DESIGN OF A HYBRID POWER/TORQUE THRUSTER CONTROLLER WITH LOSS ESTIMATION. Øyvind N. Smogeli, Asgeir J. Sørensen and Thor I. Fossen

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1 DESIGN OF A HYBRID POWER/TORQUE THRUSTER CONTROLLER WITH LOSS ESTIMATION Øyvind N. Smogeli, Asgeir J. Sørensen and Thor I. Fossen Department of Marine Technology Norwegian University of Science and Technology NO-7491 Trondheim, Norway [oyvind.smogeli, asgeir.sorensen]@marin.ntnu.no Department of Engineering Cybernetics Norwegian University of Science and Technology NO-7491 Trondheim, Norway tif@itk.ntnu.no Abstract: A hybrid power/torque thruster control scheme is proposed, together with a concept for thrust loss estimation in moderate and extreme seas. For highfidelity thruster control in extreme seas, estimates of the propeller load torque and losses due to waves, current, ventilation and in-and-out-of water e ects are of high importance for detecting the loss incidents, optimizing the thrust production, minimizing the wear and tear of the propulsion system and limiting the power consumption. The loss estimation problem is solved by designing a propeller load torque observer and by calculating an estimate of the torque loss factor based on an expected nominal load torque. The observer is proven to be globally exponentially stable, and shows good robustness subject to modelling errors. The torque loss estimation is shown to capture the main loss e ects. The hybrid controller is shown to have excellent performance in the entire operation regime, combining the best properties of torque and power control. Simulation results are presented to validate the performance of the load torque observer, the loss estimation and the hybrid controller. Copyright c 24 IFAC Keywords: Propulsion control, thrust losses, state estimation. 1. INTRODUCTION High performance control systems for positioning of ships and platforms with high tolerance to fault situations are critical for performing complex marine operations during extreme environmental conditions. Presently, dynamic positioning (DP) systems have limitations in rough seas. The reasons for this are limitations in the thrust capability and reduced performance of the control system due to losses and unmodelled nonlinearities. The limitation of power in the propulsion system and loss of thrust due to ventilation, in-and-out-of water e ects, transverse and in-line velocity fluctuations, cavitation and thruster-hull interactions will give poor control system performance, see for example Blanke (1981) and Minsaas et al. (1987). For a marine vehicle, the purpose of low-level thruster control is to fulfill the high-level control commands from e.g. the DP system or a joystick. This is essential for the total performance CAMS 24 49

2 of the operation, but thruster control has nevertheless received relatively little attention both in academia and in industry. For some recent results on thruster control see Fossen and Blanke (2), Whitcomb and Yoerger (1999) and references therein. The current industrial practice in thruster control is shaft speed control for fixed pitch (FP) propellers and pitch control for controllable pitch (CP) propellers, where the DP system produces the shaft speed or pitch setpoints respectively. More sophisticated thruster control schemes based on power and torque control were introduced by Sørensen et al. (1997). This was extended by introducing anti-spin thruster control to handle extreme conditions caused by ventilation and in-and-out-of water e ects in Smogeli et al. (23). Power and torque control were shown to give increased performance in moderate seas. In addition, the power control concept was shown to have desirable properties in connection with severe thrust losses in extreme seas. In this paper, shortcomings of the existing power and torque control concepts are discussed, and a hybrid power-torque control scheme that is valid for a larger range of operating conditions is proposed. In anti-spin thruster control, an estimate of the thrust loss due to the various loss e ects is essential for optimizing the thrust production. A thruster observer providing estimates of the propeller load torque and the torque loss factor using feedback from motor torque and propeller shaft speed is presented, and simulation results are shown to validate the performance. 2. THRUSTER MODELING A thruster may be modelled as an electric motor, a shaft with friction and a hydrodynamically loaded propeller: = 1 ( ) (1) = (2) = ( ) (3) Here, is the motor time constant, is the commanded torque from the local thruster controller, is the motor torque, is the rotational inertia of the propeller including added mass, shaft, gears and motor, is a linear friction coe cient, =2 is the rotational speed of the propeller, and is the propeller load torque. The load torque is modelled as a nonlinear function of fixed thruster parameters (i.e. propeller diameter, position, number of propeller blades, pitch ratio, propeller blade expanded-area ratio) and variables (i.e. shaft speed, relative submergence, advance speed). The actually produced propeller thrust is modelled in a similar manner as: = ( ) (4) The nominal thrust and torque are typically found from open-water tests with a deeply submerged thruster, expressed by the thrust and torque coe cients and or other equivalent mappings (Carlton, 1994): = 4 (5) = 5 (6) Here, is the propeller diameter, and is the density of water. and are in general functions of many parameters and variables, with velocity of advance,shaftspeed and thrust direction being the most important. The nominal thrust and torque coe cients and are the values for zero advance speed, =,which are commonly used in control systems for DP since information on usually is unavailable. In station-keeping, zero advance speed is a good approximation. Thrust losses may be expressed by the thrust and torque loss factors and, which express the ratio of actual to nominal thrust and torque (Minsaas et al., 1987): = = (7) This means that the actual propeller thrust and torque may be expressed as: = ( ) = = ( ) = (8) 3. WEIGHT FUNCTION A weight function satisfying certain criteria is needed in the following. The smooth weighting function ( ) : R R, where is a variable, must satisfy the following properties: lim ( ) =1 lim ( ) = (9) Additionally, ( ) should be close to zero for large. Based on this requirement, the following weight function is proposed: ( ) = (1) where, and are positive constants. Figure 1 shows ( ) for varying, and. The parameter will act as a scaling factor for the variable. A small will widen the weighting function, giving a wider transition between and 1. Increasing the parameter sharpens the peak about =, whereas increasing widens it and makes the transition from to 1 more steep. The weight function must show smooth behavior for all.the derivative of with respect to is: 41 CAMS 24

3 α(x) r = 2, p = 1, k = 1 r = 2, p =.5, k = 1 r = 8, p = 1, k = 1 r = 2, p = 1, k = x Fig. 1. Weight function ( ) for varying,, ( ) = ( 1 ) = 2 21 = ( ) (11) which is smooth in. Particularly, =for =and ±. 4. THRUSTER OBSERVER 4.1 Propeller load torque observer The actual propeller load torque in (3) may not be measured explicitly, but may be estimated by an observer. Based on (2), the following control plant model of the thruster dynamics is proposed: = = 1 + (12) is here treated as an exogenous disturbance, and modelled as a first order Markov process with time constant driven by white noise with zero mean. With the motor torque as input = and the shaft speed as measurement =, the state-space formulation, which is observable from, iswritten: = + + = (13) using the following vectors and matrices: 1 = = 1 = = 1 = 1 (14) Motivatedby(13)andaddinganinjectionterm, the following propeller load torque observer is proposed: ˆ = ˆ + + ˆ = ˆ = = 1 (15) 2 Here, = ˆ and = ˆ. The error dynamics found by subtracting (15) from (13) becomes: = ˆ =() + = + (16) where the matrix is defined as: + =()= 1 1 (17) 1 The equilibrium point = of the observer estimation error is globally exponentially stable (GES) if the measurement noise =and is Hurwitz, which is implied by positive definiteness of ( ). By Sylvester s theorem ( ) is positive definite if: 2 ) ) ( + 1 ) 2 2 ( + 1 ) (18) If is a Gaussian white noise process the trajectories will converge to ball around the origin =and uniform ultimately boundedness (UUB) follows. The equations for implementation are: ˆ = 1 ( ˆ ˆ + )+ 1 ( ˆ ) ˆ = 1 ˆ + 2 ( ˆ ) (19) In an alternative formulation, the propeller load torque can be modelled as a Wiener process, i.e. = in (12). The load torque observer is then implemented by replacing ( 1 ) with in (14), and choosing the gains 1 and 2 according to 1 and 2. Simulations indicate that this latter formulation also is adequate. 4.2 Torque loss calculation For DP operation the expected nominal propeller load torque may be calculated from (6) by feedback from the propeller shaft speed as: = 5 (2) where the nominal torque coe cient for advance speed =is used. Based on (7), the estimated torque loss factor ˆ is calculated from ˆ (19) and (2) as: ˆ = ˆ ˆ = 5 (21) The estimate is singular for =,whichmeans that special precautions must be taken for the CAMS

