Stochastic Port-Hamiltonian Systems

Authors

Francesco Cordoni, Luca Di Persio, Riccardo Muradore

Abstract

In the present work we formally extend the theory of port-Hamiltonian systems to include random perturbations. In particular, suitably choosing the space of flow and effort variables we will show how several elements coming from possibly different physical domains can be interconnected in order to describe a dynamic system perturbed by general continuous semimartingale. Relevant enough, the noise does not enter into the system solely as an external random perturbation, since each port is itself intrinsically stochastic. Coherently to the classical deterministic setting, we will show how such an approach extends existing literature of stochastic Hamiltonian systems on pseudo-Poisson and pre-symplectic manifolds. Moreover, we will prove that a power-preserving interconnection of stochastic port-Hamiltonian systems is a stochastic port-Hamiltonian system as well.

Keywords

Stochastic geometric mechanics; Port-Hamiltonian systems; Stochastic equations on manifold; Dirac manifold; 34G20; 34F05; 37N35

Citation

• Journal: Journal of Nonlinear Science
• Year: 2022
• Volume: 32
• Issue: 6
• Pages:
• Publisher: Springer Science and Business Media LLC
• DOI: 10.1007/s00332-022-09853-2

BibTeX

@article{Cordoni_2022,
  title={{Stochastic Port-Hamiltonian Systems}},
  volume={32},
  ISSN={1432-1467},
  DOI={10.1007/s00332-022-09853-2},
  number={6},
  journal={Journal of Nonlinear Science},
  publisher={Springer Science and Business Media LLC},
  author={Cordoni, Francesco and Di Persio, Luca and Muradore, Riccardo},
  year={2022}
}

