Parallel simultaneous stabilization of two systems governed by partial differential equations subject to actuator saturation

Authors

Abstract

A parallel simultaneous stabilization of two systems governed by partial differential equation (PDE), with conservation law subject to actuator saturation, is considered, and a method on the control design is proposed. Based on the orthogonal decomposition for a Port-Controlled Hamiltonian system, an approach to the parallel simultaneous stabilization of two systems governed by partial differential equation, with conservation law subject to actuator saturation, is established. The study shows that the parallel simultaneous stabilization controller obtained in this paper is valid in the case of the systems governed by the PDEs.

Keywords

actuator saturation, conservation law, parallel simultaneous stabilization, partial differential equation

Citation

Journal: Nonlinear Analysis: Theory, Methods & Applications
Year: 2009
Volume: 71
Issue: 3-4
Pages: 829–837
Publisher: Elsevier BV
DOI: 10.1016/j.na.2008.10.111

BibTeX

@article{Zong_2009,
  title={{Parallel simultaneous stabilization of two systems governed by partial differential equations subject to actuator saturation}},
  volume={71},
  ISSN={0362-546X},
  DOI={10.1016/j.na.2008.10.111},
  number={3–4},
  journal={Nonlinear Analysis: Theory, Methods &amp; Applications},
  publisher={Elsevier BV},
  author={Zong, Xiju and Han, Zhenlai},
  year={2009},
  pages={829--837}
}

