A Boundary Backstepping Control Design for 2-D, 3-D and N-D Heat Equation
We consider the problem of stabilization of an unstable
heat equation in a 2-D, 3-D and generally n-D domain by deriving a
generalized backstepping boundary control design methodology. To
stabilize the systems, we design boundary backstepping controllers
inspired by the 1-D unstable heat equation stabilization procedure.
We assume that one side of the boundary is hinged and the other
side is controlled for each direction of the domain. Thus, controllers
act on two boundaries for 2-D domain, three boundaries for 3-D
domain and ”n” boundaries for n-D domain. The main idea of the
design is to derive ”n” controllers for each of the dimensions by
using ”n” kernel functions. Thus, we obtain ”n” controllers for the
”n” dimensional case. We use a transformation to change the system
into an exponentially stable ”n” dimensional heat equation. The
transformation used in this paper is a generalized Volterra/Fredholm
type with ”n” kernel functions for n-D domain instead of the one
kernel function of 1-D design.
[1] M. Krstic, A. Smyshlyaev, ”Explicit State and Output Feedback Boundary
Controllers for Partial Differential Equations”, Journal of Automatic
Control, University of Belgrade, 13(2), pp. 1–9, 2003.
[2] M. Krstic, A. Smyshlyaev, Boundary Control of PDEs, A Course on
Backstepping Designs,Siam, 2008.
[3] M. Krstic, A. Smyshlyaev, ”Adaptive control of PDEs”, Annual Reviews
in Control , 32, pp. 149–160 , 2008.
[4] M. Krstic, B. Z. Guo, A. Balogh, A. Smyshlyaev, ”Output-feedback
stabilization of an unstable wave equation”, Automatica , 44, pp. 63–74,
2008.
[5] M. Krstic, ”Dead-Time Compensation for Wave/String PDEs”, Journal
of Dynamic Systems, Measurement, and Control , 133, pp. 031004/1–13,
2011.
[6] A. Sezgin, M. Krstic, ”Boundary Backstepping Control of Flow-Induced
Vibrations of a Membrane at High Mach Numbers”, Journal of Dynamic
Systems, Measurement, and Control, doi: 10.1115/1.4029468, 2015.
[7] M. Krstic, A. Smyshlyaev, ”Backstepping boundary control for first-order
hyperbolic PDEs and application to systems with actuator and sensor
delays”, Systems and Control Letters, 57, pp. 750–758, 2008.
[8] N. Bekiaris-Liberis, M. Krstic, ”Compensating the distributed effect of
a wave PDE in the actuation or sensing path of MIMO LTI systems”,
Systems and Control Letters, 59, pp. 713–719, 2010.
[9] M.B. Cheng, V. Radisavljevic, W.C. Su, ”Sliding mode boundary control
of a parabolic PDE system with parameter variations and boundary
uncertainties”, Automatica, 47(2), pp. 381–387, 2011.
[10] J. Ng, S. Dubljevic, ”Optimal boundary control of a
diffusionconvection-reaction PDE model with time-dependent spatial
domain: Czochralski crystal growth process”, Chemical Engineering
Science, 67(1), pp. 111–119, 2012.
[11] K. Chrysafinos, M.D. Gunzburger, L.S. Hou, ”Semidiscrete
approximations of optimal Robin boundary control problems constrained
by semilinear parabolic PDE”, Journal of Mathematical Analysis and
Applications, 323(2), pp. 891–912, 2006.
[12] S. Tang, C. Xie, ”State and output feedback boundary control for a
coupled PDE-ODE system”, Systems and Control Letters, 60(8), pp.
540–545, 2011.
[13] N. B. Liberis, and M. Krstic, ”Compensating the Distributed Effect of
Diffusion and Counter-Convection in Multi-Input and Multi-Output LTI
Systems”, IEEE Transactions on Automatic Control, 56(3), pp. 637–643,
2011.
[14] J.A. Ramirez, H. Puebla, J.A. Ochoa-Tapia, ”Linear boundary control
for a class of nonlinear PDE processes”, Systems and Control Letters,
44(5), pp. 395–403, 2001.
[15] W.S. Cheung, ”Some New Poincare–Type inequalities”, Bull. Austral.
Math. Soc., 63, pp. 321–327, 2001.
[16] M. Krstic, ”Adaptive Control of an Anti-Stable Wave PDE”, Dynamics
of Continuous, Discrete and Impulsive System , 17, pp. 853–882, 2010.
[17] S. Cox, and E. Zuazua, ”The rate at which energy decays in a string
damped at one end”, Comm. Partial Differential Equations, 19, pp.
213–243, 1994.
[18] A. Smyshlyaev, M. Krstic, ”Backstepping observers for a class of
parabolic PDEs”, Systems and Control Letters , 54, pp. 613–625, 2005.
[19] M. Krstic, I. Kanellakopoulos and P. Kokotovic, Nonlinear and Adaptive
Control Design, Wiley, 1995.
[20] M. Krstic, ”Control of an unstable reaction–diffusion PDE with long
input delay”, Systems and Control Letters, 58(10-11), pp. 773–782, 2009.
