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Yu, Xin; Xu, Chao; Jiang, Huacheng; Ganesan, Arthi; Zheng, Guojie

Backstepping synthesis for feedback control of first-order hyperbolic PDEs with spatial-temporal actuation. (English) Zbl 1474.93184

Abstr. Appl. Anal. 2014, Article ID 643640, 13 p. (2014).
Summary: This paper deals with the stabilization problem of first-order hyperbolic partial differential equations (PDEs) with spatial-temporal actuation over the full physical domains. We assume that the interior actuator can be decomposed into a product of spatial and temporal components, where the spatial component satisfies a specific ordinary differential equation (ODE). A Volterra integral transformation is used to convert the original system into a simple target system using the backstepping-like procedure. Unlike the classical backstepping techniques for boundary control problems of PDEs, the internal actuation can not eliminate the residual term that causes the instability of the open-loop system. Thus, an additional differential transformation is introduced to transfer the input from the interior of the domain onto the boundary. Then, a feedback control law is designed using the classic backstepping technique which can stabilize the first-order hyperbolic PDE system in a finite time, which can be proved by using the semigroup arguments. The effectiveness of the design is illustrated with some numerical simulations.

Cited in 2 Documents

MSC:

93D15 Stabilization of systems by feedback
93C20 Control/observation systems governed by partial differential equations
93B52 Feedback control
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\textit{X. Yu} et al., Abstr. Appl. Anal. 2014, Article ID 643640, 13 p. (2014; Zbl 1474.93184)
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References:

[1] Bensoussan, A.; da Prato, G.; Delfour, M.; Mitter, S. K., Representation and Control of Infinite Dimensional Systems (2007), Birkhäuser · Zbl 1117.93002
[2] Curtain, R. F.; Zwart, H., An Introduction to In finite-Dimensional Linear Systems Theory (1995), Springer
[3] Glowinski, R.; Lions, J.; He, J., Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach (2007), Cambridge University Press · Zbl 1142.93002
[4] Lasiecka, I.; Triggiani, R., Control Theory for Partial Differential Equations (2000), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0983.35032
[5] Luo, Z.; Guo, B.; Morgul, O., Stability and Stabilization of Infinite Dimensional Systems with Applications. Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering (1999), London, UK: Springer, London, UK · Zbl 0922.93001
[6] Coron, J.; d’Andrea Novel, B.; Bastin, G., A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Transactions on Automatic Control, 52, 1, 2-11 (2007) · Zbl 1366.93481
[7] Krstic, M.; Smyshlyaev, A., Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays, Systems and Control Letters, 57, 9, 750-758 (2008) · Zbl 1153.93022
[8] Sano, H., Exponential stability of a mono-tubular heat exchanger equation with output feedback, Systems and Control Letters, 50, 5, 363-369 (2003) · Zbl 1157.93460
[9] Yu, X.; Liu, K., Eventual regularity of the semigroup associated with the mono-tubular heat exchanger equation with output feedback, Systems and Control Letters, 55, 10, 859-862 (2006) · Zbl 1100.93027
[10] Krstic, M.; Kanellakopoulos, I.; Kokotovic, P., Nonlinear and Adaptive Control Design (1995), New York, NY, USA: John Wiley and Sons, New York, NY, USA · Zbl 0763.93043
[11] Coron, J.; d’Andrea-Novel, B., Stabilization of a rotating body beam without damping, IEEE Transactions on Automatic Control, 43, 5, 608-618 (1998) · Zbl 0908.93055
[12] Krstic, M.; Guo, B.; Balogh, A.; Smyshlyaev, A., Output-feedback stabilization of an unstable wave equation, Automatica, 44, 1, 63-74 (2008) · Zbl 1138.93046
[13] Krstic, M.; Smyshlyaev, A., Boundary Control of PDEs. A Course on Backstepping Designs (2008), Society for Industrial and Applied Mathematics · Zbl 1149.93004
[14] Liu, W., Boundary feedback stabilization of an unstable heat equation, SIAM Journal on Control and Optimization, 42, 3, 1033-1043 (2003) · Zbl 1047.93042
[15] Meurer, T.; Kugi, A., Tracking control for boundary controlled parabolic PDEs with varying parameters: combining backstepping and differential flatness, Automatica, 45, 5, 1182-1194 (2009) · Zbl 1162.93016
[16] Xu, C.; Schuster, E.; Vazquez, R.; Krstic, M., Stabilization of linearized 2D magnetohydrodynamic channel flow by backstepping boundary control, Systems & Control Letters, 57, 10, 805-812 (2008) · Zbl 1162.93031
[17] Vazquez, R.; Krstic, M., A closed-form feedback controller for stabilization of the linearized 2-D Navier-Stokes poiseuille system, IEEE Transactions on Automatic Control, 52, 12, 2298-2312 (2007) · Zbl 1366.93532
[18] Vazquez, R.; Schuster, E.; Krstic, M., Magnetohydrodynamic state estimation with boundary sensors, Automatica, 44, 10, 2517-2527 (2008) · Zbl 1155.93326
[19] Aamo, O. M., Disturbance rejection in \(2 \times 2\) linear hyperbolic systems, IEEE Transactions on Automatic Control, 58, 5, 1095-1106 (2013) · Zbl 1369.93268
[20] Meurer, T., Control of Higher-Dimensional PDEs: Flatness and Backstepping Designs (2013), New York, NY, USA: Springer, New York, NY, USA · Zbl 1253.93004
[21] Cerpa, E.; Coron, J., Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Transactions on Automatic Control, 58, 7, 1688-1695 (2013) · Zbl 1369.93480
[22] Coron, J.; Vazquez, R.; Krstic, M.; Bastin, G., Local exponential \(H^{\text{2}}\) stabilization of a 2×2 quasilinear hyperbolic system using backstepping, SIAM Journal on Control and Optimization, 51, 3, 2005-2035 (2013) · Zbl 1408.35009
[23] Lasiecka, I.; Triggiani, R., The regulator problem for parabolic equations with Dirichlet boundary control. I. Riccati’s feedback synthesis and regularity of optimal solution, Applied Mathematics and Optimization, 16, 2, 147-168 (1987) · Zbl 0639.49002
[24] Lasiecka, I.; Triggiani, R., The regulator problem for parabolic equations with Dirichlet boundary control. II. Galerkin approximation, Applied Mathematics and Optimization, 16, 3, 187-216 (1987) · Zbl 0652.49026
[25] Liu, W., Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation (2009), New York, NY, USA: Springer, New York, NY, USA
[26] Tsubakino, D.; Krstic, M.; Hara, S., Backstepping control for parabolic PDEs with in-domain actuation, Proceedings of the American Control Conference (ACC ’12)
[27] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44 (1983), New York, NY, USA: Springer, New York, NY, USA · Zbl 0516.47023
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