Output regulation of a reaction-diffusion PDE with long time delay using backstepping approach

Abstract In this paper, we are concerned with the output regulation of an unstable reaction-diffusion PDE with regulator delay and unmatched disturbances, which are generated by an exosystem. A backstepping regulator design procedure is presented by mapping the reaction-diffusion PDE cascaded with a transport equation into an error system, which is shown to be exponentially stable with a prescribed rate in a suitable Hilbert space. The output regulation is then verified by solving cascaded regulator equations. The solvability condition of the cascaded regulator equations is characterized by a transfer function and eigenvalues of the exosystem. Finally, the numerical simulations are provided to illustrate the effect of the regulator.

[1]  Boumediène Chentouf,et al.  Stabilization of a One-Dimensional Dam-River System: Nondissipative and Noncollocated Case , 2007 .

[2]  Miroslav Krstic,et al.  Wave Equation Stabilization by Delays Equal to Even Multiples of the Wave Propagation Time , 2011, SIAM J. Control. Optim..

[3]  Christopher I. Byrnes,et al.  Output regulation for linear distributed parameter systems , 2000, IEEE Trans. Autom. Control..

[4]  Shuzhi Sam Ge,et al.  Adaptive Control of a Flexible Crane System With the Boundary Output Constraint , 2014, IEEE Transactions on Industrial Electronics.

[5]  Miroslav Krstic,et al.  Adaptive Observer for a Class of Output-Delayed Systems with Parameter Uncertainty - A PDE Based Approach , 2016 .

[6]  Miroslav Krstic,et al.  Closed-form boundary State feedbacks for a class of 1-D partial integro-differential equations , 2004, IEEE Transactions on Automatic Control.

[7]  Mark W. Spong,et al.  Bilateral control of teleoperators with time delay , 1988, Proceedings of the 1988 IEEE International Conference on Systems, Man, and Cybernetics.

[8]  R. Triggiani,et al.  Control Theory for Partial Differential Equations: Continuous and Approximation Theories , 2000 .

[9]  Miroslav Krstic,et al.  Control of an unstable reaction-diffusion PDE with long input delay , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[10]  Emilia Fridman,et al.  Output regulation of nonlinear systems with delay , 2003, Syst. Control. Lett..

[11]  Shuzhi Sam Ge,et al.  Cooperative control of a nonuniform gantry crane with constrained tension , 2016, Autom..

[12]  Ron J. Patton,et al.  An observer design for linear time-delay systems , 2002, IEEE Trans. Autom. Control..

[13]  Amit Ailon,et al.  Stability analysis of a rigid robot with output-based controller and time delay , 2000 .

[14]  H. Ozbay,et al.  A solution to the robust flow control problem for networks with multiple bottlenecks , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[15]  Joachim Deutscher,et al.  A backstepping approach to the output regulation of boundary controlled parabolic PDEs , 2015, Autom..

[16]  Joachim Deutscher,et al.  Backstepping design of robust state feedback regulators for second order hyperbolic PIDEs , 2016 .

[17]  Emilia Fridman,et al.  Sliding-mode control of uncertain systems in the presence of unmatched disturbances with applications , 2010, Int. J. Control.

[18]  Jun-Min Wang,et al.  A Riesz Basis Methodology for Proportional and Integral Output Regulation of a One-Dimensional Diffusive-Wave Equation , 2008, SIAM J. Control. Optim..

[19]  Bao-Zhu Guo,et al.  Output Feedback Stabilization for One-Dimensional Wave Equation Subject to Boundary Disturbance , 2015, IEEE Transactions on Automatic Control.

[20]  M. Krstić.  Boundary Control of PDEs: A Course on Backstepping Designs , 2008 .

[21]  Eugenio Aulisa,et al.  A Practical Guide to Geometric Regulation for Distributed Parameter Systems , 2015 .