Download the bib file

References

• Armstrong, J., Brigo, D.: Intrinsic stochastic differential equations as jets. Proc. R. Soc. A Math. Phys. Eng. Sci. 474(2210), 20170559 (2018)
• Barbu V, Cordoni F, Persio LD (2015) Optimal control of stochastic FitzHugh–Nagumo equation. International Journal of Control 89(4):746–756. https://doi.org/10.1080/00207179.2015.1096023 – 10.1080/00207179.2015.1096023
• Bessaih H, Flandoli F (1999) 2-D Euler equation perturbed by noise. NoDEA : Nonlinear Differential Equations and Applications 6(1):35–54. https://doi.org/10.1007/s000300050063 – 10.1007/s000300050063
• Bismut JM (1982) Mecanique aleatoire. In: Lecture Notes in Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 1–100 – 10.1007/bfb0095618
• Cordoni, F., Di Persio, L.: Small noise asymptotic expansion for a infinite dimensional stochastic reaction-diffusion forced van der pol equation. Int. J. Math. Models Method Appl. Sci., 9, 43–49 (2015)
• Cordoni F, Persio LD (2018) Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable. EECT 7(4):571–585. https://doi.org/10.3934/eect.2018027 – 10.3934/eect.2018027
• Cordoni F, Persio LD, Muradore R (2020) A variable stochastic admittance control framework with energy tank. IFAC-PapersOnLine 53(2):9986–9991. https://doi.org/10.1016/j.ifacol.2020.12.2716 – 10.1016/j.ifacol.2020.12.2716
• Cordoni F, Di Persio L, Muradore R (2021) Bilateral teleoperation of stochastic port‐Hamiltonian systems using energy tanks. Intl J Robust & Nonlinear 31(18):9332–9357. https://doi.org/10.1002/rnc.5780 – 10.1002/rnc.5780
• Cordoni F, Di Persio L, Muradore R (2021) Stabilization of bilateral teleoperators with asymmetric stochastic delay. Systems & Control Letters 147:104828. https://doi.org/10.1016/j.sysconle.2020.104828 – 10.1016/j.sysconle.2020.104828
• Cordoni FG, Di Persio L, Muradore R (2022) Discrete stochastic port-Hamiltonian systems. Automatica 137:110122. https://doi.org/10.1016/j.automatica.2021.110122 – 10.1016/j.automatica.2021.110122
• Cordoni FG, Di Persio L, Muradore R (2022) Discrete stochastic port-Hamiltonian systems. Automatica 137:110122. https://doi.org/10.1016/j.automatica.2021.110122 – 10.1016/j.automatica.2021.110122
• Courant TJ (1990) Dirac manifolds. Trans Amer Math Soc 319(2):631–661. https://doi.org/10.1090/s0002-9947-1990-0998124-1 – 10.1090/s0002-9947-1990-0998124-1
• Dalsmo M, van der Schaft AJ (1997) A Hamiltonian framework for interconnected physical systems. In: 1997 European Control Conference (ECC). IEEE, pp 2792–2797 – 10.23919/ecc.1997.7082532
• Dalsmo M, van der Schaft A (1998) On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM J Control Optim 37(1):54–91. https://doi.org/10.1137/s0363012996312039 – 10.1137/s0363012996312039
• de Wit, C.C., Siciliano, B., Bastin, G.: Theory of Robot Control. Springer Science & Business Media, Berlin (2012)
• Elworthy KD (1982) Stochastic Differential Equations on Manifolds. Cambridge University Press – 10.1017/cbo9781107325609
• Emery, M.: An invitation to second-order stochastic differential geometry (2007) https://hal.archives-ouvertes.fr/hal-00145073/
• Émery, M.: Stochastic Calculus in Manifolds. Springer Science & Business Media, Berlin (2012)
• Eyink GL (2001) Dissipation in turbulent solutions of 2D Euler equations. Nonlinearity 14(4):787–802. https://doi.org/10.1088/0951-7715/14/4/307 – 10.1088/0951-7715/14/4/307
• Flandoli F (2018) Weak vorticity formulation of 2D Euler equations with white noise initial condition. Communications in Partial Differential Equations 43(7):1102–1149. https://doi.org/10.1080/03605302.2018.1467448 – 10.1080/03605302.2018.1467448
• Gay-Balmaz F, Ratiu TS (2008) Affine Lie–Poisson reduction, Yang–Mills magnetohydrodynamics, and superfluids. J Phys A: Math Theor 41(34):344007. https://doi.org/10.1088/1751-8113/41/34/344007 – 10.1088/1751-8113/41/34/344007
• Gay-Balmaz F, Yoshimura H (2018) Dirac structures in nonequilibrium thermodynamics. Journal of Mathematical Physics 59(1):012701. https://doi.org/10.1063/1.5017223 – 10.1063/1.5017223
• Gay-Balmaz F, Yoshimura H (2020) Dirac structures in nonequilibrium thermodynamics for simple open systems. Journal of Mathematical Physics 61(9):092701. https://doi.org/10.1063/1.5120390 – 10.1063/1.5120390
• Haddad WM, Rajpurohit T, Jin X (2018) Energy-based feedback control for stochastic port-controlled Hamiltonian systems. Automatica 97:134–142. https://doi.org/10.1016/j.automatica.2018.07.031 – 10.1016/j.automatica.2018.07.031
• Holm DD (2008) Geometric Mechanics. IMPERIAL COLLEGE PRESS – 10.1142/p557
• Holm DD (2008) Geometric Mechanics. IMPERIAL COLLEGE PRESS – 10.1142/p549
• Holm, D.D.: Applications of poisson geometry to physical problems. Geom. Topol. Monogr 17, 221–384 (2011)
