dynamic positioning control using hamilton-jacobi techniques

Qian ZhongDavid Fernández

ME C236 Control and Optimization of Distributed Parameters Systems

Berkeley, May 08th 2014

Dynamic Positioning Control Using Hamilton-Jacobi Techniques

2Qian Zhong & David Fernández 8th May, 2014

Outline1. Background

2. Method

Hamilton-Jacobi-Isaacs Based Optimal Control

3. Result and Analysis

Kinematic Model (2 Dimensions)

Time-dependent Kinematic Model (3 Dimensions)

Dynamic Model (5 Dimensions)




2. Simulation Method

Hamilton-Jacobi-Isaacs Based Optimal Control







Semisubmersible Platform

Floating offshore structure for oil exploration

Fixed position required

Motion due to Environmental Forces

Source: worldmaritimenews.com



Dynamic Positioning (DP)

𝑀 𝑥 = 𝐹𝑒𝑛𝑣. + 𝐹𝑐𝑜𝑛𝑡𝑟.

Large Fuel Consumption

Applying control force: Thruster



Objective: Reduce Energy Consumption

Less time for thruster in action

1. Minimize the time to reach the required position

Minimum-time-to-reach problem

2. Inactivate the thruster in “safe region”

Reachability problem




2. Method

Hamilton-Jacobi-Isaacs (HJI) Based Optimal Control







HJI Based Optimal Control

Model 𝑥 = 𝑓(𝑥, 𝑎, 𝑏)

Cost Function𝑉 𝑥0 = inf

𝑎 ⋅ ∈𝐴sup𝑏 ⋅ ∈𝐵

𝑡∗(𝑥0, 𝑎 ⋅ , 𝑏 ⋅ )

HJI Equation

𝐷𝑡𝜙 +min 0, 𝐻 𝑥, 𝐷𝑥𝜙 𝑥, 𝑡 = 0

Optimal Input, 𝑎∗ and 𝑏∗, and Trajectory




2. Method








Kinematic Model (2D)

𝑥 = 𝑉𝑐 𝑥 + 𝑎 𝑥, 𝑡 + 𝑏 𝑥, 𝑡

𝑥: Position

𝑉𝑐: Current velocity

𝑎: Thruster control input, | 𝑎| ≤ 5𝑚/𝑠

𝑏: Disturbance input, |𝑏| ≤ 1𝑚/𝑠




Velocity Field (Defined)

(a) Magnitude of Velocity Field (b) Quiver Plot with 𝑑𝛼 = 0.1




Velocity Field (Defined)

(a) Magnitude of Velocity Field (b) Quiver Plot with 𝑑𝛼 = 0.1

𝑑𝛼Vortex



Figure: Minimum time to reach the center (0, 0) in defined area

Minimum Time to Reach the Center



Figure: The trajectory of the platform from (-3, -3) to the Center

Trajectory



(a) Quiver Plot with 𝑑𝛼 =𝜋

2(b) Minimum time to reach the Center

Reachability



(a) Quiver Plot with 𝑑𝛼 =𝜋

2(b) Minimum time to reach the Center

Reachability

Not reachable

Reachable




2. Method








Time-dependent Kinematic Model (3D)

𝑑

𝑑𝑡

𝑥𝑦𝑡

=𝑈𝑐 𝑥, 𝑦, 𝑡 + 𝑎𝑥 𝑥, 𝑦, 𝑡 + 𝑏𝑥(𝑥, 𝑦, 𝑡)

𝑉𝑐 𝑥, 𝑦, 𝑡 + 𝑎𝑦 𝑥, 𝑦, 𝑡 + 𝑏𝑦 𝑥, 𝑦, 𝑡

−1



Velocity Field (Changing with Time)

Figure: Illustration Quiver plot, 𝑑𝛼 is changing from 0 to 𝜋

2

𝑑𝛼



Minimum Time to Reach



(a) Trajectory with initial position at (4, -4):Reachable

(b) Trajectory with initial position at (-6, -6):Not reachable

Trajectory



Dynamic Positioning (DP)



CONCEPT



Goal

Three main areas :

State observer

Controller

Thrust allocation

We simplify it by generatingdirectly the environmentalconditions.

Dynamic Model Inertial forces Added mass Wind loads Current loads Wave loads

REQUIREMENTS:

Extend the use of the Hamilton-Jacobi equation to calculate theoptimum thrust commands that minimize the operating time, and

consequently the fuel consumption.



Dynamics

Wind

Current

Waves

Less straightforward



DynamicsAssumptions Only drift forces are

compensated Only planar movements are

controlled: No restoring forces Low speed: Wave resistance is

not dominant

Quadratic Transfer Functions



Hamilton - JacobiMinimum Time To Reach (MTTR) problem



Practical application

Semisubmersible platform to install offshore wind mills

Displacement = 55800 ton

Length = 100 m

Height = 55 m

Depth = 30 m

Thrusters = 380 ton



Environmental conditions



Time domain simulation



Conclusions



Conclusions We have successfully applied the Hamilton-Jacobi formulation

to obtain the optimum control for reachable set and minimum time to reach target of floating bodies.

Inclusion of dynamic model in the simulations, although with a high computational cost.

The optimum control for large inertial systems shows relevant oscillations, which suggests that this method is not entirely effective close to the target.

A set of new functions to predict the loads on the floating bodies has been added to the toolbox.

dynamic positioning control using hamilton-jacobi techniques

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