The difference between port-Hamiltonian, passive and positive real descriptor systems

Karim Cherifi1, Hannes Gernandt2, Dorothea Hinsen1
1Institut für Mathematik, Technische Universität Berlin, Berlin, Germany
2Fraunhofer IEG, Fraunhofer Research Institution for Energy Infrastructures and Geothermal Systems IEG, Cottbus, Cottbus, Germany

Tóm tắt

The relation between passive and positive real systems has been extensively studied in the literature. In this paper, we study their connection to the more recently used notion of port-Hamiltonian descriptor systems. It is well-known that port-Hamiltonian systems are passive and that passive systems are positive real. Hence it is studied under which assumptions the converse implications hold. Furthermore, the relationship between passivity and KYP inequalities is investigated.

Tài liệu tham khảo

Beattie C, Mehrmann V, Xu H, Zwart H (2018) Linear port-Hamiltonian descriptor systems. Math Control Signals Syst 30. https://doi.org/10.1007/s00498-018-0223-3 Beattie C, Mehrmann V, Dooren PV (2019) Robust port-Hamiltonian representations of passive systems. Automatica 100:182–186 Jacob B, Zwart H (2012) Linear Port-Hamiltonian systems on Infinite-dimensional Spaces. Operator Theory: Advances and Applications, 223. Birkhäuser/Springer Basel AG, Basel Mehrmann V, Unger B (2023) Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numer 32:395–515. https://doi.org/10.1017/S0962492922000083 Ortega R, van der Schaft AJ, Mareels Y, Maschke BM (2001) Putting energy back in control. Control Syst Mag 21:18–33 van der Schaft AJ (2004) Port-Hamiltonian systems: network modeling and control of nonlinear physical systems. In: Advanced dynamics and control of structures and machines. CISM Courses and Lectures, vol 444. Springer, New York van der Schaft AJ, Jeltsema D (2014) Port-Hamiltonian systems theory: an introductory overview. Found Trends Syst Control 1(2–3):173–378 Mehl C, Mehrmann V, Sharma P (2016) Stability radii for linear Hamiltonian systems with dissipation under structure-preserving perturbations. SIAM J Matrix Anal Appl 37(4):1625–1654 Mehrmann V, Van Dooren P (2020) Optimal robustness of port-Hamiltonian systems. SIAM J Matrix Anal Appl 41(1):134–151. https://doi.org/10.1137/19M1259092 Mehl C, Mehrmann V, Wojtylak M (2021) Distance problems for dissipative Hamiltonian systems and related matrix polynomials. Linear Algebra Appl 623:335–366. https://doi.org/10.1016/j.laa.2020.05.026 Mehrmann V, van der Schaft AJ (2022) Differential-algebraic systems with dissipative Hamiltonian structure. arXiv:2208.02737 van der Schaft AJ, Maschke B (2018) Generalized port-Hamiltonian DAE systems. Syst Control Lett 121:31–37 Gernandt H, Haller F, Reis T (2021) A linear relation approach to port-Hamiltonian differential-algebraic equations. SIAM J Matrix Anal Appl 42:1011–1044 Morandin R, Mehrmann V (2019) Structure-preserving discretization for port-Hamiltonian descriptor systems. In: IEEE 58th conference on decision and control (CDC), 6863–6868 Reis T, Voigt M (2019) Linear-quadratic optimal control of differential-algebraic systems: the infinite time horizon problem with zero terminal state. SIAM J Control Optim 57:1567–1596 Brogliato B, Lozano R, Maschke B, Egeland O (2007) Dissipative systems analysis and control. Springer, Cham Freund R, Jarre F (2004) An extension of the positive real lemma to descriptor systems. Optim Methods Softw 19(1):69–87. https://doi.org/10.1080/10556780410001654232 Zhang L, Lam J, Xu S (2002) On positive realness of descriptor systems. IEEE Trans Circuits Syst-I: Fundam Theory Appl 49(3):401–407 Anderson BDO (1967) A system theory criterion for positive real matrices. J Control 5(2):171–182 Anderson BDO, Vongpanitlerd S (1973) Network analysis and synthesis. Prentice-Hall Inc, Englewood Cliffs Dai L (1989) Singular Control Systems, vol 118. Lecture notes in control and information sciences. Springer, Berlin Verghese G, Lévy B, Kailath T (1981) A generalized state-space for singular systems. IEEE Trans Autom Control 26(4):811–831 van der Schaft AJ (2009) Port-Hamiltonian systems. In: Modeling and control of complex physical systems: the port-Hamiltonian approach. Springer, Netherlands. https://doi.org/10.1007/978-3-642-03196-0 Hughes TH, Branford EH (2022) Dissipativity, reciprocity, and passive network synthesis: from the seminal dissipative dynamical systems articles of Jan Willems to the present day. IEEE Control Syst Mag 42(3):36–57. https://doi.org/10.1109/MCS.2022.3157135 Camlibel MK, Frasca R (2009) Extension of Kalman–Yakubovich–Popov lemma to descriptor systems. Syst Control Lett 58(12):795–803. https://doi.org/10.1016/j.sysconle.2009.08.010 Masubuchi I (2006) Dissipativity inequalities for continuous-time descriptor systems with applications to synthesis of control gains. Syst Control Lett 55(2):158–164. https://doi.org/10.1016/j.sysconle.2005.06.007 Reis T, Rendel O, Voigt M (2015) The Kalman–Yakubovich–Popov inequality for differential-algebraic systems. Linear Algebra Appl 485:153–193. https://doi.org/10.1016/j.laa.2015.06.021 Reis T, Stykel T (2010) Positive real and bounded real balancing for model reduction of descriptor systems. Int J Control 83(1):74–88 Reis T, Voigt M (2015) The Kalman–Yakubovich–Popov inequality for differential-algebraic systems: existence of nonpositive solutions. Syst Control Lett 86:1–8. https://doi.org/10.1016/j.sysconle.2015.09.003 Cherifi K, Mehrmann V, Hariche K (2019) Numerical methods to compute a minimal realization of a port-Hamiltonian system. arXiv:1903.07042 Gillis N, Sharma P (2018) Finding the nearest positive-real system. SIAM J Numer Anal 56(2):1022–1047 Willems JC (1971) Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans Autom Control AC-16(6):621–634 Hughes TH (2017) A theory of passive linear systems with no assumptions. Autom J IFAC 86:87–97 Gernandt H, Haller FE (2021) On the stability of port-Hamiltonian descriptor systems. IFAC-Papers OnLine 54(19):137–142. https://doi.org/10.1016/j.ifacol.2021.11.068 Iwasaki T, Hara S (2005) Generalized KYP lemma: unified frequency domain inequalities with design applications. IEEE Trans Autom Control 50:41–59 Ferrante A (2005) Positive real lemma: necessary and sufficient conditions for the existence of solutions under virtually no assumptions. IEEE Trans Autom Control 50(5):720–724. https://doi.org/10.1109/TAC.2005.847036 Breiten T, Schulze P (2021) Structure-preserving linear quadratic Gaussian balanced truncation for port-Hamiltonian descriptor systems. arXiv:2111.05065v1 Camlibel M, Iannelli L, Vasca F (2014) Passivity and complementarity. Math Program 145(1–2):531–563. https://doi.org/10.1007/s10107-013-0678-4 Berger T, Reis T, Trenn S (2017) Observability of linear differential-algebraic systems: a survey. In: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations IV. Differential-algebraic equations forum, pp 161–219. Springer, Berlin. https://doi.org/10.1007/978-3-319-46618-7_4 Scherer R, Wendler W (1994) A generalization of the positive real lemma. IEEE Trans Autom Control 39(4):882–886 Xiao C, Hill DJ (1999) Generalizations and new proof of the discrete-time positive real lemma and bounded real lemma. IEEE Trans Circuits Syst-I: Fundam Theory Appl 46(6):740–743 Banaszuk A, Kociȩcki M, Lewis FL (1992) Kalman decomposition for implicit linear systems. IEEE Trans Autom Control 37:1509–1514 Bunse-Gerstner A, Byers R, Mehrmann V, Nichols NK (1999) Feedback design for regularizing descriptor systems. Linear Algebra Appl 299:119–151 Mehl C, Mehrmann V, Wojtylak M (2018) Linear algebra properties of dissipative Hamiltonian descriptor systems. SIAM J Matrix Anal Appl 39:1489–1519 Golub GH, Van Loan CF (2013) Matrix computations, 4th edn. The Johns Hopkins University Press, Baltimore Beattie C, Gugercin S, Mehrmann V (2019) Structure-preserving interpolatory model reduction for port-Hamiltonian differential-algebraic systems. arXiv:1910.05674 Benner P, Goyal P, Van Dooren P (2020) Identification of port-Hamiltonian systems from frequency response data. Syst Control Lett 143:104741. https://doi.org/10.1016/j.sysconle.2020.104741 van der Schaft AJ, Jeltsema D (2021) On energy conversion in port-Hamiltonian systems. In: 2021 60th IEEE conference on decision and control (CDC), pp 2421–2427. https://doi.org/10.1109/CDC45484.2021.9683292 Willems JC (1972) Dissipative dynamical systems—part 2: linear systems with quadratic supply rates. Soviet J Opt Technol (English translation of Optiko-Mekhanicheskaya Promyshlennost) 45(5):352–393 Ilchmann A, Reis T (2017) Outer transfer functions of differential-algebraic systems. ESAIM: COCV 23(2):391–425. https://doi.org/10.1051/cocv/2015051 Byers R, Geerts T, Mehrmann V (1997) Descriptor systems without controllability at infinity. SIAM J Control Optim 35(2):462–479. https://doi.org/10.1137/S0363012994269818