1. PARAMETER DEPENDENT LYAPUNOV FUNCTIONS FOR STABILITY OF LINEAR PARAMETER VARYING SYSTEMS Nedia Aouani, Salah Salhi, Germain Garcia, Mekki Ksouri Research Unit of System Analysis and Control ACS, National Engineering School of Tunis University of Toulouse, LAAS-CNRS

2. OUTLINE  Motivations  Problem formulation  New representation of the time derivative of the parameter  New LMI based conditions for stability analysis of LPV polytopic systems  Numerical example  Conclusion ICECS’10 Athens GreeceN. AOUANI2

3. MOTIVATIONS Over the last two decades, LPV systems has undergone a wealth of practical and theoretical developments [CAO&al, 2004], [Geromel&al, 2006], [Montagner&al, 2009]. All these works treat the problem of stability and establish conditions for analysis purposes. As to the uncertain parameters, they can be modeled under different structures : affine, polytopic or rational dependence. One difficulty remains how to represent the time derivative of the uncertainty in the case it is assumed to vary in a polytopic domain with bounded rates. Parameter Dependent Lyapunov Functions (PDLF) are ivestigated the last ten years [Peaucelle&al, 2000], and the LMIs became a powerful skill to deal with such problems. ICECS’10 Athens GreeceN. AOUANI3

4. MOTIVATIONS As to Parameter Dependent Lyapunov Functions, some specific forms have been investigated all along the littérature: the class of Polynomial PDLFs [Chesi, 2003, 2004, 2005, 2007], [Oliveira,2005], the class of rational ones [Scorletti, 1995], [Lu, 1996] and the class of affine ones [Feron, 1996], [Gahinet, 1996], [Peaucelle, 2001]. Main Idea: Use of PDLFs of particular forms that have been used for LTI systems by [Ebihara, 2005], for the case of LPV systems. Reduction of conservatism in the proposed conditions ICECS’10 Athens GreeceN. AOUANI4

5. ICECS’10 Athens Greece PROBLEM FORMULATION N. AOUANI5 Linear Parameter Varying system The system matrices The time varying parameter varies in a vertex such that The parameter’s rate of variation Objectives Assess robust stability of the system

6. NEW REPRESENTATION OF THE TIME DERIVATIVE OF THE PARAMETER Previous representations of the parameter’s rate of variation [Cao & al, 2004] [Geromel & al, 2006] [ Xie & al, 2005] Lemma1 ICECS’10 Athens GreeceN. AOUANI6

7. Assumptions: Stability condition: With: The time derivative of Lyapunov matrix is given by: Where and ICECS’10 Athens GreeceN. AOUANI7 NEW LMI BASED CONDITIONS FOR STABILITY ANALYSIS OF LPV POLYTOPIC SYSTEMS

8. The system (1) is asymptotically stable if there exist positive definite symmetric matrices, matrices, and such that the following LMI holds: ICECS’10 Athens GreeceN. AOUANI8 Proposition NEW LMI BASED CONDITIONS FOR STABILITY ANALYSIS OF LPV POLYTOPIC SYSTEMS

9. Theorem The system (1) is asymptotically stable if there exist positive definite symmetric matrices, matrices and such that the following LMI holds: And ICECS’10 Athens GreeceN. AOUANI9 Where are given by NEW LMI BASED CONDITIONS FOR STABILITY ANALYSIS OF LPV POLYTOPIC SYSTEMS

10. We consider the following LPV system [Geromel & al, 2006] Such that the matrices Ai are taken: The uncertain parameter The time derivative of the uncertain parameter is bounded such that ; Purpose: Delimit the region of the plane ( ,  ) with and Such that the global asymptotic stability is preserved. N. AOUANI10 NUMERICAL EXAMPLE ICECS’10 Athens Greece

11. NUMERICAL EXAMPLE ICECS’10 Athens GreeceN. AOUANI11

12. NEUMERICAL EXAMPLE If we consider the system (1) with N=3: And We numerically verify the feasibility of the point ICECS’10 Athens GreeceN. AOUANI12

13. CONCLUSION We have proposed in this paper a new stability condition formulated in terms of LMI constraints, for an LPV continuous system under polytopic uncertainty structure. Further analysis conditions can be deduced following the same ideas and increasing redundancy ICECS’10 Athens GreeceN. AOUANI13

14. THANK YOU FOR YOUR ATTENTION ICECS’10 Athens GreeceN. AOUANI14