Sage Journals: Discover world-class research

Abstract

This article presents a backstepping boundary control technique to stabilize the temperature of a rod. The rod is modeled by a parabolic partial differential equation with a Neumann boundary condition at one end of rod and a Dirichlet boundary condition at the other end. The rod also includes an internal heat generator to emulate an unstable behavior. Based on backstepping boundary control, a partial differential equation gain kernel of the system is determined. It is calculated numerically by using a finite difference method and then used in a control law. Since the control law requires temperatures along the rod for feedback, a Luenberger-like observer is used to estimate the temperatures. A copper rod with a heater installed inside the rod is used as an experimental plant. The control setup is anticollocation. The temperature signal at one end is sent to the observer for estimation of the temperatures along the rod. The estimated values are then fed to the controller. Simulation results using a finite difference method and experimental results are compared and used to confirm the effectiveness of the control method.

Keywords

Parabolic partial differential equation backstepping boundary control finite difference equation

Get full access to this article

View all access options for this article.

References

Rahimian

Tavazoei

. Stabilizing fractional-order PI and PD controllers: an integer-order implemented system approach. Proc IMechE, Part I: J Systems and Control Engineering 2010; 224: 893–902.

Hulsing

. Methods for computing functional gains for LQR control of partial differential equations. PhD Dissertation, Virginia Polytechnic Institute and State University, Blacksburg, VA.

Banks

Kunisch

. The linear regulator problem for parabolic systems. SIAM J Control Optim 1984; 22(5): 684–698.

Babuska

Aziz

. The mathematical foundation of the finite element method with applications to partial differential equations. New York: Academic Press, 1972.

Lasiecka

. Control of Systems governed by Partial differential Equations: A Historical Perspective. In: Proceedings of the 34th Conference on Decision & Control, New Orleans, LA, December 1995. New York: IEEE.

Padhi

Ali

. An account of chronological developments in control of distributed parameter systems. Annu Rev Control 2009; 33: 59–68.

Boskovic

Krstic

Liu

. Boundary control of an unstable heat equation via measurement of domain-averaged temperature. IEEE T Automat Contr 2001; 46: 2022–2028.

Balogh

Krstic

. Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability. Eur J Control 2002; 8: 165–175.

Smyshlyaev

Miroslav

. Closed form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE T Automat Contr 2004; 49(12): 2185–2202.

10.

Smyshlyaev

Miroslav

. Backstepping observers for a class of parabolic PDEs. Syst Control Lett 2005; 54: 613–625.

11.

Krstic

Smyshlyaev

. Adaptive control of PDEs. Annu Rev Control 2008; 32: 149–160.

12.

Liu

. Boundary feedback stabilization of an unstable heat equation. SIAM J Control Optim 2003; 42(3): 1033–1043.

13.

Cheng

. Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties. Automatica 2011; 47: 381–387.

14.

Morgul

. Dynamic boundary control of the Timoshenko beam. Automatica 1992; 28: 1255–1260.

15.

Fard

. Passivity analysis of nonlinear Euler-Bernoulli beams. Model Ident Control 2002; 23(4): 239–258.

16.

Krstic

Balogh

Smyshlyaev

. Backstepping boundary controller and observer for the undamped shear beam. In: Proceedings of the 17th international symposium on mathematical theory of networks and systems, Kyoto, Japan, 24–28 July 2006.

17.

Krstic

Siranosian

Smyshlyaev

. Backstepping boundary controllers and observers for the slender Timoshenko beam: part I—design. In: Proceedings of the 2006 American control conference, Minneapolis, MN, 14–16 June 2006. New York: IEEE.

18.

Krstic

Siranosian

Smyshlyaev

. Backstepping boundary controllers and observers for the slender Timoshenko beam: part II—stability and simulations. In: Proceedings of the 45th IEEE conference on decision and control, San Diego, CA, 13–15 December 2006. New York: IEEE.

19.

Ali

Padhi

. Active vibration suppression of non-linear beams using optimal dynamic inversion. Proc IMechE, Part I: J Systems and Control Engineering 2009; 223(15): 657–672.

20.

Asmar

. Partial differential equations and boundary value problems. Upper Saddle River, NJ: Prentice Hall, 2000.

21.

Carslaw

. Introduction to the mathematical theory of the conduction of heat in solids. London: Macmillan, 1921.

22.

Mathews

Fink

. Numerical methods using MATLAB. Upper Saddle River, NJ: Prentice Hall, 1999.

Backstepping boundary control: An application to rod temperature control with Neumann boundary condition

Abstract

Keywords

Get full access to this article

References