Reduced Order LQG Control Design for Infinite Dimensional Port Hamiltonian Systems

Authors

Yongxin Wu, Boussad Hamroun, Yann Le Gorrec, Bernhard Maschke

Abstract

This article proposes a method that combines linear quadratic Gaussian (LQG) control design and structure preserving model reduction for the reduced order control of infinite dimensional port Hamiltonian systems (IDPHS).For that purpose the weighting operators used in LQG control design are chosen such that the resulting dynamic controller is passive and the closed-loop system equivalent to control by interconnection. The method of Petrov–Galerkin is then used to approximate the balanced realization of the IDPHS by a finite dimensional port Hamiltonian system and to provide the associated reduced order LQG controller. The main advantages of the proposed method are that, first, both control and reduction are driven by closed-loop performances and that, second, due to the passivity properties of the controller the closed-loop stability is guaranteed when the finite dimensional controller is applied to the infinite dimensional system.

Citation

• Journal: IEEE Transactions on Automatic Control
• Year: 2021
• Volume: 66
• Issue: 2
• Pages: 865–871
• Publisher: Institute of Electrical and Electronics Engineers (IEEE)
• DOI: 10.1109/tac.2020.2997373

BibTeX

@article{Wu_2021,
  title={{Reduced Order LQG Control Design for Infinite Dimensional Port Hamiltonian Systems}},
  volume={66},
  ISSN={2334-3303},
  DOI={10.1109/tac.2020.2997373},
  number={2},
  journal={IEEE Transactions on Automatic Control},
  publisher={Institute of Electrical and Electronics Engineers (IEEE)},
  author={Wu, Yongxin and Hamroun, Boussad and Le Gorrec, Yann and Maschke, Bernhard},
  year={2021},
  pages={865--871}
}

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References

• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• Le Gorrec, Y., Zwart, H. & Maschke, B. Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM Journal on Control and Optimization vol. 44 1864–1892 (2005) – 10.1137/040611677
• Jacob, B. & Zwart, H. J. Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces. (Springer Basel, 2012). doi:10.1007/978-3-0348-0399-1 – 10.1007/978-3-0348-0399-1
• Macchelli, A. & Melchiorri, C. Modeling and Control of the Timoshenko Beam. The Distributed Port Hamiltonian Approach. SIAM Journal on Control and Optimization vol. 43 743–767 (2004) – 10.1137/s0363012903429530
• Curtain, R. F. & Zwart, H. An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics (Springer New York, 1995). doi:10.1007/978-1-4612-4224-6 – 10.1007/978-1-4612-4224-6
• Brogliato, B., Maschke, B., Lozano, R. & Egeland, O. Dissipative Systems Analysis and Control. Communications and Control Engineering (Springer London, 2007). doi:10.1007/978-1-84628-517-2 – 10.1007/978-1-84628-517-2
• Ortega, R., van der Schaft, A., Castanos, F. & Astolfi, A. Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 53 2527–2542 (2008) – 10.1109/tac.2008.2006930
• Curtain, R. F. & Sasane, A. J. Compactness and nuclearity of the Hankel operator and internal stability of infinite-dimensional state linear systems. International Journal of Control vol. 74 1260–1270 (2001) – 10.1080/00207170110061059
• Pazy, A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences (Springer New York, 1983). doi:10.1007/978-1-4612-5561-1 – 10.1007/978-1-4612-5561-1
• villegas, A port-Hamiltonian approach to distributed parameter systems. (2007)
• King, B. B., Hovakimyan, N., Evans, K. A. & Buhl, M. Reduced order controllers for distributed parameter systems: LQG balanced truncation and an adaptive approach. Mathematical and Computer Modelling vol. 43 1136–1149 (2006) – 10.1016/j.mcm.2005.05.031
• camp, A comparison of balanced truncation techniques for reduced order controllers. Proc Math Theory Netw Syst Notre Dame IN USA (0)
• Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
• Curtain, R. F. 4. Model Reduction for Control Design for Distributed Parameter Systems. Research Directions in Distributed Parameter Systems 95–121 (2003) doi:10.1137/1.9780898717525.ch4 – 10.1137/1.9780898717525.ch4
• liu, Semigroups Associated with Dissipative Systems (1999)
• Harkort, C. & Deutscher, J. Stability and passivity preserving Petrov–Galerkin approximation of linear infinite-dimensional systems. Automatica vol. 48 1347–1352 (2012) – 10.1016/j.automatica.2012.04.010
• Wu, Y., Hamroun, B., Le Gorrec, Y. & Maschke, B. Reduced order LQG control design for port Hamiltonian systems. Automatica vol. 95 86–92 (2018) – 10.1016/j.automatica.2018.05.003
• Balas, M. J. Active control of flexible systems. Journal of Optimization Theory and Applications vol. 25 415–436 (1978) – 10.1007/bf00932903
• van der Schaft, A. L2 - Gain and Passivity Techniques in Nonlinear Control. Communications and Control Engineering (Springer London, 2000). doi:10.1007/978-1-4471-0507-7 – 10.1007/978-1-4471-0507-7
• kreyszig, Introductory Functional Analysis With Applications (1989)
• Braun, P., Hernández, E. & Kalise, D. Reduced-order LQG control of a Timoshenko beam model. Bulletin of the Brazilian Mathematical Society, New Series vol. 47 143–155 (2016) – 10.1007/s00574-016-0128-z
• Kato, T. Perturbation Theory for Linear Operators. Classics in Mathematics (Springer Berlin Heidelberg, 1995). doi:10.1007/978-3-642-66282-9 – 10.1007/978-3-642-66282-9
• Morris, K. A. Convergence of controllers designed using state-space techniques. IEEE Transactions on Automatic Control vol. 39 2100–2104 (1994) – 10.1109/9.328802
• Golo, G., Talasila, V., van der Schaft, A. & Maschke, B. Hamiltonian discretization of boundary control systems. Automatica vol. 40 757–771 (2004) – 10.1016/j.automatica.2003.12.017