CONTROL OF SYSTEMS ON SPATIAL DOMAINS WITH MOVING BOUNDARIES: 3D PRINTING AND TRAFFIC

Miroslav Krstić

DOI Number
https://doi.org/10.2298/FUEE1904503K
First page
503
Last page
512

Abstract


Until roughly the year 2000, control algorithms (of the kind that can be physically implemented and provided guarantees of stability and performance) were mostly available only for systems modeled by ordinary differential equations. In other words, while controllers were available for finite-dimensional systems, such as robotic manipulators of vehicles, they were not available for systems like fluid flows. With the emergence of the “backstepping” approach, it became possible to design control laws for systems modeled by partial differential equations (PDEs), i.e., for infinite dimensional systems, and with inputs at the boundaries of spatial domains. But, until recently, such backstepping controllers for PDEs were available only for systems evolving on fixed spatial PDE domains, not for systems whose boundaries are also dynamical and move, such as in systems undergoing transition of phase of matter (like the solid-liquid transition, i.e., melting or crystallization). In this invited article we review new control designs for moving-boundary PDEs of both parabolic and hyperbolic types and illustrate them by applications, respectively, in additive manufacturing (3D printing) and freeway traffic.

Keywords

PDE backstepping, Stefan problem, 3D printing, traffic

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References


A. Isidori, Nonlinear Control Systems, Springer, 1989.

M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, 1995.

M. Krstić and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, 2008.

M. Krstić, Delay Compensation for Nonlinear, Adaptive, and PDE Systems, Boston, MA: Birkhauser, 2009.

J. Stefan, “Uber die Theorie der Eisbildung, insbesondere uber die Eisbildung im Polarmeere,” Annalen der Physik, vol. 278, pp. 269–286, 1891.

A. Armaou and P.D. Christofides, “Robust control of parabolic PDE systems with time-dependent spatial domains,” Automatica, vol. 37, pp. 61–69, 2001.

N. Petit, “Control problems for one-dimensional fluids and reactive fluids with moving interfaces,” In Advances in the theory of control, signals and systems with physical modeling, volume 407 of Lecture notes in control and information sciences, pages 323–337, Lausanne, Dec 2010.

B. Petrus, J. Bentsman, and B.G. Thomas, “Feedback control of the two-phase Stefan problem, with an application to the continuous casting of steel,” In Proceedings of the 49th IEEE Conference on Decision and Control (CDC), 2010, pp. 1731–1736.

M. Izadi and S. Dubljevic, “Backstepping output feedback control of moving boundary parabolic PDEs,” European Journal of Control, vol. 21, pp. 27–35, 2015.

S. Koga, M. Diagne, and M. Krstić, “Control and state estimation of the one-phase Stefan problem via backstepping design,” IEEE Transactions on Automatic Control, vol. 64, pp. 510–525, 2019.

S. Koga, I. Karafyllis, and M. Krstić, “Input-to-state stability for the control of Stefan problem with respect to heat loss at the interface,” In Proceedings of the 2018 American Control Conference. Milwaukee, WI, 2018.

A. Friedman “Free boundary problems for parabolic equations I. Melting of solids,” Journal of Mathematics and Mechanics, vol. 8, no. 4, pp. 499–517, 1959.

S. Gupta, The Classical Stefan Problem. Basic Concepts, Modelling and Analysis. North-Holland: Applied mathematics and Mechanics, 2003.

M. J. Lighthill and G. B. Whitham, “On kinematic waves. II. A theory of traffic flow on long crowded roads,” Proc. Roy. Soc. London. Ser. A., 229 317–345, 1955.

P. I. Richards, “Shock waves on the highway,” Operations Res., 4, 42–51, 1956

H. Yu, L.-G. Zhang, M. Diagne, and M. Krstic, “Bilateral boundary control of moving traffic shockwave,” IFAC Symposium on Nonlinear Control Systems, 2019.

I. Karafyllis, N. Bekiaris-Liberis, & M. Papageorgiou. “Feedback Control of Nonlinear Hyperbolic PDE Systems Inspired by Traffic Flow Models”. IEEE Transactions on Automatic Control, 2018.


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