Approximate Robust Output Regulation of Boundary Control Systems

We extend the internal model principle for systems with boundary control and boundary observation, and construct a robust controller for this class of systems. However, as a consequence of the internal model principle, any robust controller for a plant with infinite-dimensional output space necessarily has infinite-dimensional state space. We proceed to formulate the approximate robust output regulation problem and present a finite-dimensional controller structure to solve it. Our main motivating example is a wave equation on a bounded multidimensional spatial domain with force control and velocity observation at the boundary. In order to illustrate the theoretical results, we construct an approximate robust controller for the wave equation on an annular domain and demonstrate its performance with numerical simulations.

[1]  Stevan Dubljevic,et al.  Output regulation problem for a class of regular hyperbolic systems , 2016, Int. J. Control.

[2]  George Weiss,et al.  Regular linear systems with feedback , 1994, Math. Control. Signals Syst..

[3]  Lassi Paunonen,et al.  Robust Output Regulation for Continuous-Time Periodic Systems , 2016, IEEE Transactions on Automatic Control.

[4]  Timo Hämäläinen,et al.  Robust regulation of distributed parameter systems with infinite-dimensional exosystems , 2010, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[5]  Eero Immonen.  On the Internal Model Structure for Infinite-Dimensional Systems: Two Common Controller Types and Repetitive Control , 2007, SIAM J. Control. Optim..

[6]  Olof J. Staffans,et al.  Conservative boundary control systems , 2006 .

[7]  Christopher I. Byrnes,et al.  Regular Linear Systems Governed by a Boundary Controlled Heat Equation , 2002 .

[8]  Lassi Paunonen,et al.  Robust Regulation of Infinite-Dimensional Port-Hamiltonian Systems , 2017, IEEE Transactions on Automatic Control.

[9]  J. Malinen,et al.  Impedance Passive and Conservative Boundary Control Systems , 2007 .

[10]  On Boundary Stability of Wave Equations with Variable Coefficients , 2002 .

[11]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[12]  Mikael Kurula,et al.  The duality between the gradient and divergence operators on bounded Lipschitz domains , 2012 .

[13]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[14]  Bruce A. Francis,et al.  The internal model principle of control theory , 1976, Autom..

[15]  Vũ Quôc Phóng,et al.  The operator equationAX−XB=C with unbounded operatorsA andB and related abstract Cauchy problems , 1991 .

[16]  Seppo Pohjolainen,et al.  Feedback and Feedforward Output Regulation of Bounded Uniformly Continuous Signals for Infinite-Dimensional Systems , 2006, SIAM J. Control. Optim..

[17]  Hans Zwart,et al.  Linear wave systems on n-D spatial domains , 2014, Int. J. Control.

[18]  Seppo Pohjolainen,et al.  The Internal Model Principle for Systems with Unbounded Control and Observation , 2014, SIAM J. Control. Optim..

[19]  Seppo Pohjolainen,et al.  Internal Model Theory for Distributed Parameter Systems , 2010, SIAM J. Control. Optim..

[20]  David S. Gilliam,et al.  The State Feedback Regulator Problem for Regular Linear Systems , 2014, IEEE Transactions on Automatic Control.

[21]  W. Wonham,et al.  The internal model principle for linear multivariable regulators , 1975 .

[22]  E. Davison.  The robust control of a servomechanism problem for linear time-invariant multivariable systems , 1976 .

[23]  C. Bardos,et al.  Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary , 1992 .

[24]  Lassi Paunonen.  Robust Controllers for Regular Linear Systems with Infinite-Dimensional Exosystems , 2017, SIAM J. Control. Optim..

[25]  Lassi Paunonen,et al.  Controller Design for Robust Output Regulation of Regular Linear Systems , 2015, IEEE Transactions on Automatic Control.

[26]  Richard Rebarber,et al.  Internal model based tracking and disturbance rejection for stable well-posed systems , 2003, Autom..

[27]  Ulrich Eggers,et al.  Introduction To Infinite Dimensional Linear Systems Theory , 2016 .

[28]  Peng-Fei Yao,et al.  On The Observability Inequalities for Exact Controllability of Wave Equations With Variable Coefficients , 1999 .

[29]  G. Weiss,et al.  Observation and Control for Operator Semigroups , 2009 .

[30]  H. Nounou,et al.  The PI-Controller for infinite dimensional linear systems in Banach state spaces , 2009, 2009 American Control Conference.

[31]  Timo Hämäläinen,et al.  A finite-dimensional robust controller for systems in the CD-algebra , 2000, IEEE Trans. Autom. Control..

[32]  Timo Hämäläinen,et al.  A Self-Tuning Robust Regulator for Infinite-Dimensional Systems , 2011, IEEE Transactions on Automatic Control.

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

[34]  Ruth F. Curtain,et al.  Exponential stabilization of well-posed systems by colocated feedback , 2006, SIAM J. Control. Optim..

[35]  O. Staffans.  Well-Posed Linear Systems , 2005 .