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Steady-State Sensitivities of Marine Propulsion Control Techniques

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Abstract

This paper presents a steady-state sensitivities analysis of four well-known propulsion control techniques: shaft speed, torque, power and integrative torque–power controllers applied to a ducted marine propeller driven by an electrical motor. Also, this work proposes a variation in the latter technique and analyzes its sensitivities as well. The control loop scheme follows the structure of propulsion control in a dynamic positioning system. The sensitivity concept is used to perform the analysis, which consists of a normalization of the equilibrium point with respect to its reference value. The effect of inaccuracies between the control model and plant parameters over the equilibrium points is considered and analyzed in three scenarios for each controller. Simulations with a ducted propeller model are performed to provide further information about sensitivities, and the results are compared by using performance measurements based on the mean absolute error. The results provide detailed information on each controller’s characteristics in steady state in the whole operation range regarding shaft speed, torque, power and thrust.

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Abbreviations

\(\alpha \) :

Transition function from torque to power control for iQP and iQ-P controller

\({\bar{n}}\) :

Shaft speed steady-state value for a given controller and for a certain reference value

\({\bar{P}}\) :

Power steady-state value for a given controller and for a certain reference value

\({\bar{Q}}\) :

Torque steady-state value for a given controller and for a certain reference value

\({\bar{T}}\) :

Thrust steady-state value for a given controller and for a certain reference value

\(\varDelta _{q1}\) :

Discriminant of the system equilibrium points for the torque controller for positive shaft speeds

\(\varDelta _{q2}\) :

Discriminant of the system equilibrium points for the torque controller for negative shaft speeds

\(\epsilon \) :

Sufficiently small number for the power control law switching

\({\mathbb {T}}\) :

Shaft speed range of operation

\({\mathcal {C}}_{2,\mathrm{iqp}}\) :

Set in which mixed torque–power control is active (iQP controller)

\({\mathcal {C}}_{\mathrm{iq\hbox {-}p}}\) :

Set in which mixed torque–power control is active (iQ-P controller)

\({\mathcal {P}}_{\mathrm{iq\hbox {-}p}}\) :

Set where power control operates (iQ-P controller)

\({\mathcal {P}}_{\mathrm{iqp}}\) :

Set where power control operates (iQP controller)

\({\mathcal {Q}}_{i,\mathrm{iq\hbox {-}p}}\) :

Set in which the torque with integral shaft speed error (iQ-P controller)

\({\mathcal {Q}}_{i,\mathrm{iqp}}\) :

Set in which the torque with integral of shaft speed error (iQP controller)

\({\mathcal {Q}}_{\mathrm{iqp}}\) :

Set where torque control is active without integral of shaft speed error (iQP controller)

\(\omega _r\) :

Filter natural frequency

\(\rho \) :

Water density

\(\text {AUV}\) :

Autonomous underwater vehicle

\(\text {DP}\) :

Dynamic positioning

\(\text {MAE}\) :

Mean absolute error

\(\text {PMS}\) :

Power management system

\(\text {ROV}\) :

Remotely operated underwater vehicle

\(\xi \) :

Filter damping

D :

Propeller blade diameter

\(D_f\) :

Viscous friction

\(D_f^*\) :

Thruster nominal viscous friction

\(D_{\mathrm{fc}}\) :

Estimated viscous friction used in the controller

\(e_i\) :

Shaft speed error in the iQP and iQ-P controllers

\(I_s\) :

Thruster moment of inertia seen from the propeller

\(I_{\mathrm{sc}}\) :

Estimated rotational inertia seen from the propeller and used in the controller

k :

Adjustable coefficient for \(\alpha \)

\(K_i\) :

Integrative gain of shaft speed integral error

\(K_p\) :

Proportional gain of the PI controller

\(K_Q\) :

Torque coefficient

\(K_Q^*\) :

Thruster nominal torque coefficient

\(k_{{\mathcal {R}}n}\) :

Shaft speed coefficient for the MAE calculated over a range \({\mathcal {R}}\)

\(k_{{\mathcal {R}}p}\) :

Power coefficient for the MAE calculated over a range \({\mathcal {R}}\)

\(k_{{\mathcal {R}}q}\) :

Torque coefficient for the MAE calculated over a range \({\mathcal {R}}\)

\(k_{{\mathcal {R}}t}\) :

Thrust coefficient for the MAE calculated over a range \({\mathcal {R}}\)

\(K_{\mathrm{QCr}}\) :

Estimated torque coefficient for reverse thrusts used in the controller

\(K_{\mathrm{QC}}\) :

Estimated torque coefficient used in the controller

\(K_{\mathrm{Qr}}\) :

Torque coefficient for reverse thrusts

\(K_{\mathrm{TCr}}\) :

Estimated thrust coefficient for reverse thrusts used in the controller

\(K_{\mathrm{TC}}\) :

Estimated thrust coefficient for the controller

\(K_{\mathrm{Tr}}\) :

Thrust coefficient for reverse thrusts

\(\mathrm{MAE}_n\) :

Shaft speed sensitivity mean absolute error over the whole range of operation

\(\mathrm{MAE}_p\) :

Power sensitivity mean absolute error over the whole range of operation

\(\mathrm{MAE}_q\) :

Torque sensitivity mean absolute error over the whole range of operation

\(\mathrm{MAE}_t\) :

Thrust sensitivity mean absolute error over the whole range of operation

\(\mathrm{MAE}_x\) :

Mean absolute error calculated for variable x (shaft speed, torque, power or thrust)

