Final Exam TTK4190 Guidance and Control

Transcription

1 Trondheim Department of engineering Cybernetics Contact person: Professor Thor I. Fossen Phone: Cell: Final Exam TTK4190 Guidance and Control Friday May 15, 009 Hours: 09:00-13:00 On the Exam Day: All printed and handwritten materials are allowed Calculators approved by NTNU can be used Language: English No of pages: 5 The exams will be graded by Fr iday June 5, 009. You may write your answers in Norwegian (bokmål, nynorsk), German or English. Perfect score amounts to 100%.

2 Problem 1: Modeling of Marine Vessels (30%) 1. (4%) Explain briefly the difference of first-principle and black box modeling.. (6%) Explain briefly why we need different models for simulation, feedback control and state estimation. 3. Consider a box shaped tanker with main dimensions: L pp (length between the perpendiculars), B (width) and T (draft). Assume that the drag coefficient does not vary along the x-axis of the ship and use the following formula 1 Lpp / D Y = ρ ( ) ( ) ( ) T x C Lpp / d x v + xr v + xr dx to compute an estimate of: A. (6%) The maneuvering coefficient Y vvfor a straight-line maneuver (no yaw motion) B. (6%) The maneuvering coefficient Y rrfor a pure yaw motion (no transverse motion). The derived expressions should only be functions of the cross-flow drag parameters as defined by the formula for Y. 4. (8%) Set up the longitudinal coupled equations of motion for a completely submerged torpedo shaped AUV. Assume that the damping matrix is nonlinear and decoupled. All parameters in the model must be defined (one sentence description for each parameter). Set up a list of assumptions needed for the model to be valid. The actuator models can be neglected that is, use forces and moments as control inputs. Problem : Modeling and Control of Roll Motions (40%) The roll motion of a box shaped tanker with main dimensions: L pp (length between the perpendiculars), B (width) and T (draft), can be modeled as Ix Kp) φ K pφ+ W GM sinφ = τ ( T where φ denotes the roll angle and τ is the control input. Assume that the center of gravity (CG) is located in the water line. The distance between the transverse metacenter and the center of buoyancy (CB) is given BM I T T =

3 where I T denotes the transverse moment of inertia about the water plane and is the volume displacement. 1. (3%) Find expressions for IT, and GM T as functions of the ship main dimensions.. (%) What is the relationship between the potential coefficients and the hydrodynamic derivatives K and K in the roll equation? p p 3. Assume that L pp = 46.0m, B = 46.0m and T = 10.0m while the radius of gyration R44 is 17.1m. In addition, assume that: Density of sea water: ρ = 105kg/m 3 Acceleration of gravity: g = 9.81m/s Based on the data above, A. (%) Compute the mass of the tanker. B. (%) Compute the roll moment of inertia. C. (%) Compute the transverse metacentric height for φ = 0 deg. D. (%) Plot the righting arm GZ = GM T sinφ as a function of the roll angle and explain how the nonlinearity affects the rolling motion of the ship. Also indicate how you can find the your figure. E. (%) Is the ship metacenteric stable (why/why not)? GM T -value in 4. (5%) Find an expression for the natural roll period that is valid for small roll angles. Compute an estimate of the natural roll period of the vessel based on this formula. Hint: The potential coefficients in roll as a function of the wave excitation frequency are given by Figure (5%) What is the relative damping ratio at the natural frequency? Viscous effects can be neglected. 6. (5%) Assume that a payload with mass m = tons is lifted onboard the ship and placed h = 0m vertically above the ship s CG. You may further assume that the CB and the displacement are unchanged. Compute the change in the transverse metacentric height GM T due to the payload. 7. (10%) Assume that the tanker is equipped with a stabilizing water tank that generates the control input τ and neglect all external disturbances. Derive a control law for roll damping and use a Lyapunov function to prove/conclude what kind of stability you have (uniform, asymptotic, exponential, local, global etc.). 3

4 1. x 1010 A 44 (U=0 m/s) frequency (rad/s) 14 x 108 B 44 (U=0 m/s) frequency (rad/s) Figure 1. A 44 and B 44 as a function of the wave excitation frequency. 4

5 Problem 3: Guidance and Control (30%) 1. (4%) In what kind of marine operations is it necessary to use wave filtering? Also give one example of a motion control system that not uses wave filtering.. Consider the yaw motion of ship given by ( Iz Nr ) r+ H( r) r = τ 3 where H() r = ar+ ar is a nonlinear function andτ is the yaw moment due to 1 rudders and environmental disturbances. The parameters satisfy a 1 < 0 and a > 0. Assume the ship is equipped with two rudders as shown in Figure. The rudders produce two yaw moments τi = k δi δ i, i = 1, where k>0 is constant. Figure. Ship equipped with two aft rudders. Design a nonlinear heading controller for tracking of a time-varying smooth reference signal and prove stability for the following two cases (neglect the wave filter and assume full state feedback): A. (6%) Both rudders have the same deflections: δ1 δ. B. (10 %) The rudders are individually controlled and should not be identically equal. Prove stability of the closed loop system and give the controller gain requirements as a function of the model parameters. 3. (5%) Explain how wind disturbances can be compensated by the control system. 4. (5%) Explain how current disturbances can be compensated by the control system. 5