4 zero-crossing of. Since the thrust loss is undefined for zero thrust, it makes sense to require lim ˆ ( ) =1. The torque loss estimate ˆ to be implemented should therefore be defined as: ˆ = ( )+(1 ( ))ˆ (22) where ( ) is a weight function given by (1). Additionally, ( ) must be such that ˆ ˆ for, where is a small positive number. The quality of the estimate will depend on the accuracy of the thruster model. From (19) and (2), the two most important modelling parameters for ˆ are the rotational inertia and the nominal torque coe cient. A modelling error in will to a large extent be compensated for by the robustness of the observer, but an error in will appear as a scaling error in ˆ. 5. THRUSTER CONTROL For control purposes, the nominal thrust and torque of a propeller are commonly expressed by (5) and (6), replacing and with the nominal coe cients and (Fossen, 1994). The signed power is accordingly given as: 5.1 Shaft speed control = 2 (23) The conventional thruster control scheme is shaft speed control, which utilizes shaft speed feedback from the thruster to set the commanded motor torque by e.g. a PID algorithm operating on the shaft speed error. The inverse thrust characteristics, relating the desired shaft speed to the desired thrust is expressed by inverting (5): 5.2 Torque control = ( )s 4 (24) The object of torque control is to control the setpoint of the motor torque instead of the shaft speed. A mapping from desired thrust to desired torque can be found from (5) and (6). The commanded motor torque is set equal to the desired torque, see Sørensen et al. (1997): = = (25) 5.3 Power control Power control is based on controlling the power from the drive system. The desired power is found by inserting the desired torque from (25) and the desired shaft speed from (24) in (23). The commanded motor torque is calculated from using feedback from the measured shaft speed 6= according to (23), see Sørensen et al. (1997): = 2 = 2 2 = Hybrid power/torque control ( ) 3 2 (26) A significant shortcoming of the power control scheme in (26) is the singularity for zero shaft speed, =. This means that power control should not be used close to the singular point, e.g. when commanding low thrust or changing thrust direction. For low thrust commands, torque control shows better performance in terms of constant thrust production than power control, since the mapping between thrust and torque is more direct than the mapping between thrust and power. For high thrust commands, it is essential to avoid large power transients, as these lead to higher fuel consumption and possible danger of power blackout and harmonic distortion of the power plant network. Power control is hence a natural choice for high thrust commands. This motivates the design of a hybrid power/torque control scheme, utilizing the best properties of both controllers. The commanded motor torque from the hybrid power/torque controller is defined as: = ( ) +(1 ( )) (27) where ( ) is a weight function given by (1), which defines the dominant regimes of the two control schemes. The shaft speed is physically limited to some max value max, such that ( max ) should be close to zero. The control law must show smooth behavior for all. The derivative of the commanded torque with respect to is: = ( ) + ( ) + (1 ( )) +(1 ( )) = ( ) ( ) ( ) (1 ( )) ( ) ( ) 412 CAMS 24

5 = ( ) ( ) ( ) ( ) ( ) 3 2 (28) 3 2 The first two terms contain no singularities for 1, however it remains to investigate the term: 1 ( ) = 1 The limit of ( ) as tends to zero is:, ( ) (29) 1 lim ( ) = lim (1 ) = lim ( ) = lim 2 = lim = for 1. The hybrid controller hence shows smooth behavior with respect to as long as 1 in the function ( ) defined in (1). 6. RESULTS In order to investigate the performance of the thruster observer and the hybrid controller, simulations were performed in Simulink r with MCSim (Sørensen et al., 23). A typical DP case was simulated for a supply vessel with main particulars [ ] = [ ] equipped with dieselelectric ducted propulsors in the aft ship. The thruster submergence relative to the free surface and the water velocity relative to the thruster was found by accounting for waves, current and vessel motion,withanassumptionofundisturbedwaves. Figure 2 shows the actual and estimated propeller torque found by using the observer given by (19) when the thruster is experiencing moderate losses due to in-line and cross-flow velocity fluctuations. The estimate is highly accurate, and shows good robustness subject to modelling errors. Figure 3 shows the actual and estimated propeller torque when the thruster is subject to high thrust losses due to ventilation and in-and-out-of water e ects in extreme seas. The propeller torque estimate clearly follows the fast transients due to the ventilation incidents, which can be seen as sudden losses of load torque followed by fast transients