Structure-preserving \( H^\infty \) control for port-Hamiltonian systems

Authors

Abstract

We study H ∞ control design for linear time-invariant port-Hamiltonian systems. By a modification of the two central algebraic Riccati equations, we ensure that the resulting controller will be port-Hamiltonian. Using these modified equations, we proceed to show that a corresponding balanced truncation approach preserves port-Hamiltonian structure. We illustrate the theoretical findings using numerical examples and observe that the chosen representation of the port-Hamiltonian system can have an influence on the approximation qualities of the reduced order model.

Keywords

Port-Hamiltonian systems; \( H^\infty \) control design; Model order reduction

Citation

Journal: Systems & Control Letters
Year: 2023
Volume: 174
Issue:
Pages: 105493
Publisher: Elsevier BV
DOI: 10.1016/j.sysconle.2023.105493

BibTeX

@article{Breiten_2023,
  title={{Structure-preserving $H^\infty$ control for port-Hamiltonian systems}},
  volume={174},
  ISSN={0167-6911},
  DOI={10.1016/j.sysconle.2023.105493},
  journal={Systems &amp; Control Letters},
  publisher={Elsevier BV},
  author={Breiten, Tobias and Karsai, Attila},
  year={2023},
  pages={105493}
}

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References

Kalman, R. E. & Bucy, R. S. New Results in Linear Filtering and Prediction Theory. Journal of Basic Engineering vol. 83 95–108 (1961) – 10.1115/1.3658902
Doyle, J. C., Glover, K., Khargonekar, P. P. & Francis, B. A. State-space solutions to standard H/sub 2/ and H/sub infinity / control problems. IEEE Transactions on Automatic Control vol. 34 831–847 (1989) – 10.1109/9.29425
Doyle, J. Guaranteed margins for LQG regulators. IEEE Transactions on Automatic Control vol. 23 756–757 (1978) – 10.1109/tac.1978.1101812
van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
Mehrmann, (2022)
van der Schaft, Port-Hamiltonian systems: An introductory survey. (2006)
Van der Schaft, (2000)
Breiten, Error bounds for port-Hamiltonian model and controller reduction based on system balancing. Comput. Math. Appl. (2021)
Lozano-Leal, On the design of the dissipative LQG-type controllers. (1988)
Haddad, Dissipative H2/H∞ controller synthesis. (1993)
Bernstein, D. S. & Haddad, W. M. LQG control with an H/sup infinity / performance bound: a Riccati equation approach. IEEE Transactions on Automatic Control vol. 34 293–305 (1989) – 10.1109/9.16419
Polyuga, R. V. & van der Schaft, A. J. Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems. Systems & Control Letters vol. 61 412–421 (2012) – 10.1016/j.sysconle.2011.12.008
Breiten, (2021)
Gugercin, S., Polyuga, R. V., Beattie, C. & van der Schaft, A. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica vol. 48 1963–1974 (2012) – 10.1016/j.automatica.2012.05.052
Mamunuzzaman, (2022)
Polyuga, Moment matching for linear port-Hamiltonian systems. (2009)
Polyuga, R. V. & van der Schaft, A. Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity. Automatica vol. 46 665–672 (2010) – 10.1016/j.automatica.2010.01.018
Schwerdtner, (2020)
Wen, J. T. Time domain and frequency domain conditions for strict positive realness. IEEE Transactions on Automatic Control vol. 33 988–992 (1988) – 10.1109/9.7263
Hakimi-Moghaddam, M. Positive real and strictly positive real MIMO systems: theory and application. International Journal of Dynamics and Control vol. 8 448–458 (2019) – 10.1007/s40435-019-00550-9
Anderson, B. A simplified viewpoint of hyperstability. IEEE Transactions on Automatic Control vol. 13 292–294 (1968) – 10.1109/tac.1968.1098910
Reis, T. Lur’e equations and even matrix pencils. Linear Algebra and its Applications vol. 434 152–173 (2011) – 10.1016/j.laa.2010.09.005
Willems, J. C. Dissipative dynamical systems Part II: Linear systems with quadratic supply rates. Archive for Rational Mechanics and Analysis vol. 45 352–393 (1972) – 10.1007/bf00276494
Boyd, (1994)
Anderson, B. D. O. A System Theory Criterion for Positive Real Matrices. SIAM Journal on Control vol. 5 171–182 (1967) – 10.1137/0305011
Willems, J. Least squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on Automatic Control vol. 16 621–634 (1971) – 10.1109/tac.1971.1099831
Guiver, C. & Opmeer, M. R. Error bounds in the gap metric for dissipative balanced approximations. Linear Algebra and its Applications vol. 439 3659–3698 (2013) – 10.1016/j.laa.2013.09.032
Ober, R. Balanced Parametrization of Classes of Linear Systems. SIAM Journal on Control and Optimization vol. 29 1251–1287 (1991) – 10.1137/0329065
Beattie, C. A., Mehrmann, V. & Van Dooren, P. Robust port-Hamiltonian representations of passive systems. Automatica vol. 100 182–186 (2019) – 10.1016/j.automatica.2018.11.013
Mehl, C., Mehrmann, V. & Sharma, P. Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations. SIAM Journal on Matrix Analysis and Applications vol. 37 1625–1654 (2016) – 10.1137/16m1067330
Benhabib, R. J., Iwens, R. P. & Jackson, R. L. Stability of Large Space Structure Control Systems Using Positivity Concepts. Journal of Guidance and Control vol. 4 487–494 (1981) – 10.2514/3.56100
Haddad, Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle and Popov theorems and their application to robust stability. (1991)
Zames, G. Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Transactions on Automatic Control vol. 26 301–320 (1981) – 10.1109/tac.1981.1102603
Zhou, (1996)
Dullerud, (2013)
Stengel, (1994)
Mustafa, D. & Glover, K. Controller reduction by H/sub infinity /-balanced truncation. IEEE Transactions on Automatic Control vol. 36 668–682 (1991) – 10.1109/9.86941
Chen, S. Necessary and sufficient conditions for the existence of positive solutions to algebraic Riccati equations with indefinite quadratic term. Applied Mathematics & Optimization vol. 26 95–110 (1992) – 10.1007/bf01218397
Antoulas, (2005)
Breiten, Balancing-related model reduction methods. (2021)
Harshavardhana, Stochastic balancing and approximation - stability and minimality. (1983)
Meyer, D. & Franklin, G. A connection between normalized coprime factorizations and linear quadratic regulator theory. IEEE Transactions on Automatic Control vol. 32 227–228 (1987) – 10.1109/tac.1987.1104569