Final Exam TTK 4190 Guidance and Control

Transcription

1 Page 1 of 8 Contact person during the exam: University lecturer Morten Breivik, Department of Engineering Cybernetics, Gløshaugen Phone: 73 5( ) Cell: Final Exam TTK 4190 Guidance and Control Wednesday May 23rd 2007 Hours: Aids: Language: A All printed and handwritten materials are allowed. Calculators approved by NTNU can be used. English. No. of pages: 8. The exams will be graded by June 13th The percentages represent the relative weight each problem is attributed at the evaluation.

2 Page 2 of 8 1 Multiple Choice (15%) Only one alternative is correct. 1. (2%) Which dimensionless expression holds true? (a) Kq 0 = 1 K 1 2 ρl3 U q (b) Kq 0 = 1 1 K 2 ρl3 U 2 q (c) Kq 0 = 1 K 1 2 ρl4 U q 2. (3%) Consider a floating cube where all the sides have length L. Assume that the water density is ρ while the cube density is ρ c. The hydrodynamically added mass in heave is Zẇ = m where m isthecubemass(uniformly distributed). What is the undamped resonance frequency in heave for the floating cube? (a) ω z = q 1 2 ( ρ c ρ ) g L (b) ω z = q 1 2 ( ρ ρ c ) g L 2 (c) ω z = q 1 2 ( ρ ρ c ) g L 3. (4%) Given the nonlinear system ψ = r (1) (I z Nṙ)ṙ + H(r)r = Kδ + d (2) d = 0, (3) where d is a disturbance, H(r) represents the nonlinear maneuvering characteristics, and the rudder angle δ is the control input. Considering e = ψ ψ d and r d = ψ d, which of the suggested controllers delivers the best overall performance? h R i (a) δ = 1 t (I K z Nṙ)(ṙ d K p e K d ė K i e(τ)dτ)+h(r)r o h R i (b) δ = 1 t (I K z Nṙ)(ṙ K p e K d ė K i e(τ)dτ)+h(r)r o h R i (c) δ = 1 t (I K z Nṙ)(ṙ d K p e K d ė K i e(τ)dτ) o 1

3 Page 3 of 8 4. (4%) Employing the following sliding mode controller δ = 1 Z t (I z Nṙ)(ṙ d 2λė λ 2 e)+h(r)(r d 2λe λ 2 K where Z t s = ė +2λe + λ 2 e(τ)dτ, λ>0,andk>0, 0 what is the resulting error dynamics of the nonlinear system (1)-(2)? (a) (I z Nṙ)ṡ + H(r)s + K s = d (b) (I z Nṙ)ṡ +[H(r)+K]sgn(s) =d (c) (I z Nṙ)ṡ + H(r)s + Ksgn(s) =d 5. (2%) Which statement is true (regarding the control design paradigms Maneuvering and Trajectory Tracking)? (a) Maneuvering and Trajectory Tracking are equally flexible. (b) Maneuvering is more flexible than Trajectory Tracking. (c) Trajectory Tracking is more flexible than Maneuvering. 2 Guidance System (20%) 0 e(τ)dτ) Ksgn(s), Consider Figure 1, where a kinematic model of a marine surface vessel is represented by the position vector p =[x, y] > and the velocity vector ṗ =[ẋ, ẏ] > (in NED coordinates). Also, consider a straight line segment (track) represented by its orientation angle α (with respect to the NED frame) and an arbitrary point p l =[x l,y l ] > somewhere along the track. 1. (2%) Show that the cross-track error e (see Figure 1) can be calculated by e = (x x l )sinα +(y y l )cosα. 2. (5%) Consider the Control Lyapunov Function (CLF) for the cross-track error V = 1 2 e2, (4) and show that U V = p e e2 (5) when employing the velocity vector guidance law 2

4 Page 4 of 8 p χ p X N e Δ α p l Y N Figure 1: An ideal, kinematic guidance consideration. µ χ = α +tan 1 e, (6) 4 where 4 > 0 (the so-called lookahead distance), and ṗ = U with U> 0 (Hint: Consider the trigonometry of Figure 1). What global and local stability properties do(4)and(5)givefor the origin of the cross-track error? 3. (5%) How can you combine this guidance concept with a heading (and speed) autopilot in order to achieve track following? How will such a track following concept depend on the lookahead distance 4 (Hint: Relate 4 to what you do when you drive a car along a road)? 4. (3%) What is the relationship between the guidance approach of Figure 1 and the modified LOS guidance scheme? 5. (5%) A ship guided by the LOS (or the modified LOS algorithm) will drift off track when subjected to a constant environmental disturbance. Why? What can be done to minimize the cross-track error in such a situation? Moreover, how can perfect track following be achieved in such a situation (Hint: Distinguish between the course and the heading of a ship)? 3