when the thruster re-enters the water and ventilation terminates. Figure 4 shows the actual and estimated torque loss factors and ˆ when the thruster is subject to high thrust losses. The torque loss factor estimate clearly captures the Q p [Nm] 9.4 x time [s] Fig. 2. Actual (solid) and estimated (dotted) propeller torque in moderate seas. Q p [Nm] x time [s] Fig. 3. Actual (solid) and estimated (dotted) propeller torque when subject to high thrust losses. β Q time [s] Fig. 4. Actual (solid) and estimated (dotted) torque loss factor when subject to high thrust losses. main loss events. The peaks in ˆ occurring at the termination of ventilation are due to the small time delay in ˆ. Figure 5 shows the commanded torque from the hybrid controller, the torque controller and the power controller for a varying thrust reference, including a zero-crossing of the shaft speed. The weight function parameters were =5 = 12 and =4, such that pure power control was achieved for high shaft speed and pure torque control was achieved for low shaft speed. The results illustrate that the power controller singularity in (26) is avoided by using the hybrid CAMS

6 Commanded torque Controller comparison Hybrid Torque Power Time Fig. 5. Commanded torque for hybrid, torque and power controller with changing thrust reference. Motor Power 1 x Controller comparison Hybrid Torque Power Time Fig. 6. Motor power for hybrid, torque and power controller with changing thrust reference. controller (27). The corresponding motor power is plotted in Fig. 6, illustrating the advantages of the hybrid controller. For high shaft speeds, and hence high power consumption, the hybrid controller coincides with the power controller, assuring small variations in power consumption. For low shaft speeds, the controller coincides with the torque controller, assuring best possible thrust production. 7. CONCLUSIONS The equilibrium point = of the observer estimation error was proven to be GES under the assumption of no measurement disturbances. For the case when plant was exposed to white noise disturbances the observer estimation error was UUB implying that the system trajectories converge to a ball around the origin. The observer showed good performance even in transient situations, and the torque loss calculation clearly captured the main e ects of the thrust losses experienced in moderate and severe loss situations. The hybrid power/torque thruster controller showed good performance for the complete operating range. The best properties of both the power and the torque control schemes were utilized, and the singularity for power control at zero shaft speed was avoided. The hybrid controller and loss estimation scheme will be of high importance for further work on thruster anti-spin and fault-tolerant thruster control. 8. ACKNOWLEDGMENT This work has been carried out at the Centre for Ships and Ocean Structures (CESOS)atNTNUin cooperation with the research project on Energy- E cient All-Electric Ship (EEAES). The Norwegian Research Council is acknowledged as the main sponsor of CESOS and EEAES. 9. REFERENCES Blanke, M., (1981). Ship Propulsion Losses Related to Automated Steering and Prime Mover Control. PhD dissertation, The Technical University of Denmark, Lyngby, Denmark. Carlton, J.S. (1994). Marine Propellers & Propulsion. Butterworth-Heinemann Ltd. Fossen, T.I. and M. Blanke (2). Nonlinear Output Feedback Control of Underwater Vehicle Propellers Using Feedback From Estimated Axial Flow Velocity. IEEE Journal of Oceanic Engineering, 25 (2). Fossen, T.I. (1994). Guidance and Control of Ocean Vehicles. John Wiley and Sons Ltd. Minsaas, K.J., H.J. Thon and W. Kauczynski (1987). Estimation of Required Thruster Capacity for Operation of O shore Vessels under Severe Weather Conditions. PRADS Smogeli, Ø.N., L. Aarseth, E.S. Overå, A.J. Sørensen and K.J. Minsaas (23). Anti-spin thruster control in extreme seas. Proceedings of 6 IFAC Conference on Manoeuvring and Control of Marine Craft (MCMC 3), Girona, Spain. Sørensen, A.J., A.K. Ådnanes, T.I. Fossen and J.P. Strand (1997). A new method of thruster control in positioning of ships based on power control. Proceedings of the 4 IFAC Conference on Manoeuvring and Control of Marine Craft (MCMC 97), Brijuni,Croatia. Sørensen, A.J., E. Pedersen and Ø.N. Smogeli (23). Simulation-Based Design and Testing of Dynamically Positioned Marine Vessels. Proceedings of International Conference on Marine Simulation and Ship Maneuverability, MARSIM 3, August 25-28, Kanazawa, Japan. Whitcomb, L.L. and D.R. Yoerger (1999). Preliminary Experiments in Model-Based Thruster Control for Underwater Vehicle Positioning. IEEE Journal of Oceanic Engineering, 24 (4). 414 CAMS 24