Robust design of backstepping controllers for systems of linearhyperbolic PDEs

Abstract : Linear First-Order Hyperbolic Partial Differential Equations (LFOH PDEs) represent systems of conservation and balance law and are predominant in modeling of traffic flow, heat exchanger, open channel flow or multiphase flow. Different control approaches have been tackled for the stabilization or observation of such systems. Among them, the backstepping method consists to map the original system to a simpler system for which the control design is easier. The resulting controllers are explicit.In the first part of this thesis, we develop some general results in control theory. More precisely, we solve the problem of finite-time stabilization of a general class of LFOH PDEs using the backstepping methodology. The minimum stabilization time reachable may change depending on the number of available actuators. The corresponding boundary observers (crucial to envision industrial applications) are obtained through a dual approach. An important by-product of the proposed approach is to derive an explicit mapping from the space generated by the solutions of the considered LFOH PDEs to the space generated by the solutions of a general class of neutral systems with distributed delays. This mapping opens new prospects in terms of stability analysis for LFOH PDEs, extending the stability analysis methods developed for neutral systems.In the second part of the thesis, we prove the necessity of a change of strategy for robust control while considering industrial applications, for which the major limitation is known to be the robustness of the resulting control law to uncertainties in the parameters, delays in the loop, neglected dynamics or disturbances and noise acting on the system. In some situations, one may have to renounce to finite-time stabilization to ensure the existence of robustness margins. We propose some adjustments in the previously designed control laws by means of several degrees of freedom enabling trade-offs between performance and robustness. The robustness analysis is fulfilled using the explicit mapping between LFOH PDEs and neutral systems previously introduced.
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Jean Auriol. Robust design of backstepping controllers for systems of linearhyperbolic PDEs. Optimization and Control [math.OC]. PSL Research University, 2018. English. ⟨NNT : 2018PSLEM011⟩. ⟨tel-01904844⟩

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