• Holm, D.D.: Variational principles for stochastic fluid dynamics. Proc. R. Soc. Math. Phys. Eng. Sci. 471(2176), 20140963 (2015)
• Holm DD, Schmah T, Stoica C, Ellis DCP (2009) Geometric Mechanics and Symmetry. Oxford University PressOxford – 10.1093/oso/9780199212903.001.0001
• Holm, D.D., Tyranowski, T.M.: Variational principles for stochastic soliton dynamics. Proc. R. Soc. Math. Phys. Eng. Sci. 472(2187), 20150827 (2016)
• Hsu, E.P.: Stochastic Analysis on Manifolds, vol. 38. American Mathematical Society, Ann Arbor (2002)
• Leung TP, Qin HS (2001) Advanced Topics in Nonlinear Control Systems. WORLD SCIENTIFIC – 10.1142/4541
• Lázaro-Camí J-A, Ortega J-P (2008) Stochastic hamiltonian dynamical systems. Reports on Mathematical Physics 61(1):65–122. https://doi.org/10.1016/s0034-4877(08)80003-1 – 10.1016/s0034-4877(08)80003-1
• Meyer PA (1981) Geometrie stochastique sans larmes. In: Lecture Notes in Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 44–102 – 10.1007/bfb0088360
• Morselli R, Zanasi R (2008) Control of port Hamiltonian systems by dissipative devices and its application to improve the semi-active suspension behaviour. Mechatronics 18(7):364–369. https://doi.org/10.1016/j.mechatronics.2008.05.008 – 10.1016/j.mechatronics.2008.05.008
• Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications. Springer Science & Business Media, Berlin (2013)
• Olver, P.J.: Applications of Lie Groups to Differential Equations, vol. 107. Springer Science & Business Media, Berlin (2000)
• Ortega J-P, Planas-Bielsa V (2004) Dynamics on Leibniz manifolds. Journal of Geometry and Physics 52(1):1–27. https://doi.org/10.1016/j.geomphys.2004.01.002 – 10.1016/j.geomphys.2004.01.002
• Ortega R, van der Schaft A, Maschke B, Escobar G (2002) Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38(4):585–596. https://doi.org/10.1016/s0005-1098(01)00278-3 – 10.1016/s0005-1098(01)00278-3
• Protter PE (2005) Stochastic Differential Equations. In: Stochastic Modelling and Applied Probability. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 249–361 – 10.1007/978-3-662-10061-5_6
• Ramirez H, Maschke B, Sbarbaro D (2013) Irreversible port-Hamiltonian systems: A general formulation of irreversible processes with application to the CSTR. Chemical Engineering Science 89:223–234. https://doi.org/10.1016/j.ces.2012.12.002 – 10.1016/j.ces.2012.12.002
• Satoh S (2017) Input‐to‐state stability of stochastic port‐Hamiltonian systems using stochastic generalized canonical transformations. Intl J Robust & Nonlinear 27(17):3862–3885. https://doi.org/10.1002/rnc.3769 – 10.1002/rnc.3769
• Satoh S, Fujimoto K (2010) Stabilization of Time-varying Stochastic Port-Hamiltonian Systems Based on Stochastic Passivity. IFAC Proceedings Volumes 43(14):611–616. https://doi.org/10.3182/20100901-3-it-2016.00057 – 10.3182/20100901-3-it-2016.00057
• Satoh S, Fujimoto K (2013) Passivity Based Control of Stochastic Port-Hamiltonian Systems. IEEE Trans Automat Contr 58(5):1139–1153. https://doi.org/10.1109/tac.2012.2229791 – 10.1109/tac.2012.2229791
• Satoh S, Saeki M (2014) Bounded stabilisation of stochastic port-Hamiltonian systems. International Journal of Control 87(8):1573–1582. https://doi.org/10.1080/00207179.2014.880127 – 10.1080/00207179.2014.880127
• Schwartz L (1982) Geometrie differentielle du 2ème ordre, semi-martingales et equations differentielles stochastiques sur une variete differentielle. In: Lecture Notes in Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 1–148 – 10.1007/bfb0092647
• Secchi, C., Stramigioli, S., Fantuzzi, C.: Control of Interactive Robotic Interfaces: A Port-Hamiltonian Approach, vol. 29. Springer Science & Business Media, Berlin (2007)
• Tabuada P, Pappas GJ (2003) Abstractions of Hamiltonian control systems. Automatica 39(12):2025–2033. https://doi.org/10.1016/s0005-1098(03)00235-8 – 10.1016/s0005-1098(03)00235-8
• Tsionas EG (2002) Stochastic frontier models with random coefficients. J of Applied Econometrics 17(2):127–147. https://doi.org/10.1002/jae.637 – 10.1002/jae.637
• Vaisman, I.: Lectures on the Geometry of Poisson Manifolds, vol 118. Birkhäuser (2012)
• van der Schaft A, Jeltsema D (2014) Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control 1(2-3):173–378. https://doi.org/10.1561/2600000002 – 10.1561/2600000002
• van der Schaft AJ, Maschke BM (2002) Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics 42(1-2):166–194. https://doi.org/10.1016/s0393-0440(01)00083-3 – 10.1016/s0393-0440(01)00083-3
• van der Schaft A (2000) L2 - Gain and Passivity Techniques in Nonlinear Control. Springer London, London – 10.1007/978-1-4471-0507-7
• Yu, H., Yu, J., Liu, J., Wang, Y.: Energy-shaping and l2 gain disturbance attenuation control of induction motor. Int. J. Innov. Comput. Inf. Control 8(7), 5011–5024 (2012)