In domain dissipation assignment of boundary controlled Port-Hamiltonian systems using backstepping

Authors

Jeanne Redaud, Jean Auriol, Yann Le Gorrec

Abstract

In this paper, we develop a systematic approach to stabilize a general class of hyperbolic systems while assigning them a specified closed-loop behavior with a clear energy interpretation. More precisely, we address in-domain dissipation assignment for boundary-controlled Port Hamiltonian systems. The controller is designed so that the closed-loop system behaves like a target system with a specified energy decay rate. The PHS framework is used to take advantage of the natural physical properties of the system to define well-posed, exponentially stable, and easily parametrizable target system candidates, thus resulting in modular controllers. Under some generic structural assumptions, we rewrite the considered Port Hamiltonian system in the Riemann coordinates. The control approach is then based on the backstepping methodology. We combine classical Volterra transformations with an innovative time-affine transform to map the original system to the desired target system. The proposed approach is applied to two test cases: a clamped string and a clamped Timoshenko beam. Both are illustrated in numerical simulations.

Keywords

backstepping-based control design, distributed parameter systems, hyperbolic pde systems, port-hamiltonian systems

Citation

Journal: Systems & Control Letters
Year: 2024
Volume: 185
Issue:
Pages: 105722
Publisher: Elsevier BV
DOI: 10.1016/j.sysconle.2024.105722

BibTeX

@article{Redaud_2024,
  title={{In domain dissipation assignment of boundary controlled Port-Hamiltonian systems using backstepping}},
  volume={185},
  ISSN={0167-6911},
  DOI={10.1016/j.sysconle.2024.105722},
  journal={Systems \&amp; Control Letters},
  publisher={Elsevier BV},
  author={Redaud, Jeanne and Auriol, Jean and Gorrec, Yann Le},
  year={2024},
  pages={105722}
}

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References

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- – 10.1016/s0393-0440(01)00083-3
Le Gorrec, A semigroup approach to Port Hamiltonian systems associated with linear skew symmetric operator. (2004)
Le Gorrec Y, Zwart H, Maschke B (2005) Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators. SIAM J Control Optim 44(5):1864–1892. https://doi.org/10.1137/04061167 – 10.1137/040611677
Villegas, (2007)
Duindam, (2009)
Serban, Modelling and control of fluid power systems using the port-Hamiltonian approach. Int. J. Fluid Power (2011)
van der Schaft, Port-Hamiltonian formulation of distributed parameter systems: Convective transport and phase transitions. J. Math. Anal. Appl. (2002)
Wu, Port-Hamiltonian modeling and control of beam vibration. Mech. Syst. Signal Process. (2017)
Dunn, Optimization-based design of a high-performance hybrid vehicle powertrain using the port-Hamiltonian framework. IEEE Trans. Control Syst. Technol. (2016)
Augner B, Jacob B (2014) Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. EECT 3(2):207–229. https://doi.org/10.3934/eect.2014.3.20 – 10.3934/eect.2014.3.207
Bagherpour, Stability and energy-preserving boundary control of flexible mechanical systems with distributed parameters using the port-Hamiltonian approach. IEEE Trans. Control Syst. Technol. (2015)
Ortega R, van der Schaft A, Castanos F, Astolfi A (2008) Control by Interconnection and Standard Passivity-Based Control of Port-Hamiltonian Systems. IEEE Trans Automat Contr 53(11):2527–2542. https://doi.org/10.1109/tac.2008.200693 – 10.1109/tac.2008.2006930
Macchelli A, Le Gorrec Y, Ramirez H, Zwart H (2017) On the Synthesis of Boundary Control Laws for Distributed Port-Hamiltonian Systems. IEEE Trans Automat Contr 62(4):1700–1713. https://doi.org/10.1109/tac.2016.259526 – 10.1109/tac.2016.2595263
Zwart H, Le Gorrec Y, Maschke B, Villegas J (2009) Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain. ESAIM: COCV 16(4):1077–1093. https://doi.org/10.1051/cocv/200903 – 10.1051/cocv/2009036
Macchelli A, Le Gorrec Y, Ramirez H (2020) Exponential Stabilization of Port-Hamiltonian Boundary Control Systems via Energy Shaping. IEEE Trans Automat Contr 65(10):4440–4447. https://doi.org/10.1109/tac.2020.300479 – 10.1109/tac.2020.3004798
Krstic, (2008)
Anfinsen H, Aamo OM (2017) Adaptive Output Feedback Stabilization of $n + m$ Coupled Linear Hyperbolic PDEs with Uncertain Boundary Conditions. SIAM J Control Optim 55(6):3928–3946. https://doi.org/10.1137/16m109966 – 10.1137/16m1099662
Wang J, Pi Y, Krstic M (2018) Balancing and suppression of oscillations of tension and cage in dual-cable mining elevators. Automatica 98:223–238. https://doi.org/10.1016/j.automatica.2018.09.02 – 10.1016/j.automatica.2018.09.027
Redaud J, Auriol J, Niculescu S-I (2021) Output-feedback control of an underactuated network of interconnected hyperbolic PDE–ODE systems. Systems & Control Letters 154:104984. https://doi.org/10.1016/j.sysconle.2021.10498 – 10.1016/j.sysconle.2021.104984
Bou Saba D, Bribiesca-Argomedo F, Auriol J, Di Loreto M, Di Meglio F (2020) Stability Analysis for a Class of Linear $2 imes 2$ Hyperbolic PDEs Using a Backstepping Transform. IEEE Trans Automat Contr 65(7):2941–2956. https://doi.org/10.1109/tac.2019.293438 – 10.1109/tac.2019.2934384
Li T, Rao B (2010) Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems. Chin Ann Math Ser B 31(5):723–742. https://doi.org/10.1007/s11401-010-0600- – 10.1007/s11401-010-0600-9
Coron J-M, Hu L, Olive G (2017) Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation. Automatica 84:95–100. https://doi.org/10.1016/j.automatica.2017.05.01 – 10.1016/j.automatica.2017.05.013
Auriol J, Di Meglio F (2019) An explicit mapping from linear first order hyperbolic PDEs to difference systems. Systems & Control Letters 123:144–150. https://doi.org/10.1016/j.sysconle.2018.11.01 – 10.1016/j.sysconle.2018.11.012
Logemann H, Rebarber R, Weiss G (1996) Conditions for Robustness and Nonrobustness of the Stability of Feedback Systems with Respect to Small Delays in the Feedback Loop. SIAM J Control Optim 34(2):572–600. https://doi.org/10.1137/s036301299325070 – 10.1137/s0363012993250700
Auriol J, Di Meglio F (2020) Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs. Automatica 115:108896. https://doi.org/10.1016/j.automatica.2020.10889 – 10.1016/j.automatica.2020.108896
Chen, Rapid stabilization of Timoshenko beam by PDE backstepping. IEEE Trans. Automat. Control (2023)
Chen G, Vazquez R, Krstic M (2023) Backstepping-based Rapid Stabilization of Two-layer Timoshenko Composite Beams. IFAC-PapersOnLine 56(2):8159–8164. https://doi.org/10.1016/j.ifacol.2023.10.99 – 10.1016/j.ifacol.2023.10.992
Wei C, Li J (2023) Prescribed-time stabilization of uncertain heat equation via boundary time-varying feedback and disturbance estimator. Systems & Control Letters 171:105419. https://doi.org/10.1016/j.sysconle.2022.10541 – 10.1016/j.sysconle.2022.105419
Steeves, Input delay compensation in prescribed-time of boundary-actuated reaction-diffusion PDEs. (2021)
Espitia N, Polyakov A, Efimov D, Perruquetti W (2019) Boundary time-varying feedbacks for fixed-time stabilization of constant-parameter reaction–diffusion systems. Automatica 103:398–407. https://doi.org/10.1016/j.automatica.2019.02.01 – 10.1016/j.automatica.2019.02.013
Ramirez, On backstepping boundary control for a class of linear port-Hamiltonian systems. (2017)
Trang VU NM, LEFÈVRE L, NOUAILLETAS R (2015) Distributed and backstepping boundary controls to achieve IDA-PBC design. IFAC-PapersOnLine 48(1):482–487. https://doi.org/10.1016/j.ifacol.2015.05.03 – 10.1016/j.ifacol.2015.05.034
Redaud J, Auriol J, Gorrec YL (2022) Distributed Damping Assignment for a Wave Equation in the Port-Hamiltonian Framework. IFAC-PapersOnLine 55(26):155–161. https://doi.org/10.1016/j.ifacol.2022.10.39 – 10.1016/j.ifacol.2022.10.393
Redaud, In-domain damping assignment of a Timoshenko beam using state feedback boundary control. (2022)
Hu L, Vazquez R, Meglio FD, Krstic M (2019) Boundary Exponential Stabilization of 1-Dimensional Inhomogeneous Quasi-Linear Hyperbolic Systems. SIAM J Control Optim 57(2):963–998. https://doi.org/10.1137/15m101271 – 10.1137/15m1012712
Bastin, (2016)
Auriol J, Di Meglio F (2016) Minimum time control of heterodirectional linear coupled hyperbolic PDEs. Automatica 71:300–307. https://doi.org/10.1016/j.automatica.2016.05.03 – 10.1016/j.automatica.2016.05.030
Jacob, (2012)
Curtain, (2012)
Coron J-M, Vazquez R, Krstic M, Bastin G (2013) Local Exponential $H^2$ Stabilization of a $2 imes2$ Quasilinear Hyperbolic System Using Backstepping. SIAM J Control Optim 51(3):2005–2035. https://doi.org/10.1137/12087573 – 10.1137/120875739
Mattioni, A Lyapunov approach for the exponential stability of a damped Timoshenko beam. IEEE Trans. Automat. Control (2023)
Lamare, Robust output regulation of 2×2 hyperbolic systems: Control law and input-to-state stability. (2018)
Auriol J, Aarsnes UJF, Martin P, Meglio FD (2018) Delay-Robust Control Design for Two Heterodirectional Linear Coupled Hyperbolic PDEs. IEEE Trans Automat Contr 63(10):3551–3557. https://doi.org/10.1109/tac.2018.279881 – 10.1109/tac.2018.2798818
Auriol J, Bribiesca Argomedo F, Di Meglio F (2023) Robustification of stabilizing controllers for ODE-PDE-ODE systems: A filtering approach. Automatica 147:110724. https://doi.org/10.1016/j.automatica.2022.11072 – 10.1016/j.automatica.2022.110724
Di Meglio F, Argomedo FB, Hu L, Krstic M (2018) Stabilization of coupled linear heterodirectional hyperbolic PDE–ODE systems. Automatica 87:281–289. https://doi.org/10.1016/j.automatica.2017.09.02 – 10.1016/j.automatica.2017.09.027
LeVeque, (2002)
Wu, Optimal actuator location for electro-active polymer actuated endoscope. (2018)
Timoshenko, (1974)