\(\mathrm{MAE}_{{\mathcal {R}}}\) :

Mean absolute error for a given range \({\mathcal {R}}\)

\(\mathrm{MAE}_{\mathrm{hi}}\) :

Combined mean absolute error of shaft speed, torque, power and thrust for high thrusts

\(\mathrm{MAE}_{\mathrm{low}}\) :

Combined mean absolute error of shaft speed, torque, power and thrust for low thrusts

\(\mathrm{MAE}_{\mathrm{med}}\) :

Combined mean absolute error of shaft speed, torque, power and thrust for medium thrusts

N :

Number of points in the dataset

\(n_d\) :

Shaft speed reference before filter application

\(n_N\) :

Nominal shaft speed

\(n_r\) :

Shaft speed reference

\(N_{{\mathcal {R}}}\) :

Number of points in the dataset for a given range \({\mathcal {R}}\)

\(n_{s1}\) :

Shaft speed reference threshold for disabling the shaft speed integral error term

\(n_{s2}\) :

Shaft speed reference threshold for transition from torque to power controller

p :

Adjustable coefficient for \(\alpha \)

\(P_r\) :

Signed power reference

\(P_{\mathrm{max}}\) :

Power limitation

\(Q_c\) :

Input torque from the controller to the system (with torque and power limitation)

\(Q_j\) :

Output torque of controller j

\(Q_p\) :

Power controller output

\(Q_q\) :

Torque controller output

\(Q_r\) :

Torque reference

\(Q_s\) :

Coulomb friction

\(Q_s^*\) :

Thruster nominal Coulomb friction

\(Q_{c0}\) :

Torque signal before the application of torque and power limitation

\(Q_{\mathrm{ff}}\) :

Friction compensation output

\(Q_{\mathrm{if}}\) :

Inertia compensation output

\(Q_{\mathrm{iQP}}\) :

iQP controller torque output

\(Q_{i}\) :

Shaft speed integral error torque

\(Q_{\mathrm{max}}\) :

Torque limitation

\(Q_{\mathrm{res}}\) :

Resultant torque

\(Q_{\mathrm{sc}}\) :

Estimated Coulomb friction used in the controller

r :

Adjustable coefficient for \(\alpha \)

\(\mathrm{sn}_i\) :

Shaft speed sensitivity for controller i

\(\mathrm{sn}_n\) :

Shaft speed sensitivity for the shaft speed controller

\(\mathrm{sn}_p\) :

Shaft speed sensitivity for the power controller

\(\mathrm{sn}_q\) :

Shaft speed sensitivity for the torque controller

\(\mathrm{sn}_{\mathrm{iQ\hbox {-}P}}\) :

Shaft speed sensitivity for the iQ-P controller

\(\mathrm{sn}_{\mathrm{iQP}}\) :

Shaft speed sensitivity for the iQP controller

\(\mathrm{sp}_i\) :

Power sensitivity for controller i

\(\mathrm{sp}_n\) :

Power sensitivity for the shaft speed controller

\(\mathrm{sp}_p\) :

Power sensitivity for the power controller

\(\mathrm{sp}_q\) :

Power sensitivity for the torque controller

\(\mathrm{sp}_{\mathrm{iQ\hbox {-}P}}\) :

Power sensitivity for the iQ-P controller

\(\mathrm{sp}_{\mathrm{iQP}}\) :

Power sensitivity for the iQP controller

\(\mathrm{sq}_i\) :

Torque sensitivity for controller i

\(\mathrm{sq}_n\) :

Torque sensitivity for the shaft speed controller

\(\mathrm{sq}_p\) :

Torque sensitivity for the power controller

\(\mathrm{sq}_q\) :

Torque sensitivity for the torque controller

\(\mathrm{sq}_{\mathrm{iQ\hbox {-}P}}\) :

Torque sensitivity for the iQ-P controller

\(\mathrm{sq}_{\mathrm{iQP}}\) :

Torque sensitivity for the iQP controller

\(\mathrm{st}_i\) :

Thrust sensitivity for controller i

\(\mathrm{st}_n\) :

Thrust sensitivity for the shaft speed controller

\(\mathrm{st}_p\) :

Thrust sensitivity for the power controller

\(\mathrm{st}_q\) :

Thrust sensitivity for the torque controller

\(\mathrm{st}_{\mathrm{iQ\hbox {-}P}}\) :

Thrust sensitivity for the iQ-P controller

\(\mathrm{st}_{\mathrm{iQP}}\) :

Thrust sensitivity for the iQP controller

\(\mathrm{sx}_k(i)\) :

i-th point of shaft speed, torque, power or thrust (variable x) sensitivity for a controller k

\(T_d\) :

Thrust reference from the thrust allocation algorithm

\(T_i\) :

Integral time constant of the PI controller

\(T_r\) :

Filtered thrust reference

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Author notes

  1. Matheus C. Hidalgo

    Present address: Escola Politécnica - University of São Paulo, São Paulo, Brazil

Authors and Affiliations

  1. Department of Telecommunications and Control Engineering, Escola Politécnica - University of São Paulo, São Paulo, Brazil

    Matheus C. Hidalgo & Fuad Kassab Junior

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  2. Fuad Kassab Junior
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Correspondence to Matheus C. Hidalgo.

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Hidalgo, M.C., Kassab Junior, F. Steady-State Sensitivities of Marine Propulsion Control Techniques. J Control Autom Electr Syst 32, 543–562 (2021). https://doi.org/10.1007/s40313-021-00710-3

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