American Institute of Mathematical Sciences

Advanced
doi: 10.3934/mcrf.2020029

Finite-dimensional controllers for robust regulation of boundary control systems

Duy Phan 1,, and  Lassi Paunonen 2,

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.

Keywords: Distributed parameter systems, robust output regulation, finite-dimensional controllers, feedback boundary controls, Galerkin approximation.
Mathematics Subject Classification: Primary: 93C05, 93B52, 93D09; Secondary: 35K10.
Citation: Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 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

Figure 1.  Boundary controls located on red segments and regions of observations (blue)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Two extensions of boundary actuators
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Output tracking of the boundary control of the 2D parabolic equation
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Hankel singular values
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Solutions of ODEs
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Output tracking of the boundary controlled beam equation with two different extensions
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434
[2]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120
[3]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270
[4]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340
[5]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351
[6]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[7]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463
[8]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454
[9]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076
[10]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078
[11]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158
[12]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[13]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453
[14]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319
[15]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[16]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[17]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[18]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349
[19]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031
[20]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (35)
  • HTML views (172)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences