{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,9]],"date-time":"2026-09-09T09:06:16Z","timestamp":1788944776017,"version":"build-2803163510"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T00:00:00Z","timestamp":1788739200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper aims to describe a synthesis procedure for discrete-time, energy-based regulators for continuous-time port-Hamiltonian systems. The methodology consists of three steps. The first deals with the definition of a discrete-time approximation of the plant, which is subsequently employed in the development of the control law. The discrete-time model is obtained from the continuous-time dynamics by replacing the gradient of the Hamiltonian function with a discrete gradient. In this way, passivity, with the energy as storage function, is preserved, although the resulting state equation is in implicit form. The second step concerns the control synthesis and extends the continuous-time energy-shaping plus damping injection design technique to the proposed class of discrete-time port-Hamiltonian systems. Finally, the last step addresses the interconnection between the digital controller and the continuous-time plant. The coupling is implemented via a zero-order hold and relies on the solution of an optimization problem that determines the \u201cbest\u201d and \u201cminimal\u201d correction to be applied to the nominal control action in order to achieve the same performance as that obtained when the regulator is connected in closed loop with the discrete-time model of the plant. This is the reference scenario used to develop and tune the control law. The complete procedure (time discretisation, control design, and coupling implementation) is illustrated through an example.<\/jats:p>","DOI":"10.3390\/e28091002","type":"journal-article","created":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T16:30:02Z","timestamp":1788798602000},"page":"1002","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modeling and Control Design of Port-Hamiltonian Systems in Discrete-Time"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2258-9699","authenticated-orcid":false,"given":"Alessandro","family":"Macchelli","sequence":"first","affiliation":[{"name":"Department of Electrical, Electronic, and Information Engineering\u2014\u201cGuglielmo Marconi\u201d (DEI), University of Bologna, viale del Risorgimento 2, 40136 Bologna, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,9,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"van der Schaft, A. (2017). L2-Gain and Passivity Techniques in Nonlinear Control, Springer International Publishing AG. [3rd ed.]. Communication and Control Engineering.","DOI":"10.1007\/978-3-319-49992-5"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1109\/37.915398","article-title":"Putting energy back in control","volume":"21","author":"Ortega","year":"2001","journal-title":"Control Syst. Mag. IEEE"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/S0005-1098(01)00278-3","article-title":"Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems","volume":"38","author":"Ortega","year":"2002","journal-title":"Automatica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/S0167-6911(00)00091-8","article-title":"Canonical transformation and stabilization of generalized Hamiltonian systems","volume":"42","author":"Fujimoto","year":"2001","journal-title":"Syst. Control Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2669","DOI":"10.1109\/TNS.1983.4332919","article-title":"A Canonical Integration Technique","volume":"30","author":"Ruth","year":"1983","journal-title":"Nucl. Sci. IEEE Trans."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/0045-7825(96)01009-2","article-title":"On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry","volume":"134","author":"Gonzalez","year":"1996","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1017\/S096249290100006X","article-title":"Discrete mechanics and variational integrators","volume":"10","author":"Marsden","year":"2001","journal-title":"Acta Numer."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"104530","DOI":"10.1016\/j.sysconle.2019.104530","article-title":"Discrete-time port-Hamiltonian systems: A definition based on symplectic integration","volume":"113","author":"Kotyczka","year":"2019","journal-title":"Syst. Control Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"109842","DOI":"10.1016\/j.automatica.2021.109842","article-title":"Symplectic discrete-time energy-based control for nonlinear mechanical systems","volume":"133","author":"Kotyczka","year":"2021","journal-title":"Automatica"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/BF02440162","article-title":"Time integration and discrete Hamiltonian systems","volume":"6","author":"Gonzalez","year":"1996","journal-title":"J. Nonlinear Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"L341","DOI":"10.1088\/0305-4470\/29\/13\/006","article-title":"Discrete gradient methods for solving ODEs numerically while preserving a first integral","volume":"29","author":"Quispel","year":"1996","journal-title":"J. Phys. Math. Gen."},{"key":"ref_12","unstructured":"G\u00f6ren-S\u00fcmer, L., and Yal\u00e7in, Y. (2008, January 6\u201311). Gradient Based Discrete-Time Modeling and Control of Hamiltonian Systems. Proceedings of the 17th IFAC World Congress, Seoul, Repulic of Korea."