no. 4
An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems
Lamare, Pierre-Olivier1 ; Girard, Antoine2 ; Prieur, Christophe3
1 BIOCORE project-team, Inria Sophia Antipolis – Méditerranée, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis cedex, France.
2 Laboratoire des signaux et systèmes (L2S), CNRS, Centrale Supélec, Université Paris-Sud, Université Paris-Saclay, 3, rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France.
3 Department of Automatic Control, Gipsa-lab, 11 rue des Mathématiques, BP 46, 38402 Saint Martin d’Hères cedex, France.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1236-1263.

In this paper, we consider the problems of stability analysis and control synthesis for first-order hyperbolic linear Partial Differential Equations (PDEs) over a bounded interval with spatially varying coefficients. We propose Linear Matrix Inequalities (LMI) conditions for the stability and for the design of boundary and distributed control for the system. These conditions involve an infinite number of LMI to solve. Hence, we show how to overapproximate these constraints using polytopic embeddings to reduce the problem to a finite number of LMI. We show the effectiveness of the overapproximation with several examples and with the Saint-Venant equations with friction.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2016038
Classification : 49J20, 37N35, 93B52
Mots clés : Hyperbolic PDE, Lyapunov method, LMI
Lamare, Pierre-Olivier 1 ; Girard, Antoine 2 ; Prieur, Christophe 3

1 BIOCORE project-team, Inria Sophia Antipolis – Méditerranée, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis cedex, France.
2 Laboratoire des signaux et systèmes (L2S), CNRS, Centrale Supélec, Université Paris-Sud, Université Paris-Saclay, 3, rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France.
3 Department of Automatic Control, Gipsa-lab, 11 rue des Mathématiques, BP 46, 38402 Saint Martin d’Hères cedex, France. 
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     title = {An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1236--1263},
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Lamare, Pierre-Olivier; Girard, Antoine; Prieur, Christophe. An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1236-1263. doi : 10.1051/cocv/2016038. http://archive.numdam.org/articles/10.1051/cocv/2016038/

G. Bastin and J.-M. Coron, On boundary feedback stabilization of non-uniform linear hyperbolic systems over a bounded interval. Syst. Control Lett. 60 (2011) 900–906. | DOI | Zbl

G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems. PNLDE Subseries in Control. Springer (2016).

S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Society for industrial and applied mathematics, Philadelphia (1994). | Zbl

F. Castillo, E. Witrant, C. Prieur and L. Dugard, Boundary observers for linear and quasi-linear hyperbolic systems with application to flow control. Automatica 49 (2013) 3180–3188. | DOI | Zbl

F. Castillo, E. Witrant, C. Prieur, V. Talon and L. Dugard, Fresh air fraction control in engines using dynamic boundary stabilization of LPV hyperbolic systems. IEEE Trans. Control Syst. Technol. 23 (2015) 963–974. | DOI

J.-M. Coron, B. D’Andréa Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans. Automat. Control 52 (2007) 2–11. | DOI | Zbl

J.-M. Coron, G. Bastin and B. D’Andréa Novel, Dissipative boundary conditions for one dimensional nonlinear hyperbolic systems. SIAM J. Control Optim. 47 (2008) 1460–1498. | DOI | Zbl

J. Daafouz, M. Tucsnak and J. Valein, Nonlinear control of a coupled PDE/ODE system modeling a switched power converter with a transmission line. Syst. Control Lett. 70 (2014) 92–99. | DOI | Zbl

F. Di Meglio, R. Vazquez and M. Krstic, Stabilization of a system of n+1 coupled first-order hyperbolic linear PDEs with a single boundary input. IEEE Trans. Automat. Control 58 (2013) 3097–3111. | DOI | Zbl

A. Diagne, G. Bastin and J.-M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws. Automatica 48 (2012) 109–114. | DOI | Zbl

E. Fridman and Y. Orlov, An LMI approach to H ∞ boundary control of semilinear parabolic and hyperbolic systems. Automatica 45 (2009) 2060–2066. | DOI | Zbl

I. Karafyllis and M. Krstic, On the relation of delay equations to first-order hyperbolic partial differential equations. ESAIM: COCV 20 (2014) 894–923. | Numdam | Zbl

I. Karafyllis, M. Malisoff and M. Krstic, Ergodic theorem for stabilization of a hyperbolic PDE inspired by age-structured chemostat. Preprint . | arXiv

J. Löfberg, YALMIP: A toolbox for modeling and optimization in MATLAB. In IEEE International Symposium on Computer Aided Control Systems Design (2004).

C. Prieur, A. Girard and E. Witrant, Stability of switched linear hyperbolic systems by Lyapunov techniques. IEEE Trans. Automat. Control 59 (2014) 2196–2202. | DOI | Zbl

L.F. Shampine, Solving hyperbolic PDEs in MATLAB. Appl. Numer. Anal. Comput. Math. 2 (2005) 346–358. | DOI | Zbl

C.Z. Xu and G. Sallet, Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems. ESAIM: COCV 7 (2002) 421–442. | Numdam | Zbl

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