Optimal robustness of passive discrete-time systems

Authors

V Mehrmann, P Van Dooren

Abstract

We study different representations of a given rational transfer function that represents a passive (or positive real) discrete-time system. When the system is subject to perturbations, passivity or stability may be lost. To make the system robust, we use the freedom in the representation to characterize and construct optimally robust representations in the sense that the distance to non-passivity is maximized with respect to an appropriate matrix norm. We link this construction to the solution set of certain linear matrix inequalities defining passivity of the transfer function. We present an algorithm to compute a nearly optimal representation using an eigenvalue optimization technique. We also briefly consider the problem of finding the nearest passive system to a given non-passive one.

Citation

• Journal: IMA Journal of Mathematical Control and Information
• Year: 2020
• Volume: 37
• Issue: 4
• Pages: 1248–1269
• Publisher: Oxford University Press (OUP)
• DOI: 10.1093/imamci/dnaa013

BibTeX

@article{Mehrmann_2020,
  title={{Optimal robustness of passive discrete-time systems}},
  volume={37},
  ISSN={1471-6887},
  DOI={10.1093/imamci/dnaa013},
  number={4},
  journal={IMA Journal of Mathematical Control and Information},
  publisher={Oxford University Press (OUP)},
  author={Mehrmann, V and Van Dooren, P},
  year={2020},
  pages={1248--1269}
}

Download the bib file

References

• Bankmann, Computation of the analytic center of the solution set of the linear matrix inequality arising in continuous- and discrete-time passivity analysis. (2019)
• 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
• Benner, P. & Mitchell, T. Faster and More Accurate Computation of the $\mathcal{H}_\infty$ Norm via Optimization. SIAM Journal on Scientific Computing vol. 40 A3609–A3635 (2018) – 10.1137/17m1137966
• Boyd, S. & Balakrishnan, V. A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L∞-norm. Systems & Control Letters vol. 15 1–7 (1990) – 10.1016/0167-6911(90)90037-u
• Boyd, S., El Ghaoui, L., Feron, E. & Balakrishnan, V. Linear Matrix Inequalities in System and Control Theory. (1994) doi:10.1137/1.9781611970777 – 10.1137/1.9781611970777
• Byers, Symplectic, BVD, and palindromic eigenvalue problems and their relation to discrete-time control problems. Collection of Papers Dedicated to the 60-th Anniversary of Mihail Konstantinov (2009)
• Freiling, G., Mehrmann, V. & Xu, H. Existence, Uniqueness, and Parametrization of Lagrangian Invariant Subspaces. SIAM Journal on Matrix Analysis and Applications vol. 23 1045–1069 (2002) – 10.1137/s0895479800377228
• Gillis, N., Mehrmann, V. & Sharma, P. Computing the nearest stable matrix pairs. Numerical Linear Algebra with Applications vol. 25 (2018) – 10.1002/nla.2153
• 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
• Higham, N. J. Computing the Polar Decomposition—with Applications. SIAM Journal on Scientific and Statistical Computing vol. 7 1160–1174 (1986) – 10.1137/0907079
• Kotyczka, Discrete-time port-Hamiltonian systems based on Gauss–Legendre collocation. Syst. Control Lett.
• Kressner, D. & Vandereycken, B. Subspace Methods for Computing the Pseudospectral Abscissa and the Stability Radius. SIAM Journal on Matrix Analysis and Applications vol. 35 292–313 (2014) – 10.1137/120869432
• Lancaster, The Theory of Matrices (1985)
• Mehrmann, The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution. Volume 163 of Lecture Notes in Control and Inform Sci (1991)
• Mehrmann, V. & Van Dooren, P. M. Optimal Robustness of Port-Hamiltonian Systems. SIAM Journal on Matrix Analysis and Applications vol. 41 134–151 (2020) – 10.1137/19m1259092
• Overton, On computing the complex passivity radius. 44th IEEE Conference on Decision and Control, and the European Control Conference (2005)
• Prajna, S., van der Schaft, A. & Meinsma, G. An LMI approach to stabilization of linear port-controlled Hamiltonian systems. Systems & Control Letters vol. 45 371–385 (2002) – 10.1016/s0167-6911(01)00195-5
• Seslija, Port-Hamiltonian systems on discrete manifolds. IFAC Proceedings (2012)
• Talasila, V., Clemente-Gallardo, J. & van der Schaft, A. J. Discrete port-Hamiltonian systems. Systems & Control Letters vol. 55 478–486 (2006) – 10.1016/j.sysconle.2005.10.001
• 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
• Willems, J. C. Dissipative dynamical systems part I: General theory. Archive for Rational Mechanics and Analysis vol. 45 321–351 (1972) – 10.1007/bf00276493
• 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