# American Institute of Mathematical Sciences

March  2021, 11(1): 95-117. doi: 10.3934/mcrf.2020029

## Finite-dimensional controllers for robust regulation of boundary control systems

 1 Institut für Mathematik, Leopold-Franzens-Universität Innsbruck, Technikerstraße 13/7, A-6020 Innsbruck, Austria 2 Mathematics, Faculty of Information Technology and Communication Sciences, Tampere University, PO. Box 692, 33101 Tampere, Finland

* Corresponding author: duy.phan-duc@uibk.ac.at

Received  November 2019 Revised  March 2020 Published  June 2020

We study the robust output regulation of linear boundary control systems by constructing extended systems. The extended systems are established based on solving static differential equations under two new conditions. We first consider the abstract setting and present finite-dimensional reduced order controllers. The controller design is then used for particular PDE models: high-dimensional parabolic equations and beam equations with Kelvin-Voigt damping. Numerical examples will be presented using Finite Element Method.

Citation: Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029
##### References:
 [1] M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var., 15 (2009), 934-968.  doi: 10.1051/cocv:2008059.  Google Scholar [2] H. T. Banks and K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Control Optim., 22 (1984), 684-698.  doi: 10.1137/0322043.  Google Scholar [3] R. F. Curtain and H. Zwart, An Introduction to Infinite–Dimensional Linear Systems Theory. Texts in Applied Mathematics, Vol. 21, Springer-Verlag New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar [4] B.-Z. Guo, H.-C. Zhou, A. S. AL-Fhaid, A. M. M. Younas and A. Asiri, Stabilization of Euler-Bernoulli Beam Equation with Boundary Moment Control and Disturbance by Active Disturbance Rejection Control and Sliding Mode Control Approaches, J. Dyn. Control Syst., 20 (2014), 539-558.  doi: 10.1007/s10883-014-9241-8.  Google Scholar [5] T. Hämäläinen and S. Pohjolainen, Robust regulation for exponentially stable boundary control systems in Hilbert space, in Proceedings of the 8th IEEE International Conference on Methods and Models in Automation and Robotics, Szczecin, Poland, 2002,171–178. Google Scholar [6] T. Hämäläinen and S. Pohjolainen, Robust regulation of distributed parameter systems with infinite-dimensional exosystems, SIAM J. Control Optim., 48 (2010), 4846-4873.  doi: 10.1137/090757976.  Google Scholar [7] E. Immonen, On the internal model structure for infinite-dimensional systems: Two common controller types and repetitive control, SIAM J. Control Optim., 45 (2007), 2065-2093.  doi: 10.1137/050638916.  Google Scholar [8] K. Ito and K. Morris, An approximation theory of solutions to operator Riccati equations for $H^\infty$ control, SIAM J. Control Optim., 36 (1998), 82-99.  doi: 10.1137/S0363012994274422.  Google Scholar [9] H. Logemann and S. Townley, Low-gain control of uncertain regular linear systems, SIAM J. Control Optim., 35 (1997), 78-116.  doi: 10.1137/S0363012994275920.  Google Scholar [10] L. Paunonen and D. Phan, Reduced order controller design for robust output regulation, IEEE Transactions on Automatic Control, (2019), 1–1. doi: 10.1109/TAC.2019.2930185.  Google Scholar [11] L. Paunonen, Controller design for robust output regulation of regular linear systems, IEEE Trans. Automat. Control, 61 (2016), 2974-2986.  doi: 10.1109/TAC.2015.2509439.  Google Scholar [12] D. Phan and S. S. Rodrigues, Stabilization to trajectories for parabolic equations, Math. Control Signals Systems, 30 (2018), Art. 11, 50 pp. doi: 10.1007/s00498-018-0218-0.  Google Scholar [13] R. Rebarber and G. Weiss, Internal model based tracking and disturbance rejection for stable well-posed systems, Automatica J. IFAC, 39 (2003), 1555-1569.  doi: 10.1016/S0005-1098(03)00192-4.  Google Scholar [14] S. S. Rodrigues, Boundary observability inequalities for the 3D Oseen-Stokes system and applications, ESAIM Control Optim. Calc. Var., 21 (2015), 723-756.  doi: 10.1051/cocv/2014045.  Google Scholar [15] D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar [16] O. Staffans, Well-Posed Linear Systems, Encyclopedia of Mathematics and its Applications, Vol. 103, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.  Google Scholar [17] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

