LMI Methods for Oceanic Control Systems

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LMI Methods for Oceanic Control Systems

• Jean-Pierre Folcher
• Laboratoire Signaux et Systèmes de Sophia
  Antipolis, CNRS/UNSA
• Worshop SUMARE, Sophia Antipolis, December 18th,
  2001

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Outline

• Introduction to LMI Methods
• Linear Matrix Inequality (LMI)
• Semidefinite Programming (SDP)
• Linear-Fractional Representation
• LFR construction
• Uncertain linear constraint
• Oceanic Systems Cases Study
• LMI Control methods for AUV with saturating
  actuators

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Linear Matrix Inequality (LMI)

• decision vector,
• given matrices of
  
• LMI means that every
  eigenvalue of is positive.

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Semidefinite Programming (SDP)

• where is a raw vector.

• Important features
• non linear, non differentiable, convex problem
• amenable to efficient (polynomial time)
  interior points methods
• many applications

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Linear-Fractional Representation
Let be a matrix-valued rational function
of well-defined for
Fact there exists matrices
and integers such that
with identity matrix of order k.
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LFR construction
Addition, multiplication, inversion are
possible. Example the product, if then the
product has LFR

with
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Uncertain linear constraint
Consider a constraint between vectors
where and is a
(matrix-valued) rational fonction.
LFR model
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Outline

• Introduction to LMI Methods
• Oceanic Cases Study
• Underwater vehicle dynamic analysis
• Robust model-based fault diagnosis
• Obstacle avoidance
• LMI Control method for AUV with saturating
  actuators

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Underwater vehicule dynamics analysis
Classical analysis and control methods based on
linear system theory. A crude assumption
vehicule body motion has to be precisely
described by a linearized model. For high
manoeuvring vehicle trajectories, dynamic models
are highly non linear Analysis methodologies for
more complex systems (uncertain, non linear) are
required.
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Uncertain systems
Uncertainty, for a given signal input

• only an output a model
• a family of output possible, a family
  of models.
• Models
• Linear time invariant systems,
• Linear Parameter Varying (LPV) systems.

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LPV systems
LTI system connected to uncertain matrice
Ex spring-mass system
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LPV closed-form representation
elim. leads to
with and
which express that respect a
dissipative property.
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Stability analysis for LPV systems
Consider the system and such
that for all dissipative.

• Lyapunov function,
  ensuring quadratic stability
• Invariant ellipsoïd
• Lyapunov index

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Robust model-based fault diagnosis for underwater
vehicle
Crucial function of AUV control systems early
detection of malfunctions, faults.
Powerfull methods use the knowledge of the
vehicle dynamics.
Under stringent operating conditions, the plant
may exibit parameter variations and non
linearities, may be described by LPV systems. .
LMI methods are usefull to design robust observer
i.e. the residual vector generator.
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Bank of residual generatorsfor fault diagnosis
Residual gen. 1

AUV dynamics
Controller
Residual gen. 2
-
Residual gen. 3
Design problem find the observation gain L can
be expessed in terms of LMI constraints.
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Obstacle avoidance system

• Efficiently avoiding strategy implies
• quick observation of an extended area in the
  vicinity of the vehicle,
• to choose an avoiding trajectory (high
  manoeuvering phase).

A crucial question find a control ensuring
secure trajectories for the plant in presence of
non linearities and uncertainties.
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An LMI formulation
Uncertain discrete time system where is
the state vector and an uncertain matrice.
Control objectives find such that
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Synthesis problem cast as an LMI optimization
problem (El Ghaoui 1999)
Navigation limits allowing to define System
ouputs constraints