Laboratory: L2S (Laboratoire des Signaux et Systèmes), UMR 8506 Université Paris–Saclay, CNRS,

CentraleSupélec, Laboratoire des Signaux et Systèmes, 91190, Gif–sur–Yvette, France.

Supervisor : Jean AURIOL (jean.auriol@centralesupelec.fr)

DESCRIPTION OF THE INTERNSHIP:

General context and positioning

Distributed parameter systems allow a realistic representation of industrial processes involving physical quantities evolving in time and space. In particular, systems of hyperbolic partial differential equations (H–PDE) provide a mathematical description of physical processes involving transport phenomena (transport of matter, sound waves, information). Such equations allow, for example, to model wave propagation phenomena, electrical transmission lines [1], hydraulic channels [2], drilling devices [3], traffic flows [4], [5], intelligent materials and structures, multi–scale and multiphysics systems. These systems are at the origin of complex control engineering problems subject to strong constraints in terms of environmental safety and economic feasibility. This is the case, for example, for the problem of leak detection in gas pipes. Such leaks can cause significant environmental damage and financial losses. It is, therefore, crucial to detect them early. Controlling and estimating such hyperbolic systems in real–time is a challenging task due to their distributed nature (time and space dependency), the complex nature of the underlying propagation phenomena, and due to the physical/economic impossibility of placing sensors and actuators along the entire spatial domain. The operational requirements and the high mathematical complexity of these systems explain why traditional control methods (usually using spatial discretization) have not been satisfactory when implemented and why new approaches had to be considered [7, 8].

Many approaches have been proposed in the literature to allow the synthesis of stabilizing control laws for infinite dimensional systems. Among them, we can mention the platitude controllers [9], the optimal control methods [10], the Lyapunov type methods [5], or the backstepping method [11]. The latter consists in finding an invertible change of variable (in general, an integral transformation) which transforms the original system into a « target system » with a simpler structure, for which the analysis is thus made easier. This technique has allowed the explicit synthesis of feedback control laws in many configurations. Extensions to the case of interconnected systems (network type), to underactuated systems, or to the case of systems with delays have been the subject of recent works [12,13,14].

However, the backstepping method suffers from several practical limitations. The most important one is that there is no systematic method for choosing the target system. This target system must be both reachable (i.e., there must be an invertible transformation from it to the original system) and simplify the stability analysis. This question of the choice of the target system is all the more crucial as the structure of the original system is complex. Moreover, as the choice of the target system has direct consequences on the design of the control law; the structure of the latter directly influences the behavior of the system in closed–loop (both in transient and steady state), as well as its overall performance. Therefore, even if the original system is stabilized, the control law may be unsuitable for practical implementation, as it does not satisfy industrial criteria (e.g. robustness margins, convergence rate, input saturation, sensitivity). These issues have been mostly neglected in the literature. Indeed, the different research works focus, most of the time, only on the stabilization problem. There are also other limitations to the backstepping method (related to neglected dynamics or model reduction for implementation on real systems), but these are outside the scope of this project.

Scientific objectives

Currently, the backstepping method does not take into account the natural physical properties of the original system (such as dissipativity, passivity, or reversibility) when choosing the target system. This can lead to using the controller to compensate for naturally stabilizing terms (which induces a higher control effort fordegraded performances). In this context, combining the backstepping method with a Port Hamiltonian Systems (PHS) approach [15] can be particularly useful. The PHSs theory allows a structured representation of the physical phenomena involved in the system. It is an energy–based approach that is both multiphysical and modular (it extends perfectly to systems with a graph structure). Thus, the SHPs approach would provide a physical framework for applying the backstepping method. This would lead to a systematic approach to the choice of the target system and the introduction of « physical » degrees of freedom (such as the dissipation rate, for example) with a clear energetic interpretation, thus making it easier to respect the application performance constraints. Encouraging preliminary results have been obtained when applying such an approach to simple systems such as the wave equation or the Timoshenko beam [16, 17, 18]. In this internship, we would like to go beyond these results in order to generalize such an approach to higher dimensional systems or to networks of interconnected systems.

Thus, the main objective of this internship is to propose a structural representation of the physical phenomena involved in the dynamics of linear hyperbolic systems using Port–Hamiltonian Systems. Such a representation will facilitate the stability analysis and will allow the definition of a class of target systems for the backstepping method that are both attainable and easily parameterized. This will lead to the introduction of degrees of freedom in the design of the control law allowing trade–offs between various performance indices. Such an approach will also allow the characterization of the stability/robustness margins in terms of physical parameters.

– Required skills:

This thesis topic mainly requires good skills in control theory and mathematics (Grandes Ecoles or Master in mathematics/control). Very good results in the engineering curriculum as well as expertise in the topics related to partial differential equations will constitute strengths to the proposed subject. The candidate should also become familiar with Matlab (numerical methods, simulations) or Python.

–Application : Send

• CV

• Master’s and/or engineering studies evaluations

• A letter of recommendation from the master manager

• The coordinates of two referees

–Contact: Send your application by email to

– Jean AURIOL (jean.auriol@centralesupelec.fr)

References :

[1] T. Li, G. Ledwich, Y. Mishra, J. Chow, A. Vahidnia, “Wave aspect of power system transient stability–part ii: Control

implications”, IEEE Transactions on Power Systems, vol. 32, pp. 2501–2508, 2016.

[2] J. de Halleux, C. Prieur, J.–M. Coron, B. d’Andréa Novel, G. Bastin, “Boundary feedback control in networks of open

channels”, Automatica, vol. 39, pp. 1365–1376, 2003.

[3] U. J. F. Aarsnes, N. van de Wouw, “Axial and torsional self–excited vibrations of a distributed drill–string, Journal of Sound

and Vibration, vol. 444, pp. 127–151, 2019.

[4] M. Burkhardt, H. Yu, M. Krstic, “Stop–and–go suppression in two–class congested traffic”, Automatica, 2021.

[5] G. Bastin, J.–M. Coron, Stability and boundary stabilization of 1–D hyperbolic systems, Birkhaüser, 2016.

[6] O. M. Aamo. Leak detection, size estimation and localization in pipe flows. IEEE TAC, 61(1):246–251, 2015.

[7] J. Auriol, F. Di Meglio, “Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs”,

Automatica, vol. 115, 2020.

[8] R.F Curtain and H. Zwart, An introduction to infinite–dimensional linear systems theory, Springer, 2012.

[9] T. Meurer, Control of higher–dimensional PDEs: Flatness and backstepping designs, 2012.

[10] J. L. Lions, Optimal control of systems governed by partial differential equations, Springer, 1971.

[11] M. Krstic, A. Smyshlyaev, Boundary control of PDEs: a course on backstepping design, SIAM, 2008.

[12] J. Redaud, J. Auriol, S. I. Niculescu, “Stabilizing output–feedback control law for hyperbolic systems using a Fredholm

transformation”. IEEE Transactions on Automatic Control, 2022

[13] J. Auriol, D. Bresch–Pietri, “Robust state–feedback stabilization of an underactuated network of interconnected n+ m

hyperbolic PDE systems”, Automatica, 2022

[14] J. Auriol, S. Kong D. Bresch–Pietri, “Explicit Prediction–Based Control for Linear Difference Equations with Distributed

Delays”, IEEE Control Systems Letters, 6, 2864–2869

[15] B. Jacob and H. Zwart, Linear port–Hamiltonian systems on infinite–dimensional spaces, Springer, 2012.