dynamic positioning control using hamilton-jacobi techniques
TRANSCRIPT
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)
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Outline1. Background
2. Simulation 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)
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Semisubmersible Platform
Floating offshore structure for oil exploration
Fixed position required
Motion due to Environmental Forces
Source: worldmaritimenews.com
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Dynamic Positioning (DP)
𝑀 𝑥 = 𝐹𝑒𝑛𝑣. + 𝐹𝑐𝑜𝑛𝑡𝑟.
Large Fuel Consumption
Applying control force: Thruster
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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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
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Outline1. Background
2. Method
Hamilton-Jacobi-Isaacs (HJI) Based Optimal Control
3. Result and Analysis
Kinematic Model (2 Dimensions)
Time-dependent Kinematic Model (3 Dimensions)
Dynamic Model (5 Dimensions)
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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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
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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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
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Outline1. Background
2. Method
Hamilton-Jacobi-Isaacs (HJI) Based Optimal Control
3. Result and Analysis
Kinematic Model (2 Dimensions)
Time-dependent Kinematic Model (3 Dimensions)
Dynamic Model (5 Dimensions)
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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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
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Kinematic Model (2D)
Velocity Field (Defined)
(a) Magnitude of Velocity Field (b) Quiver Plot with 𝑑𝛼 = 0.1
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Kinematic Model (2D)
Velocity Field (Defined)
(a) Magnitude of Velocity Field (b) Quiver Plot with 𝑑𝛼 = 0.1
𝑑𝛼Vortex
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Figure: Minimum time to reach the center (0, 0) in defined area
Minimum Time to Reach the Center
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Figure: The trajectory of the platform from (-3, -3) to the Center
Trajectory
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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(a) Quiver Plot with 𝑑𝛼 =𝜋
2(b) Minimum time to reach the Center
Reachability
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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(a) Quiver Plot with 𝑑𝛼 =𝜋
2(b) Minimum time to reach the Center
Reachability
Not reachable
Reachable
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Outline1. Background
2. Method
Hamilton-Jacobi-Isaacs (HJI) Based Optimal Control
3. Result and Analysis
Kinematic Model (2 Dimensions)
Time-dependent Kinematic Model (3 Dimensions)
Dynamic Model (5 Dimensions)
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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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
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Velocity Field (Changing with Time)
Figure: Illustration Quiver plot, 𝑑𝛼 is changing from 0 to 𝜋
2
𝑑𝛼
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Minimum Time to Reach
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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(a) Trajectory with initial position at (4, -4):Reachable
(b) Trajectory with initial position at (-6, -6):Not reachable
Trajectory
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Dynamic Positioning (DP)
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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CONCEPT
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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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.
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Dynamics
Wind
Current
Waves
Less straightforward
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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DynamicsAssumptions Only drift forces are
compensated Only planar movements are
controlled: No restoring forces Low speed: Wave resistance is
not dominant
Quadratic Transfer Functions
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Hamilton - JacobiMinimum Time To Reach (MTTR) problem
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Practical application
Semisubmersible platform to install offshore wind mills
Displacement = 55800 ton
Length = 100 m
Height = 55 m
Depth = 30 m
Thrusters = 380 ton
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Environmental conditions
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Time domain simulation
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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Conclusions
Dynamic Positioning Control Using Hamilton-Jacobi Techniques
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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.