ONLINE NONLINEAR OPTIMAL CONTROL FOR SHIP POSITIONING AND TRACKING
Masanori Hamamatsu (Kawasaki Heavy Industries, Ltd., Japan)
Yukinobu Kohno (Kawasaki Heavy Industries, Ltd., Japan)
Hiroshi Ohnishi (Kawasaki Heavy Industries, Ltd., Japan)
Yasuo Saito (Kawasaki Ship Building Corp., Japan)
Abstract: In the dynamic positioning control and thrust allocation system of the vessels, it is difficult to obtain an optimal solution in real time because of the inherent non-linearity, and developments of a reliable real-time calculation method for the optimal control have been waited for. The control system that we have developed realizes a real-time optimal control. This paper presents examples of application of the real-time nonlinear Receding Horizon (RH) control for route-tracking and real-time algorithm for an optimal thrust allocation for redundant actuators. The effectiveness of the method applied to the systems is verified by computer simulation.
In recent years, the dynamic positioning system (DPS), which controls a ship's position with high accurate positioning sensors such as the Differential GPS, has been used not only for keeping a semi-submersible drilling rig in a fixed position but also for many mission as an indispensable device, for example, precise ocean investigations by marine research vessels, marine work by cable or pipe laying vessels, or fire fighting by fireboats.
As for feedback control systems, simple feedback control laws such as the PID are used in the present DPS. But these classical methods still stand improvement when to control a normal ship whose shape is designed to minimize the hydrodynamic drag toward fore direction, because her characteristics of the motion to the side or turning direction change largely at a low speed. So usually they need excessive amount of thruster powers for compensation of those uncertain characteristics. On the other hands, to satisfy the increasing demand for the controller, that is the energy saving function, more accurate positioning performance, complicated route tracking or automatic berthing function in harbor where various constraint conditions must be considered, we need nonlinear control methods which can directly deal with complex characteristics of the ship's motions.
In addition, thrust allocation problem that distributes the thrust commands of a ship from the feedback controller to each thruster or rudder is also an important. The thrust allocation algorithm is very difficult because there are many solutions how to distribute each command to redundant actuators, and the performance of the algorithm greatly affects on the running cost of the DPS.
To cope with these problems, we developed a nonlinear ship maneuvering controller with the RH control method by including nonlinear ship's hull models in it. Also in the thrust allocation problem, we developed an optimal controller that minimized pre-defined evaluation function by formulating optimization problem of redundant actuators into the RH control problem which could obtain optimal solutions on line.
2. OPTIMUM THRUST ALLOCATION
We focus on the optimal thrust allocation problem of multiple azimuth thrusters. Assuming that the ship has n thrusters and i-th thrust force is generated at the position (xsi,ysi) in the ship-fixed axis originated in the ship center ( bow direction xs, portside direction ys ). The generated thrust is Ti and the turning angle from the xs axis is θi, the thrust X, Y and N are shown in (1)〜(3).
Fig.1 Arrangement of actuators
The vector of state variables of the thrusters is defined as x(t)=[T1,T2,・・・, Tn, θ1,θ2,・・・, θn]T, and the vector of manipulated variables that is to be calculated on line is u(t) = dx/dt. The evaluation function J is defined as the following:
where ψ expresses the penalties of the state variables in the terminal time t + T, L expresses the penalty of the state and manipulated variables in [t,t + T]. Also, xri is the i-th desired value of the state variables, Xr, Yr, Nr is the thrust command to each direction, they are set by the upper level feedback controllers, sfi is the weighting coefficient acting on the state variables. SWX,Y,N is the weighting coefficient acting on the deviation of the thrust. We use the C/GMRES method for the online calculation of this optimization problem.