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References

Coutinho, D. F. & da Silva, J. M. G. Estimating the Region of Attraction of Nonlinear Control Systems with Saturating Actuators. 2007 American Control Conference 4715–4720 (2007) doi:10.1109/acc.2007.4282430 – 10.1109/acc.2007.4282430
Yong-Yan Cao & Zongli Lin. Robust stability analysis and fuzzy-scheduling control for nonlinear systems subject to actuator saturation. IEEE Trans. Fuzzy Syst. 11, 57–67 (2003) – 10.1109/tfuzz.2002.806317
Stoorvogel, A. A., Saberi, A. & Weiland, S. On external semi-global stochastic stabilization of linear systems with input saturation. 2007 American Control Conference 5845–5850 (2007) doi:10.1109/acc.2007.4282930 – 10.1109/acc.2007.4282930
Gomes, Anti-windup design with guaranteed regions of stability: An LMI-based approach. IEEE Trans. Automat. Control (2005)
Hu, T., Lin, Z. & Chen, B. M. An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica 38, 351–359 (2002) – 10.1016/s0005-1098(01)00209-6
Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38, 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM J. Control Optim. 37, 54–91 (1998) – 10.1137/s0363012996312039
Fujimoto, K., Sakurama, K. & Sugie, T. Trajectory tracking control of port-controlled Hamiltonian systems via generalized canonical transformations. Automatica 39, 2059–2069 (2003) – 10.1016/j.automatica.2003.07.005
Cheng, D., Astolfi, A. & Ortega, R. On feedback equivalence to port controlled Hamiltonian systems. Systems & Control Letters 54, 911–917 (2005) – 10.1016/j.sysconle.2005.02.005
Wang, Y., Li, C. & Cheng, D. Generalized Hamiltonian realization of time-invariant nonlinear systems. Automatica 39, 1437–1443 (2003) – 10.1016/s0005-1098(03)00132-8
Wang, Y., Feng, G. & Cheng, D. Simultaneous stabilization of a set of nonlinear port-controlled Hamiltonian systems. Automatica 43, 403–415 (2007) – 10.1016/j.automatica.2006.09.008
Degasperis, Asymptotic integrability. (1999)
Camassa, R. & Holm, D. D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993) – 10.1103/physrevlett.71.1661
JOHNSON, R. S. Camassa–Holm, Korteweg–de Vries and relatedmodels for water waves. J. Fluid Mech. 455, 63–82 (2002) – 10.1017/s0022112001007224
Constantin, A. & McKean, H. P. A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999) – 10.1002/(sici)1097-0312(199908)52:8<949::aid-cpa3>3.0.co;2-d
Constantin, A., Gerdjikov, V. S. & Ivanov, R. I. Inverse scattering transform for the Camassa–Holm equation. Inverse Problems 22, 2197–2207 (2006) – 10.1088/0266-5611/22/6/017
Constantin, A. The trajectories of particles in Stokes waves. Invent. math. 166, 523–535 (2006) – 10.1007/s00222-006-0002-5
Toland, J. F. Stokes waves. TMNA 7, 1 (1996) – 10.12775/tmna.1996.001
Constantin, A. & Escher, J. Particle trajectories in solitary water waves. Bull. Amer. Math. Soc. 44, 423–431 (2007) – 10.1090/s0273-0979-07-01159-7
Constantin, A. & Strauss, W. A. Stability of peakons. Comm. Pure Appl. Math. 53, 603–610 (2000) – 10.1002/(sici)1097-0312(200005)53:5<603::aid-cpa3>3.0.co;2-l
Constantin, A. & Molinet, L. Orbital stability of solitary waves for a shallow water equation. Physica D: Nonlinear Phenomena 157, 75–89 (2001) – 10.1016/s0167-2789(01)00298-6
Lenells, J. A Variational Approach to the Stability of Periodic Peakons. JNMP 11, 151 (2004) – 10.2991/jnmp.2004.11.2.2
El Dika, K. & Molinet, L. Exponential decay o -localized solutions and stability of the train o solitary waves for the Camassa–Holm equation. Phil. Trans. R. Soc. A. 365, 2313–2331 (2007) – 10.1098/rsta.2007.2011
Beals, R., Sattinger, D. H. & Szmigielski, J. Multi-peakons and a theorem of Stieltjes. Inverse Problems 15, L1–L4 (1999) – 10.1088/0266-5611/15/1/001
Bressan, A. & Constantin, A. Global Conservative Solutions of the Camassa–Holm Equation. Arch Rational Mech Anal 183, 215–239 (2006) – 10.1007/s00205-006-0010-z
Degasperis, A new integrable equation with peakon solitons. Theoret. Math. Phys. (2002)
Degasperis, Integrable and non-integrable equation with peakons. Nonlinear Phys.: Theory and Experiment, Gallllipoli (2002)
Hone, A. N. W. & Wang, J. P. Prolongation algebras and Hamiltonian operators for peakon equations. Inverse Problems 19, 129–145 (2002) – 10.1088/0266-5611/19/1/307
Lundmark, H. & Szmigielski, J. Multi-peakon solutions of the Degasperis–Procesi equation. Inverse Problems 19, 1241–1245 (2003) – 10.1088/0266-5611/19/6/001
Zhou, Y. Blow-up phenomenon for the integrable Degasperis–Procesi equation. Physics Letters A 328, 157–162 (2004) – 10.1016/j.physleta.2004.06.027
Yin, Z. Global existence for a new periodic integrable equation. Journal of Mathematical Analysis and Applications 283, 129–139 (2003) – 10.1016/s0022-247x(03)00250-6
Yin, Z. Global weak solutions for a new periodic integrable equation with peakon solutions. Journal of Functional Analysis 212, 182–194 (2004) – 10.1016/j.jfa.2003.07.010
Constantin, A. & Kolev, B. Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78, 787–804 (2003) – 10.1007/s00014-003-0785-6
Constantin, A. Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Annales de l’Institut Fourier 50, 321–362 (2000) – 10.5802/aif.1757
Tsuchida, T., Ujino, H. & Wadati, M. Integrable semi-discretization of the coupled nonlinear Schrödinger equations. J. Phys. A: Math. Gen. 32, 2239–2262 (1999) – 10.1088/0305-4470/32/11/016
Li-xin, T., Gang, X. & Zeng-rong, L. The concave or convex peaked and smooth soliton solutions of Camassa-Holm equation. Appl Math Mech 23, 557–567 (2002) – 10.1007/bf02437774
Vakhnenko, V. O. & Parkes, E. J. Periodic and solitary-wave solutions of the Degasperis–Procesi equation. Chaos, Solitons & Fractals 20, 1059–1073 (2004) – 10.1016/j.chaos.2003.09.043
Guo, Periodic cusp wave solution sand is single-solutions for b-equation. Chaos Solitons Fractals (2005)
Qian, Peakons and periodic cusp waves in a generalized Camassa–Holm equation. Chaos Solitons Fractals (2000)
Liu, Z., Wang, R. & Jing, Z. Peaked wave solutions of Camassa–Holm equation☆. Chaos, Solitons & Fractals 19, 77–92 (2004) – 10.1016/s0960-0779(03)00082-1
Tang, M. & Yang, C. Extension on peaked wave solutions of CH-γ equation. Chaos, Solitons & Fractals 20, 815–825 (2004) – 10.1016/j.chaos.2003.09.018
Guo, B. & Liu, Z. Peaked wave solutions of CH-r equation. Sci. China Ser. A-Math. 46, 696–709 (2003) – 10.1007/bf02942241
Daizhan Cheng, Spurgeon, S. & Jianping Xiang. On the development of generalized Hamiltonian realizations. Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187) vol. 5 5125–5130 – 10.1109/cdc.2001.914763
Khalil, (1996)