},{"key":"ref_13","unstructured":"Monaco, S., Normand-Cyrot, D., and Tiefensee, F. (2010, January 15\u201317). Nonlinear port controlled Hamiltonian systems under sampling. Proceedings of the 49th IEEE Conference on Decision and Control (CDC 2010), Atlanta, GA, USA."},{"key":"ref_14","unstructured":"Celledoni, E., and H\u00f8iseth, E. (2017). Energy-Preserving and Passivity-Consistent Numerical Discretization of Port-Hamiltonian Systems. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.sysconle.2017.10.003","article-title":"Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability","volume":"110","author":"Aoues","year":"2017","journal-title":"Syst. Control Lett."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Moreschini, A., Mattioni, M., Monaco, S., and Normand-Cyrot, D. (2019, January 11\u201313). Discrete port-controlled Hamiltonian dynamics and average passivation. Proceedings of the IEEE 58th Annual Conference on Decision and Control (CDC), Nice, France.","DOI":"10.1109\/CDC40024.2019.9029809"},{"key":"ref_17","unstructured":"Moreschini, A., Monaco, S., and Normand-Cyrot, D. (2019, January 4\u20136). Gradient and Hamiltonian dynamics under sampling. Proceedings of the 11th IFAC Symposium on Nonlinear Control Systems, NOLCOS 2019, Vienna, Austria."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1109\/LCSYS.2020.3000705","article-title":"Stabilization of discrete port-Hamiltonian dynamics via interconnection and damping assignment","volume":"5","author":"Moreschini","year":"2021","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3146","DOI":"10.1109\/LCSYS.2022.3182845","article-title":"Trajectory Tracking for Discrete-Time Port-Hamiltonian Systems","volume":"6","author":"Macchelli","year":"2022","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"8224","DOI":"10.1109\/TAC.2023.3292180","article-title":"Control Design for a Class of Discrete-Time Port-Hamiltonian Systems","volume":"68","author":"Macchelli","year":"2023","journal-title":"Autom. Control. IEEE Trans."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1109\/LCSYS.2024.3404769","article-title":"A Discrete-Time Formulation of Nonlinear Distributed-Parameter Port-Hamiltonian Systems","volume":"8","author":"Macchelli","year":"2024","journal-title":"IEEE Control Syst. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"722","DOI":"10.1109\/TAC.2025.3593062","article-title":"Port-Hamiltonian Boundary Control Systems in Discrete-Time. Modelling and Control Design","volume":"71","author":"Macchelli","year":"2026","journal-title":"Autom. Control. IEEE Trans."},{"key":"ref_23","unstructured":"G\u00f6ren-S\u00fcmer, L., and Yal\u00e7in, Y. (September, January 29). A Direct Discrete-time IDA-PBC Design Method for a Class of Underactuated Hamiltonian Systems. Proceedings of the 18th IFAC World Congress, Milan, Italy."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Aoues, S., Eberard, D., and Marquis-Favre, W. (2015, January 15\u201318). Discrete IDA-PBC control law for Newtonian mechanical port-Hamiltonian systems. Proceedings of the IEEE 54th Annual Conference on Decision and Control (CDC), Osaka, Japan.","DOI":"10.1109\/CDC.2015.7402904"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"574","DOI":"10.1109\/TRO.2004.842330","article-title":"Sampled Data Systems Passivity and Discrete Port-Hamiltonian Systems","volume":"21","author":"Stramigioli","year":"2005","journal-title":"Robot. IEEE Trans."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"583","DOI":"10.3166\/ejc.13.583-590","article-title":"On Preserving Passivity in Sampled-data Linear Systems","volume":"13","author":"Fossas","year":"2007","journal-title":"Eur. J. Control"},{"key":"ref_27","unstructured":"Macchelli, A. (2023, January 9\u201314). On the Synthesis of Discrete-time Energy-based Regulators for Port-Hamiltonian Systems. Proceedings of the 22nd IFAC World Congress, Yokohama, Japan."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1137\/1025002","article-title":"On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws","volume":"25","author":"Harten","year":"1983","journal-title":"SIAM Rev."},{"key":"ref_29","unstructured":"Ehrhardt, M., Riis, E., Ringholm, T., and Sch\u00f6nlieb, C.B. (2020). A geometric integration approach to smooth optimisation: Foundations of the discrete gradient method. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2350","DOI":"10.1109\/TAC.2015.2400663","article-title":"Constructive Interconnection and Assignment for Port-Controlled Hamiltonian Systems","volume":"60","author":"Nunna","year":"2015","journal-title":"Autom. Control. IEEE Trans."},{"key":"ref_31","unstructured":"Khalil, H. (1996). Nonlinear Systems, Prentice Hall. [2nd ed.]."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2376","DOI":"10.1109\/TCST.2024.3386062","article-title":"A Sparse ADMM-Based Solver for Linear MPC Subject to Terminal Quadratic Constraint","volume":"32","author":"Krupa","year":"2024","journal-title":"Control Syst. Technol. IEEE Trans."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/9\/1002\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,9]],"date-time":"2026-09-09T08:08:13Z","timestamp":1788941293000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/9\/1002"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,7]]},"references-count":32,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,9]]}},"alternative-id":["e28091002"],"URL":"https:\/\/doi.org\/10.3390\/e28091002","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,7]]}}}