Authors

Mohamed A. Mabrok

Abstract

In this paper, we consider the problem of robust stabilization of linear time-invariant systems with respect to unmodeled dynamics and structure uncertainties. To that end, we first present a methodology to find the nearest negative imaginary system for a given non-negative imaginary system. Then, we employ this result to construct a near optimal linear quadratic Gaussian controller achieving desired performance measures. The problem is formulated using port-Hamiltonian method and the required conditions are defined in terms of linear matrix inequalities. The technique is presented using fast gradient method to solve the problem systematically. The designed controller satisfies a negative imaginary property and guarantees a robust feedback loop. The effectiveness of the approach is demonstrated by simulation on a numerical example.

Citation

BibTeX

@inproceedings{Mabrok_2023,
  title={{Finding the Nearest Negative Imaginary System with Application to Near-Optimal Controller Design}},
  DOI={10.23919/ecc57647.2023.10178307},
  booktitle={{2023 European Control Conference (ECC)}},
  publisher={IEEE},
  author={Mabrok, Mohamed A.},
  year={2023},
  pages={1--6}
}

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References

  • Mahmood, I. A., Moheimani, S. O. R. & Bhikkaji, B. A New Scanning Method for Fast Atomic Force Microscopy. IEEE Transactions on Nanotechnology vol. 10 203–216 (2011) – 10.1109/tnano.2009.2036844
  • Xiong, J., Petersen, I. R. & Lanzon, A. A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems. IEEE Transactions on Automatic Control vol. 55 2342–2347 (2010) – 10.1109/tac.2010.2052711
  • Mabrok, M. A. & Petersen, I. R. Controller synthesis for negative imaginary systems: a data driven approach. IET Control Theory & Applications vol. 10 1480–1486 (2016) – 10.1049/iet-cta.2015.0800
  • Wilson, D. G., Robinett, III, R. D., Parker, G. G. & Starr, G. P. Journal of Intelligent and Robotic Systems vol. 34 415–430 (2002) – 10.1023/a:1019635709331
  • Feedback Control of Negative-Imaginary Systems. IEEE Control Systems vol. 30 54–72 (2010) – 10.1109/mcs.2010.937676
  • nesterov, A method of solving a convex programming problem with convergence rate o(1/k2). Soviet Mathematics Doklady (1983)
  • van der Schaft, A. Interconnections of input-output Hamiltonian systems with dissipation. 2016 IEEE 55th Conference on Decision and Control (CDC) 4686–4691 (2016) doi:10.1109/cdc.2016.7798983 – 10.1109/cdc.2016.7798983
  • Mabrok, M. A., Lanzon, A., Kallapur, A. G. & Petersen, I. R. Enforcing negative imaginary dynamics on mathematical system models. International Journal of Control vol. 86 1292–1303 (2013) – 10.1080/00207179.2013.804951
  • wang, PEDS:Passivity enforcement for descriptor systems via Hamiltonian-symplectic matrix pencil perturbation. (2010)
  • mabrok, Stabilization of conditional uncertain negative-imaginary systems using riccati equation approach. Proceedings of the 20th International Symposium on Mathematical Theory of Networks and Systems (MTNS) (2012)
  • voigt, Passivity enforcement of descriptor systems via structured perturbation of Hamiltonian matrix pencils. Talk at Meeting of the GAMM Activity Group Dynamics and Control Theory (2011)
  • Brull, T. & Schroder, C. Dissipativity Enforcement via Perturbation of Para-Hermitian Pencils. IEEE Transactions on Circuits and Systems I: Regular Papers vol. 60 164–177 (2013) – 10.1109/tcsi.2012.2215731
  • Bhikkaji, B. & Moheimani, S. O. R. Fast scanning using piezoelectric tube nanopositioners: A negative imaginary approach. 2009 IEEE/ASME International Conference on Advanced Intelligent Mechatronics 274–279 (2009) doi:10.1109/aim.2009.5230001 – 10.1109/aim.2009.5230001
  • schröder, Passivation of LTI systems. (2007)
  • Preumont, A. Vibration Control of Active Structures. Solid Mechanics and Its Applications (Springer Netherlands, 2011). doi:10.1007/978-94-007-2033-6 – 10.1007/978-94-007-2033-6
  • van der Schaft, A. J. Positive feedback interconnection of Hamiltonian systems. IEEE Conference on Decision and Control and European Control Conference 6510–6515 (2011) doi:10.1109/cdc.2011.6160395 – 10.1109/cdc.2011.6160395
  • Tran, V. P., Garratt, M. & Petersen, I. R. Formation control of multi-UAVs using negative-imaginary systems theory. 2017 11th Asian Control Conference (ASCC) 2031–2036 (2017) doi:10.1109/ascc.2017.8287487 – 10.1109/ascc.2017.8287487
  • Mabrok, M. A. Controller synthesis for negative imaginary systems using nonlinear optimisation and H2 performance measure. International Journal of Control vol. 94 579–587 (2019) – 10.1080/00207179.2019.1601773
  • Lanzon, A. & Petersen, I. R. Stability Robustness of a Feedback Interconnection of Systems With Negative Imaginary Frequency Response. IEEE Transactions on Automatic Control vol. 53 1042–1046 (2008) – 10.1109/tac.2008.919567
  • Mabrok, M. A., Kallapur, A. G., Petersen, I. R. & Lanzon, A. Generalizing Negative Imaginary Systems Theory to Include Free Body Dynamics: Control of Highly Resonant Structures With Free Body Motion. IEEE Transactions on Automatic Control vol. 59 2692–2707 (2014) – 10.1109/tac.2014.2325692
  • Fazzi, A., Guglielmi, N. & Lubich, C. Finding the Nearest Passive or Nonpassive System via Hamiltonian Eigenvalue Optimization. SIAM Journal on Matrix Analysis and Applications vol. 42 1553–1580 (2021) – 10.1137/20m1376972
  • Díaz, I. M., Pereira, E. & Reynolds, P. Integral resonant control scheme for cancelling human-induced vibrations in light-weight pedestrian structures. Structural Control and Health Monitoring vol. 19 55–69 (2010) – 10.1002/stc.423
  • Gillis, N. & Sharma, P. Finding the Nearest Positive-Real System. SIAM Journal on Numerical Analysis vol. 56 1022–1047 (2018) – 10.1137/17m1137176
  • Gillis, N. & Sharma, P. On computing the distance to stability for matrices using linear dissipative Hamiltonian systems. Automatica vol. 85 113–121 (2017) – 10.1016/j.automatica.2017.07.047