[1] M. Krstic, A. Smyshlyaev, ”Explicit State and Output Feedback Boundary
Controllers for Partial Differential Equations”, Journal of Automatic
Control, University of Belgrade, 13(2), pp. 1–9, 2003.
[2] M. Krstic, A. Smyshlyaev, Boundary Control of PDEs, A Course on
Backstepping Designs,Siam, 2008.
[3] M. Krstic, A. Smyshlyaev, ”Adaptive control of PDEs”, Annual Reviews
in Control , 32, pp. 149–160 , 2008.
[4] M. Krstic, B. Z. Guo, A. Balogh, A. Smyshlyaev, ”Output-feedback
stabilization of an unstable wave equation”, Automatica , 44, pp. 63–74,
2008.
[5] M. Krstic, ”Dead-Time Compensation for Wave/String PDEs”, Journal
of Dynamic Systems, Measurement, and Control , 133, pp. 031004/1–13,
2011.
[6] A. Sezgin, M. Krstic, ”Boundary Backstepping Control of Flow-Induced
Vibrations of a Membrane at High Mach Numbers”, Journal of Dynamic
Systems, Measurement, and Control, doi: 10.1115/1.4029468, 2015.
[7] M. Krstic, A. Smyshlyaev, ”Backstepping boundary control for first-order
hyperbolic PDEs and application to systems with actuator and sensor
delays”, Systems and Control Letters, 57, pp. 750–758, 2008.
[8] N. Bekiaris-Liberis, M. Krstic, ”Compensating the distributed effect of
a wave PDE in the actuation or sensing path of MIMO LTI systems”,
Systems and Control Letters, 59, pp. 713–719, 2010.
[9] M.B. Cheng, V. Radisavljevic, W.C. Su, ”Sliding mode boundary control
of a parabolic PDE system with parameter variations and boundary
uncertainties”, Automatica, 47(2), pp. 381–387, 2011.
[10] J. Ng, S. Dubljevic, ”Optimal boundary control of a
diffusionconvection-reaction PDE model with time-dependent spatial
domain: Czochralski crystal growth process”, Chemical Engineering
Science, 67(1), pp. 111–119, 2012.
[11] K. Chrysafinos, M.D. Gunzburger, L.S. Hou, ”Semidiscrete
approximations of optimal Robin boundary control problems constrained
by semilinear parabolic PDE”, Journal of Mathematical Analysis and
Applications, 323(2), pp. 891–912, 2006.
[12] S. Tang, C. Xie, ”State and output feedback boundary control for a
coupled PDE-ODE system”, Systems and Control Letters, 60(8), pp.
540–545, 2011.
[13] N. B. Liberis, and M. Krstic, ”Compensating the Distributed Effect of
Diffusion and Counter-Convection in Multi-Input and Multi-Output LTI
Systems”, IEEE Transactions on Automatic Control, 56(3), pp. 637–643,
2011.
[14] J.A. Ramirez, H. Puebla, J.A. Ochoa-Tapia, ”Linear boundary control
for a class of nonlinear PDE processes”, Systems and Control Letters,
44(5), pp. 395–403, 2001.
[15] W.S. Cheung, ”Some New Poincare–Type inequalities”, Bull. Austral.
Math. Soc., 63, pp. 321–327, 2001.
[16] M. Krstic, ”Adaptive Control of an Anti-Stable Wave PDE”, Dynamics
of Continuous, Discrete and Impulsive System , 17, pp. 853–882, 2010.
[17] S. Cox, and E. Zuazua, ”The rate at which energy decays in a string
damped at one end”, Comm. Partial Differential Equations, 19, pp.
213–243, 1994.
[18] A. Smyshlyaev, M. Krstic, ”Backstepping observers for a class of
parabolic PDEs”, Systems and Control Letters , 54, pp. 613–625, 2005.
[19] M. Krstic, I. Kanellakopoulos and P. Kokotovic, Nonlinear and Adaptive
Control Design, Wiley, 1995.
[20] M. Krstic, ”Control of an unstable reaction–diffusion PDE with long
input delay”, Systems and Control Letters, 58(10-11), pp. 773–782, 2009.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:71217", author = "Aziz Sezgin", title = "A Boundary Backstepping Control Design for 2-D, 3-D and N-D Heat Equation", abstract = "We consider the problem of stabilization of an unstable
heat equation in a 2-D, 3-D and generally n-D domain by deriving a
generalized backstepping boundary control design methodology. To
stabilize the systems, we design boundary backstepping controllers
inspired by the 1-D unstable heat equation stabilization procedure.
We assume that one side of the boundary is hinged and the other
side is controlled for each direction of the domain. Thus, controllers
act on two boundaries for 2-D domain, three boundaries for 3-D
domain and ”n” boundaries for n-D domain. The main idea of the
design is to derive ”n” controllers for each of the dimensions by
using ”n” kernel functions. Thus, we obtain ”n” controllers for the
”n” dimensional case. We use a transformation to change the system
into an exponentially stable ”n” dimensional heat equation. The
transformation used in this paper is a generalized Volterra/Fredholm
type with ”n” kernel functions for n-D domain instead of the one
kernel function of 1-D design.", keywords = "Backstepping, boundary control, 2-D, 3-D, n-D heat
equation, distributed parameter systems.", volume = "9", number = "8", pages = "488-6", }