Skip to main content
SpringerLink
Log in

Adaptive Control of PDEs

  • Living reference work entry
  • First Online:
Encyclopedia of Systems and Control

  • 455 Accesses

Abstract

The infinite-dimensional backstepping method has recently been used for adaptive control of partial differential equations (PDEs). We will in this article briefly explain the main ideas for the three most commonly used methods for backstepping-based adaptive control of PDEs: Lyapunov-based design, identifier-based design, and swapping-based design. Swapping-based design is also demonstrated on a simple, scalar hyperbolic PDE with an uncertain in-domain parameter, clearly showing all the steps involved in using this method.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Bibliography

  • Anfinsen H, Aamo OM (2016a) Stabilization of linear 2 × 2 hyperbolic systems with uncertain coupling coefficients – part I: identifier-based design. In: Australian control conference 2016, Newcastle

    Google Scholar 

  • Anfinsen H, Aamo OM (2016b) Stabilization of linear 2 × 2 hyperbolic systems with uncertain coupling coefficients – part II: swapping design. In: Australian control conference 2016, Newcastle

    Google Scholar 

  • Anfinsen H, Aamo OM (2017a) Adaptive output feedback stabilization of n + m coupled linear hyperbolic PDEs with uncertain boundary conditions. SIAM J Control Optim 55(6):3928–3946

    Article  MathSciNet  Google Scholar 

  • Anfinsen H, Aamo OM (2017b) Adaptive output-feedback stabilization of linear 2 × 2 hyperbolic systems using anti-collocated sensing and control. Syst Control Lett 104:86–94

    Article  MathSciNet  Google Scholar 

  • Anfinsen H, Aamo OM (2017c) Adaptive stabilization of 2 × 2 linear hyperbolic systems with an unknown boundary parameter from collocated sensing and control. IEEE Trans Autom Control 62(12):6237–6249

    Google Scholar 

  • Anfinsen H, Aamo OM (2017d) Adaptive stabilization of n + 1 coupled linear hyperbolic systems with uncertain boundary parameters using boundary sensing. Syst Control Lett 99:72–84

    Article  MathSciNet  Google Scholar 

  • Anfinsen H, Aamo OM (2018) Adaptive control of linear 2 × 2 hyperbolic systems. Automatica 87:69–82

    Article  MathSciNet  Google Scholar 

  • Anfinsen H, Aamo OM (2019) Adaptive control of hyperbolic PDEs. Springer International Publishing, Cham

    Book  Google Scholar 

  • Bernard P, Krstić M (2014) Adaptive output-feedback stabilization of non-local hyperbolic PDEs. Automatica 50:2692–2699

    Article  MathSciNet  Google Scholar 

  • Coron JM, Vazquez R, Krstić M, Bastin G (2013) Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping. SIAM J Control Optim 51(3):2005–2035

    Google Scholar 

  • Di Meglio F, Vazquez R, Krstić M (2013) Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input. IEEE Trans Autom Control 58(12):3097–3111

    Google Scholar 

  • Hu L, Di Meglio F, Vazquez R, Krstić M (2016) Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs. IEEE Trans Autom Control 61(11):3301–3314

    Article  MathSciNet  Google Scholar 

  • Krstić M, Smyshlyaev A (2008a) Adaptive boundary control for unstable parabolic PDEs – part I: Lyapunov design. IEEE Trans Autom Control 53(7):1575–1591

    Article  Google Scholar 

  • Krstić M, Smyshlyaev A (2008b) Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst Control Lett 57(9):750–758

    Article  MathSciNet  Google Scholar 

  • Krstić M, Smyshlyaev A (2008c) Boundary control of PDEs: a course on backstepping designs. Soc Ind Appl Math

    Book  Google Scholar 

  • Lamare PO, Auriol J, Aarsnes UJ, Di Meglio F (2018) Robust output regulation of 2 × 2 hyperbolic systems. In: American control conference 2018, Milwaukee

    Google Scholar 

  • Liu W (2003) Boundary feedback stabilization of an unstable heat equation. SIAM J Control Optim 42:1033–1043

    Article  MathSciNet  Google Scholar 

  • Smyshlyaev A, Krstić M (2004) Closed form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Trans Autom Control 49:2185–2202

    Article  MathSciNet  Google Scholar 

  • Smyshlyaev A, Krstić M (2005) Backstepping observers for a class of parabolic PDEs. Syst Control Lett 54:613–625

    Article  MathSciNet  Google Scholar 

  • Smyshlyaev A, Krstić M (2006) Output-feedback adaptive control for parabolic PDEs with spatially varying coefficients. In: 45th IEEE conference on decision & control, San Diego

    Google Scholar 

  • Smyshlyaev A, Krstić M (2007a) Adaptive boundary control for unstable parabolic PDEs – part II: estimation-based designs. Automatica 43:1543–1556

    Article  MathSciNet  Google Scholar 

  • Smyshlyaev A, Krstić M (2007b) Adaptive boundary control for unstable parabolic PDEs – part III: output feedback examples with swapping identifiers. Automatica 43:1557–1564

    Article  MathSciNet  Google Scholar 

  • Smyshlyaev A, Krstić M (2010) Adaptive control of parabolic PDEs. Princeton University Press, Princeton

    Book  Google Scholar 

  • Strecker T, Aamo OM (2017) Output feedback boundary control of 2 × 2 semilinear hyperbolic systems. Automatica 83:290–302

    Article  MathSciNet  Google Scholar 

  • Vazquez R, Krstić M, Coron JM (2011) Backstepping boundary stabilization and state estimation of a 2 × 2 linear hyperbolic system. In: 2011 50th IEEE conference on, decision and control and european control conference (CDC-ECC), pp 4937–4942

    Google Scholar 

  • Xu Z, Liu Y (2016) Adaptive boundary stabilization for first-order hyperbolic PDEs with unknown spatially varying parameter. Int J Robust Nonlinear Control 26(3):613–628

    Article  MathSciNet  Google Scholar 

  • Yu H, Vazquez R, Krstić M (2017) Adaptive output feedback for hyperbolic PDE pairs with non-local coupling. In: 2017 American control conference, Seattle

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering Cybernetics, NTNU, Trondheim, Norway

    Henrik Anfinsen & Ole Morten Aamo

Authors
  1. Henrik Anfinsen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ole Morten Aamo
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Electrical and Computer Engineering, Boston University, Boston, MA, USA

    John Baillieul

  2. Automation and Control Solutions, Honeywell, Golden Valley, MN, USA

    Tariq Samad

Section Editor information

  1. Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA, USA

    Miroslav Krstic

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer-Verlag London Ltd., part of Springer Nature

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Anfinsen, H., Aamo, O.M. (2020). Adaptive Control of PDEs. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_100022-1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4471-5102-9_100022-1

  • Published:

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-5102-9

  • Online ISBN: 978-1-4471-5102-9

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation