 588.5 31 BB GPC -0.0496 -41.354 37.701 599 17.5 BB GPC RRS 0.15408 -23.311 24.146 410.5 25 VSS 0.32531 -0.0008 0.0003 42 11 VSS RRS 0.02617 -9e-05 0.0028 70.5 20 Pole Plc 0.00237 -0.0476 0.1434 113 21.5 Pole RRS 0.0001 -0.0005 0.0043 71 29.5 $S GPC -0.1898 -35.549 34.262 599.5 17.5 SS GPC RRS^_ 0.30052 -25.041 25.199 599.5 25 87 Eastward Ho Chancie in Displacement Controller Final yaw angle Maximum yaw angles Settlingtime^(s) Rise Time(s) P1D -7e-06 -0.001 0.0005 63.5 12 MV -0.7322 -0.0005 0.0012 56 34.5 MV RRS -0.1813 -16.977 14.463 599.5 25 BB GPC 0.63329 -52.121 53.6 593.5 13.5 BB GPC RRS 0.19987 -21.275 10.783 398 22 VSS 0.23758 -0.0002 0.0005 40 14 VSS RRS 0.23783 -0.0004 5e-05 64.5 12.5 Pole Plc 0.00021 -0.0146 0.048 105.5 19.5 Pole RRS 0.00001 -0.0001 0.0013 64.5 26.5 SS GPC 0.61526 -52.615 52.128 599.5 13.5 SS GPC RRS -0.4353 -43.571 42.295 595 19 Eastward Ho Change in Speed and Displacement Controller Final yaw angle Maximum yaw angles - Settling time (s) Rise Time (s) PID 0.00075 -0.0133 0.0498 89 20.5 MV 0.89277 -0.001 0.0008 45 24.5 MV RRS 0.35148 -7.8573 11.842 567.5 31.5 BB GPC 0.5361 -37.423 37.03 588.5 19 BB GPC RRS 0.02525 -21.816 21.485 597 26.5 VSS 0.43468 -20.182 19.51 599.5 12 VSS RRS 0.09858 -0.0057 0.0002 75.5 22.5 Pole Plc -0.0015 -0.1442 0.2244 121.5 19.5 Pole RRS 0.00051 -0.005 0.0293 85.5 29.5 SS GPC -0.3188 -34.427 35.615 591 19 SS GPC RRS 0.41512 -31.58 29.617 593.5 27 88 Eastward Ho Rudder Servo Slows Controller Final yaw angle Maximum yaw angles Settling time^(s) Rise Time (s) PID -6e-06 -0.0007 0.0002 53 11.5 MV -0.3545 -0.0002 0.0004 58.5 34.5 MV RRS 3.77801 -172.25 165.01 599.5 24.5 BB GPC -2.0863 -92.61 93.464 599.5 12 BB GPC RRS 0.06609 -2.5976 2.3221 325.5 22 VSS -0.3546 -0.0001 0.0004 38.5 21 VSS RRS 0.09006 -0.0006 3e-05 69 13.5 Pole Plc 0.00013 -0.0162 0.0488 105.5 19 Pole RRS 0.00001 -8e-05 0.001 65.5 26.5 SS GPC 0.50356 -81.621 82.572 594.5 12 SS GPC RRS -3.0707 -60.934 62.794 599.5 19 Eastward Ho Constant Yaw Moment Controller Final yaw angle Maximum yaw angles Settling time (s) Rise Time(s) PID 2.30345 -7e-05 8e-05 56.5 2 MV -1.2041 -1.5041 1.3088 599.5 24.5 MV RRS 125.524 -113.83 161.42 599.5 351.5 BB GPC 0.19339 -0.1952 0.2128 599.5 6 BB GPC RRS -2.005 -8.6811 11.364 577 5 VSS 0.07789 -0.1144 0.0831 596.5 11.5 VSS RRS -0.5758 -0.115 0.1003 596.5 13 Pole Plc 0.54117 -0.458 0.6232 599.5 147 Pole RRS 0.62503 -6.2533 6.4285 599.5 12 SS GPC 0.11393 -0.1941 0.2111 475 7.5 SS GPC RRS -7.709 -5.606 8.5922 599.5 56.5 89 Eastward Ho 900 Seas Controller RMSrudder angle RMS yaw angle Maximum yaw angles RMS roll angle Maximum roll angles %PID PID 0.79 0.741 -1.71 2.34 4.264 -12.5 10.6 0.0% MV 4.42 0.564 -1.69 1.81 3.649 -12.3 12.6 14.4% MV RRS 23 40.66 -105 84.2 4.013 -12.6 12.4 5.9% BB GPC 22.8 19.32 -42.2 39.9 4.632 -13.1 12.7 -8.6% BB GPC RRS 26.8 23.7 -42.9 44.4 5.749 -16.1 18.7 -34.8% VSS 6.73 0.831 -4.09 2.96 3.873 -10.6 11.1 9.2% VSS RRS 5.33 2.043 -6.67 6.42 4.112 -12.1 12.1 3.6% Pole Plc 0.39 0.83 -1.81 2.9 3.748 -9.93 9.94 12.1% Pole RRS 12.8 20.51 -66.6 45.8 4.509 -11.9 11.9 -5.7% SS GPC 20.3 16.29 -37.4 38.3 4.557 -13.1 13.2 -6.9% SS GPC RRS 25.4 20.46 -39.4 37.1 4.272 -13 11 -0.2% Eastward Ho 450 Seas Controller RMSrudder angle RMS yaw angle Maximum yaw angles RMS roll angle Maximum roll angle %PID PID 1.68 1.639 -3.91 3.88 0.929 -2.66 3.14 0.0% MV 2.17 0.464 -1.26 1.45 0.97 -2.56 3.36 -4.5% MV RRS 2.63 2.821 -9.94 7.5 1.024 -3.37 3.2 -10.2% BB GPC 2.76 0.318 -1.52 0.81 1.177 -2.85 3.78 -26.7% BB GPC RRS 7.43 3.68 -18.6 15.5 2.375 -6.25 5.89 -156% VSS 2.21 0.105 -0.37 0.25 0.859 -2.59 2.69 7.5% VSS RRS 3.12 2.847 -6.39 7.61 0.828 -2.02 2.03 10.9% Pole Plc 2.16 2.341 -6.55 6.08 0.913 -3.46 2.96 1.7% Pole RRS 2.32 1.091 -3.46 3.79 0.923 -2.89 4.93 0.6% SS GPC 2.7 0.395 -1.49 1.4 1.06 -3.47 2.96 -14.1% SS GPC RRS 4.16 0.863 -3.61 3.33 1.812 -5.71 5.65 -95.1% 90 Eastward Ho 60° Seas Controller RMSrudder angle RMS yaw angle Maximum yaw angles RMS roll angle Maximum roll angles %PID PID 1.78 1.72 -4.96 4.53 1.959 -4.71 6.34 0.0% MV 2.75 0.487 -1.26 1.35 1.955 -6.03 7.03 0.2% MV RRS 2.47 2.394 -9.44 7.29 1.64 -4.25 4.74 16.3% BB GPC 3.14 0.38 -1.84 1.19 1.91 -6.32 7.02 2.5% BB GPC RRS 5.99 2.367 -8.47 8.27 2.855 -7.8 7.27 -45.8% VSS 2.71 0.118 -0.39 0.32 1.629 -5.05 5.03 16.8% VSS RRS 4.15 3.62 -9.16 9.29 1.267 -4.17 3.7 35.3% Pole Plc 2.15 2.726 -6.34 8.06 1.639 -4.55 5.27 16.3% Pole RRS 4.24 2.633 -8.05 7.95 1.634 -6.94 6.14 16.6% SS GPC 2.9 0.457 -2.06 1.92 1.663 -4.41 4.4 15.1% SS GPC RRS 4.82 1.177 -5.9 5.29 2.537 -6.66 7.68 -29.5% Eastward Ho 300 Seas Controller RMSrudder angle RMS yaw angle Maximum yaw angles RMS roll angle Maximum roll angles %PID PID 0.84 0.809 -2.08 1.68 1.004 -2.95 4.02 0.0% MV 3.03 1.393 -2.61 5.26 1.159 -3.48 3.13 -15.4% MV RRS 2.08 0.737 -2 2.81 1.004 -3.47 3.19 0.0% BB GPC 24.2 21.6 -42.1 40.5 2.474 -5.69 5.22 -146% BB GPC RRS 7.53 1.998 -6.61 6.04 1.968 -5.34 4.88 -96% VSS 4.66 0.212 -0.8 0.87 1.32 -3.54 4.06 -31.4% VSS RRS 2.85 1.199 -3.3 3.22 0.687 -1.77 1.88 31.6% Pole Plc 0.53 1.101 -3.6 2.02 0.918 -2.68 2.66 8.6% Pole RRS 5.09 1.813 -5.85 4.17 1.348 -3.51 3.72 -34.2% SS GPC 23.6 19.13 -36.6 37.1 2.269 -6.03 5.3 -125% 91 The following are results with a rudder rate of 15°/s. Kynoc 90° Seas Controller RMS rudder angle RMS yaw angle Maximum yaw angle RMS roll angle Maximum roll angle %PID VSS RR 6.26 1.79 -4.63 5.67 1.9 -5.51 6.29 27.0% Pole RR 9.41 2.42 -12.1 7 2.44 -8.81 9.71 6.3% MV RR 10.1 1.38 -3.64 7.03 4.28 -11.8 12.6 -64.4% SS GPC RR 8.35 2.2 -8.69 8.83 2.99 -11.5 12.6 -14.7% PID 2 1.79 -4.53 5.39 2.6 -9.32 9.18 0.0% Pole AP 5.91 0.56 -2.02 2 4.16 -13.9 15.1 -59.7% Kynoc 45° Seas Controller RMS rudder angle RMSyaw angle Maximum yaw angles RMS roll angle Maximum roll angles %PID VSS RR 6.46 6.12 -17.8 19.1 3.77 -11.2 10.3 -69.8% Pole RR 3.55 1.39 -4.6 13.6 2.23 -18 15.6 -0.6% MV RR 3.26 2.26 -8.57 6.53 2.56 -7.01 8.27 -15.2% SS GPC RR 2.68 0.85 -3.34 2.63 2.57 -7.81 8.3 -15.8% PID 2.11 1.98 -6.4 5.65 2.22 -7.07 6.53 0.0% Kynoc 60° Seas Controller RMS rudder angle RMS yaw angle Maximum yaw angle RMS roll angle Maximum roll angle %PID VSS RR 6.25 5.61 -19.8 17.7 3.33 -8.98 8.94 -40.6% Pole RR 3.37 0.96 -3.33 5.71 1.84 -12.4 12.2 22.3% MV RR 5.49 1.92 -6.45 6 4.34 -11.2 9.8 -83.6% SS GPC RR 3.08 0.96 -4.32 4.27 3.37 -10.8 9.89 -42.4% PID 2.17 1.96 -5.84 5.55 2.37 -5.88 8.7 0.0% Pole AP 2.27 0.59 -2.79 2.01 2.38 -7.28 7.9 -0.5% 92 Kynoc 30° Seas Controller RMSruddera ngle RMS yaw angle Maximum yaw angle RMS roll angle Maximum roll angle %PID VSS RR 4.19 1.61 -4.08 4.61 1.21 -3.52 3.73 9.5% Pole RR 2.49 0.72 -2.69 2.7 1.3 -4.28 4.34 2.4% MV RR 6.56 0.92 -4.34 3.06 2.1 -7.21 6.76 -57.5% SS GPC RR 5.67 0.98 -4.9 2.99 1.71 -7.47 6.43 -28.5% PID 1.7 1.48 -3.69 4.51 1.33 -4.79 4.07 0.0% Pole AP 3.97 0.47 -2.44 1.98 1.54 -4.87 5.02 -15.5% Eastward Ho 900 Seas Controller RMS ruddera ngle RMS yaw angle Maximum yaw angle RMS rollangle Maximum roll angle %PID VSS RR 12.6 2.13 -9.07 5.41 4.37 -12 11.8_ 0.7% Pole RR 16.4 11.3 -33 31.2 4.81 -16.6 15.6 -9.5% MV RR 15.9 8.11 -23.6 30.9 3.72 -11.1 11.8 15.4% SS GPC RR 18.4 2.54 -7.94 10.4 4.07 -11.6 11.9 7.4% PID 0.73 0.68 -2.37 1.81 4.39 -12 12.3 0.0% Pole AP 0.46 0.98 -2.52 2.69 3.92 -12.1 11.4 10.7% Eastward Ho 450 Seas, Controller RMS ruddera ngle RMS yaw angle Maximum yaw angle RMS roll angle Maximum roll angle %PID ySS RR 4.12 2.25 -6.89 7.74 0.72 -2.47 2.03 11.0% Pole RR 3.31 0.61 -2.41 1.65 0.71 -2.24 2.47 12.2% MV RR 12.8 19.4 -53.2 55.2 2.31 -6.69 6.94 -186.5% SS GPC RR 5.5 0.62 -2.65 3.3 2.12 -10.7 9.68 -162.6% PID 1.59 1.54 -3.58 4.02 0.81 -2.35 2.5 0.0% Pole AP 2.12 2.17 -6.7 4.55 0.83 -2.46 2.74 -3.6% 93 Eastward Ho 600 Seas Controller RMS ruddera ngle RMS yaw angle Maximum yaw angle RMS roll angle Maximum roll angle %PID VSS RR 5.46 3.53 -10.3 9.97 1.34 -4.1 3.74 20.1% Pole RR 5.15 0.68 -1.9 3.18 1.23 -3.42 3.78 26.8% MV RR 3.97 3.26 -12.6 8.62 1.83 -5.25 5.77 -9.0% SS GPC RR 5.9 0.84 -4.58 4.57 2.96 -11.6 11.5 -76.3% PID 2.04 1.97 -5.96 7.08 1.68 -5.43 5.67 0.0% Pole AP 2.45 2.75 -9.56 7.76 1.81 -4.22 5.03 -7.8% Eastward Ho 300 Seas Controller RMS ruddera ngle RMS yaw angle Maximum yaw angle RMS roll angle Maximum roll angle %PID VSS RR 4.65 1.38 -3.85 2.68 0.78 -2.62 2.02 19.8% Pole RR 5.54 0.99 -2.87 3.89 0.98 -5.56 4.4 0.0% MV RR 2.16 1.32 -2.69 2.89 1 -3.08 3.66 -1.8% SS GPC RR 8.65 0.7 -2.96 2.89 2.29 -8.37 8.21 -134.2% PID 0.92 0.88 -2.34 1.97 0.98 -3.06 3.38 0.0% Pole AP 0.68 1.28 -2.97 2.93 0.92 -2.6 3.05 6.3% 94

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