show all references

##### References:
 [1] M. Badra, Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system, ESAIM Control Optim. Calc. Var., 15 (2009), 934-968.  doi: 10.1051/cocv:2008059.  Google Scholar [2] H. T. Banks and K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Control Optim., 22 (1984), 684-698.  doi: 10.1137/0322043.  Google Scholar [3] R. F. Curtain and H. Zwart, An Introduction to Infinite–Dimensional Linear Systems Theory. Texts in Applied Mathematics, Vol. 21, Springer-Verlag New York, 1995. doi: 10.1007/978-1-4612-4224-6.  Google Scholar [4] B.-Z. Guo, H.-C. Zhou, A. S. AL-Fhaid, A. M. M. Younas and A. Asiri, Stabilization of Euler-Bernoulli Beam Equation with Boundary Moment Control and Disturbance by Active Disturbance Rejection Control and Sliding Mode Control Approaches, J. Dyn. Control Syst., 20 (2014), 539-558.  doi: 10.1007/s10883-014-9241-8.  Google Scholar [5] T. Hämäläinen and S. Pohjolainen, Robust regulation for exponentially stable boundary control systems in Hilbert space, in Proceedings of the 8th IEEE International Conference on Methods and Models in Automation and Robotics, Szczecin, Poland, 2002,171–178. Google Scholar [6] T. Hämäläinen and S. Pohjolainen, Robust regulation of distributed parameter systems with infinite-dimensional exosystems, SIAM J. Control Optim., 48 (2010), 4846-4873.  doi: 10.1137/090757976.  Google Scholar [7] E. Immonen, On the internal model structure for infinite-dimensional systems: Two common controller types and repetitive control, SIAM J. Control Optim., 45 (2007), 2065-2093.  doi: 10.1137/050638916.  Google Scholar [8] K. Ito and K. Morris, An approximation theory of solutions to operator Riccati equations for $H^\infty$ control, SIAM J. Control Optim., 36 (1998), 82-99.  doi: 10.1137/S0363012994274422.  Google Scholar [9] H. Logemann and S. Townley, Low-gain control of uncertain regular linear systems, SIAM J. Control Optim., 35 (1997), 78-116.  doi: 10.1137/S0363012994275920.  Google Scholar [10] L. Paunonen and D. Phan, Reduced order controller design for robust output regulation, IEEE Transactions on Automatic Control, (2019), 1–1. doi: 10.1109/TAC.2019.2930185.  Google Scholar [11] L. Paunonen, Controller design for robust output regulation of regular linear systems, IEEE Trans. Automat. Control, 61 (2016), 2974-2986.  doi: 10.1109/TAC.2015.2509439.  Google Scholar [12] D. Phan and S. S. Rodrigues, Stabilization to trajectories for parabolic equations, Math. Control Signals Systems, 30 (2018), Art. 11, 50 pp. doi: 10.1007/s00498-018-0218-0.  Google Scholar [13] R. Rebarber and G. Weiss, Internal model based tracking and disturbance rejection for stable well-posed systems, Automatica J. IFAC, 39 (2003), 1555-1569.  doi: 10.1016/S0005-1098(03)00192-4.  Google Scholar [14] S. S. Rodrigues, Boundary observability inequalities for the 3D Oseen-Stokes system and applications, ESAIM Control Optim. Calc. Var., 21 (2015), 723-756.  doi: 10.1051/cocv/2014045.  Google Scholar [15] D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar [16] O. Staffans, Well-Posed Linear Systems, Encyclopedia of Mathematics and its Applications, Vol. 103, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197.  Google Scholar [17] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar
Boundary controls located on red segments and regions of observations (blue)
Two extensions of boundary actuators
Output tracking of the boundary control of the 2D parabolic equation
Hankel singular values
Solutions of ODEs
Output tracking of the boundary controlled beam equation with two different extensions
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