1. LMI Methods for Oceanic Control Systems Jean-Pierre Folcher Laboratoire Signaux et Systèmes de Sophia Antipolis, CNRS/UNSA Worshop SUMARE, Sophia Antipolis, December 18 th, 2001

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

3. Linear Matrix Inequality (LMI) – decision vector, – given matrices of –LMI means that every eigenvalue of is positive.

4. Semidefinite Programming (SDP) where is a raw vector. minimize such that Important features – non linear, non differentiable, convex problem – amenable to efficient (polynomial time) interior points methods –many applications

5. Fact: there exists matrices and integers such that with identity matrix of order k. Linear-Fractional Representation Let be a matrix-valued rational function of well-defined for

6. Addition, multiplication, inversion are possible. Example: the product, if then the product has LFR with LFR construction

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

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

9. 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.

10. Uncertain systems Uncertainty, for a given signal input system –only an output a model ; –a family of output possible, a family of models. Models –Linear time invariant systems, –Linear Parameter Varying (LPV) systems.

11. LPV systems LTI system connected to uncertain matrice Ex: spring-mass system G pq z  w

12. LPV closed-form representation elim. leads to with and which express that respect a dissipative property.

13. Stability analysis for LPV systems Consider the system and such that for all dissipative. – Lyapunov function, ensuring quadratic stability ; –Invariant ellipsoïd –Lyapunov index

14. 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.

15. Bank of residual generators for fault diagnosis + - AUV dynamicsController Residual gen. #2 Residual gen. #3 Residual gen. #1 Design problem : find the observation gain L can be expessed in terms of LMI constraints.

16. Obstacle avoidance system Efficiently avoiding strategy implies –quick observation of an extended area in the vicinity of the vehicle, high manoeuvering phase) –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.

17. An LMI formulation Uncertain discrete time system where is the state vector and an uncertain matrice. Control objectives: find such that

18. Navigation limits allowing to define System ouputs constraints Synthesis problem cast as an LMI optimization problem (El Ghaoui 1999)