The workshop


Goal


The aim of this workshop is to give an introductory overview and some basic concepts on recent control strategies developed for distributed parameters systems. This workshop focusses on three main control design strategies: flatness-based motion planning, tracking and observation, Backstepping and energy shaping. The theoretical development will be illustrated on a set of physical examples such as transmission lines, beam equations, shallow water equations, Burgers equations or reaction-transport diffusion problems including chemical processes, diffusion equation, population dynamics, etc.


Synopsis


Open physical distributed parameter systems governed by partial differential equations (PDEs) are more and more often encountered in modern engineering applications. It is the case for example for applications involving fluids, elasticity, plasmas, and other spatially distributed phenomena modeled by PDEs. The system theoretical formulation of such phenomena, their analysis and control are of high theoretical and practical interest. This interest, even for industrial applications, has been strengthened by the recent computational and technological progresses. From the control point of view, many results have been proposed the last 20 years for the asymptotic or exponential stabilization of PDE systems. Even if the literature on this topic is quite prolific, only few contributions attempt to deal with achievable performances and system oriented control design. The aim of this workshop is to present some of these control design techniques and to highlight recent trends as well as open research questions.

    The first one is the flatness-based motion planning. It addresses the determination of the input trajectories so that the system states or outputs, respectively, follow certain prescribed paths. For solving the motion planning problem for distributed parameter systems differential flatness has in the last years evolved into a design systematics that is applicable to rather large system classes, see, e.g., the literature review in [Meurer(2013)]. Depending on the type of PDE, the dimension and shape of the spatial domain and the location of the control inputs the used techniques utilize, e.g., formal power series, spectral decomposition of the operator or formal integration. In addition to theoretical results the available experimental results obtained for flexible structures or thermal processes clearly indicate the
applicability of flatness-based motion planning for PDEs.

    The second control technique that is presented in this workshop is backstepping. The aim of backstepping is to convert PDE systems into desirable target systems using explicit transformations and feedback laws, which both employ Volterra-type integrals in spatial variables, with kernels resulting from solving linear PDEs of Goursat type on triangular domains. This strategy establishes a general methodology for stabilizing linear PDEs [Krstic(2008)], but also for adaptive control of PDEs [Smyshlyaev(2010)] with unknown parameters and for nonlinear PDEs.

    The third and last control design technique proposed in this workshop is energy shaping using the port Hamiltonian framework. This framework, initially proposed for non linear finite dimensional systems in [Maschke(1992)] and extended to distributed parameter systems in [van der Schaft(2002), Le Gorrec(2005)] is particularly suited for the modeling and control of open multiphysical or/and modular systems. Indeed this representation takes advantage of the intrinsic geometric structure arising from the internal power exchanges, the dissipation phenomena and the power exchanges of the system with his surrounding. Many results on existence of solution, stability and control of linear distributed parameter systems have been proposed in the last ten years (see [Jacob(2012)] and references therein). But it is particularly well adapted for control purpose as the energy is at the center of the model and can be easily used as Lyapunov function. The stabilization can be obtained by direct passivation of the system. But the control strategy can be drastically improved by shaping the energy, allowing to deal with closed loop performances even in the boundary control case. The application of the considered control concepts for PDE systems will be illustrated for different physical examples both providing simulation as well as experimental results.


Intended audience


The workshop is open to all researchers, industrials and aspiring scholars in any area of mathematics or engineering who wish to work on control of systems governed by PDEs of any kind (whether theoretical or applied).


Prerequisites


Basic knowledge on automatic control and PDEs.



Workshop material


  1. Presentation
  2. An introduction on Control of Distributed Parameter Systems
  3. Flatness based motion planning and tracking
  4. Introduction to Backstepping control of PDEs
  5. Energy shaping of boundary controlled Distributed Parameter Systems