5 Page 5 of 8 3 Ship Roll Control (20%) Consider the 1 degree-of-freedom (DOF) roll dynamics of a ship φ = p (7) (I x Kṗ)ṗ K p p p p K p p + W GM T sin φ = τ K, (8) where K p p < 0, K p < 0, andgm T > 0. The control objective is to track a bounded, continuous and readily available roll reference trajectory φ d, φ d, φ d. The ship is equipped with two identical fins located symmetrically on each side of the hull such that τ K = K α α,wherek α 0 and α is the effective fin angle. 1. (5%) Derive a feedback linearizing control law for α that fulfills the control objective. Will your controller be able to track the desired roll motion at all ship speeds? Justify your answer. 2. (5%) Derive a backstepping control law for α that fulfills the control objective and takes advantage of useful model properties. 3. (2%) Compare the feedback linearizing and backstepping controller expressions. 4. (3%) The hydraulic machinery that rotate the fins do not respond perfectly to the controller commands. Hence, augment the dynamic model (7)-(8) with the findynamics α = 1 T (α c α), (9) where α c represents the commanded control input and T > 0. Derive a backstepping control law for α c that fulfills the control objective (Hint: Augment your existing backstepping design). 5. (5%) Linearize the open-loop roll kinetics (8) (i.e., τ K =0) about the equillibrium point φ, =[0, 0] > and show the resulting mass-damper-spring h φi > (MDS) model. Subsequently, for a ship with the model data (I x Kṗ) = [kgm 2 ], K p = [kgm 2 /s], W = [kgm/s 2 ], and GM T =1[m], find the relative damping ratio and natural frequency associated with the MDS roll model. Based on this MDS roll model, does the vessel need a ship roll stabilization system? Justify your answer. 4

6 Page 6 of 8 4 Weather Optimal Control (10%) 1. (3%) What is the motivation behind weather optimal position control (WOPC) for marine vessels? What kind of vessels should have WOPC capability? In what situations will the employment of WOPC be an advantage/disadvantage? 2. (2%) What are the basic two rules that make the weather optimal heading control (WOHC) concept work? Justify your answer. 3. (2%) Which variable most significantly influences the rate of convergence toward the weather optimal heading and why? 4. (2%) State the full control law expression τ for the WOPC scheme and explain the purpose of each term. 5. (1%) Why is the WOPC stability result only local? Justify your answer. Figure 2: The principle of weather optimal control. 5 Vessel Motion Control (35%) The standard 3 DOF maneuvering model of a marine surface vessel is given by η = R(ψ)ν (10) M ν + C(ν)ν + D(ν)ν = τ, (11) where M = M > > 0, C(ν) = C > (ν), andd(ν) 0. Here,τ represents the control input vector (i.e., environmental disturbances are not considered). 1. (2%) Explain why you can simplify equation (11) when designing a dynamic positioning (DP) system. Write down the resulting DP model. 5

7 Page 7 of 8 2. (16%) Consider the CLF V = 1 2 (z> 1 z 1 + z > 2 Mz 2 ), (12) where z 2 = ν α (13) and α is a stabilizing function yet to be designed. Furthermore, we can consider either z 1 = η η d, (14) or z 1 = R > (ψ)(η η d ) (15) when deriving a motion control law by backstepping. Here, η d represents the desired trajectory to be tracked, and you can assume that η d, η d,and η d are all bounded, continuous and readily available. (a) (5%) Using the DP model, derive two backstepping controllers by respectively employing (14) and (15) such that the time derivative of (12) becomes V = z > 1 K 1 z 1 z > 2 K 2 z 2 (16) where K 1 = K > 1 > 0 and K 2 = K > 2 > 0. (b) (5%) What is the main qualitative difference between the controller based on (14) and the controller based on (15)? Which controller would you have chosen for implementation and why? (c) (2%) On the basis of (12) and (16), what is the stability property of the origin of the error system state z = z > 1, z > > 2? (d) (2%) Why is (12) referred to as the pseudo-kinetic energy of the error system? (e) (2%) How can you redesign your backstepping controllers to take advantage of hydrodynamic damping? How does such a redesign affect (16) and the stability property of the error system origin? 3. (9%) Now consider the complete maneuvering kinetics of (11) (representing a vessel travelling at moderate speeds). (a) (3%) Derive a backstepping controller by using (12), (13) and (15) such that the time derivative of (12) becomes V = z > 1 K 1 z 1 z > 2 (D(ν)+K 2 )z 2. 6

8 Page 8 of 8 (b) (2%) Explain the role of each term in your control law. (c) (4%) Derive the associated error system dynamics (i.e., the z-dynamics) and elaborate on the resulting model structure. 4. (3%) Consider the control input vector τ =[τ X,τ Y,τ N ] >,representingthe surge, sway and yaw control inputs, respectively. A marine surface vessel travelling at high speeds cannot independently control its sway DOF (usually because τ Y = f(τ N ) for some function f( )). Why is this so and how does this fact influence (the feasibility of) the recently developed backstepping controllers? 5. (5%)Why is it useful(and necessary)to distinguish between autopilot (AP) and dynamic positioning (DP) motion control systems? What are the main differences between AP and DP control system design? 7