Name
Title | ||
|---|---|---|
A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws |
Abstract | ||
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We present a strict Lyapunov function for hyper- bolic systems of conservation laws that can be diagonalised with Riemann invariants. The time derivative of this Lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions. It is shown that the derived boundary control allows to guarantee the local convergence of the state towards a desired set point. Furthermore, the control can be implemented as a feedback of the state only measured at the boundaries. The control design method is illustrated with an hydraulic application, namely the level and flow regulation in an horizontal open channel. exhibit a strict Lyapunov function which is an extension of the entropy and is stated in terms of Riemann invariants but whose time derivative can be made strictly negative definite by an appropriate choice of the boundary controls. This function is related to a Lyapunov function used in (8) for the stabilization of the Euler equation of incompressible fluids. It is also similar to the Lyapunov function used in (9) to analyse the stability of a general class of linear symmetric hyperbolic systems. Our contribution in this paper is to show how this kind of Lyapunov function can be extended in order to analyse the stability of nonlinear hyperbolic systems of conservation laws. For this class of systems, we give a theorem which shows that the boundary control allows to prove the local convergence (in H2(0,L)-norm) of the system trajectories towards a desired set point. Furthermore, the control can be implemented as a feedback of the state only measured at the boundaries. The considered class of conservative systems has a wide range of potential engineering applications, including for in- stance electrical transmission lines (10), gas flow pipelines (11)-(12), road traffic models (13) or heat exchangers (9). In this paper, the control design method is illustrated with an hydraulic application : the regulation of the level and the flow in an horizontal reach of an open channel. For the sake of simplicity, our presentation is limited to second order systems (i.e. systems of two scalar conservation laws). But, as we indicate in the conclusions, the method can be easily extended to higher order systems provided they can be diagonalised with Riemann invariants. |
Year | DOI | Venue |
|---|---|---|
2007 | 10.1109/TAC.2006.887903 | IEEE Transactions on Automatic Control |
Keywords | DocType | Volume |
Lyapunov method,Control systems,State feedback,Control design,Partial differential equations,Asymptotic stability,Entropy,Stability analysis,Boundary conditions,Steady-state | Journal | 52 |
Issue | ISSN | Citations |
1 | 0018-9286 | 120 |
PageRank | References | Authors |
12.21 | 5 | 3 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Jean-Michel Coron | 1 | 663 | 133.36 |
| Brigitte d'Andrea-Novel | 2 | 136 | 16.65 |
| Georges Bastin | 3 | 1039 | 177.30 |