Skip to main content
SpringerLink
Log in
Book cover

Developments in Model-Based Optimization and Control pp 339–359Cite as

Ultimate Bounds and Robust Invariant Sets for Linear Systems with State-Dependent Disturbances

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 464))

Abstract

The objective of this chapter is to present a methodology for computing robust positively invariant sets for linear, discrete time-invariant systems that are affected by additive disturbances, with the particularity that these disturbances are subject to state-dependent bounds. The proposed methodology requires less restrictive assumptions compared to similar established techniques, while it provides the framework for determining the state-dependent (parameterized) ultimate bounds for several classes of disturbances. The added value of the proposed approach is illustrated by an optimization-based problem for detecting the mode of functioning of a switching system.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Often the notion of global boundedness is complemented by the attribute uniform to emphasize the possible dependence of the bound on the initial condition \(x_0\) but not on the initial moment in time. This addition is superfluous for time-invariant dynamics and will be abandoned here.

  2. 2.

    The set S is a proper superset of \(\limsup _{k\rightarrow \infty }X_k\). The meaning of the outer limit (\(\limsup \)) is particular in this context as it is understood in a set-theoretic framework [2] as the set of cluster points of sequences \(x(x_0,\mathbf{w}^{0:k-1})\in X_k\).

  3. 3.

    A function \(f:\mathbb {R}^n\rightarrow \mathbb {R}\) is radially unbounded if \(\Vert x\Vert \rightarrow \infty \implies f(x)\rightarrow \infty \).

  4. 4.

    The unit ball is defined with respect to a predefined norm \(|.|_p\). In the present case the matrix \(B\in \mathbb {R}^{n\times 1}\) and thus the corresponding unit ball is defined in \(\mathbb {R}\) where the \(|.|_p\) are equivalent for \(p\in [1,\infty )\).

  5. 5.

    This particular function is increasing from \(\bar{x}=0\) in the sense of Definition 16.2 but not monotonic according to (16.12) and thus the hypothesis of Theorem 16.2 is not satisfied in this case.

References

  1. J.-P. Aubin, Viability Theory (Springer Science & Business Media, Heidelberg, 2009)

    Book  MATH  Google Scholar 

  2. J.-P. Aubin, H. Frankowska, Set-Valued Analysis. (Springer, Heidelberg, 2009)

    Book  MATH  Google Scholar 

  3. B.R. Barmish, G. Leitmann, On ultimate boundedness control of uncertain systems in the absence of matching assumptions. IEEE Trans. Autom. Control 27(1), 153–158 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. B.R. Barmish, J. Sankaran, The propagation of parametric uncertainty via polytopes. IEEE Trans. Autom. Control 24(2), 346–349 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. D.P. Bertsekas, I.B. Rhodes, Recursive state estimation for a set-membership description of uncertainty. IEEE Trans. Autom. Control 16(2), 117–128 (1971)

    Article  MathSciNet  Google Scholar 

  6. G. Bitsoris, Positively invariant polyhedral sets of discrete-time linear systems. Int. J. Control 47(6), 1713–1726 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Blanchini, Ultimate boundedness control for uncertain discrete-time systems via set-induced lyapunov functions, in Proceedings of the 30th IEEE Conference on Decision and Control (1991), pp. 1755–1760

    Google Scholar 

  8. F. Blanchini, S. Miani, Set-Theoretic Methods in Control (Springer, Heidelberg, 2007)

    MATH  Google Scholar 

  9. E. De Santis, Invariant sets: a generalization to constrained systems with state dependent disturbances, in Proceedings of the 37th IEEE Conference on Decision and Control, vol. 1 (1998), pp. 622–623

    Google Scholar 

  10. E. De Santis, On positively invariant sets for discrete-time linear systems with disturbance: an application of maximal disturbance sets. IEEE Trans. Autom. Control 39(1), 245–249 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. X. Feng, V. Puig, C. Ocampo-Martinez, S. Olaru, F. Stoican, Set-theoretic methods in robust detection and isolation of sensor faults. Int. J. Syst. Sci. 46(13), 2317–2334 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. J.D. Glover, F.C. Schweppe, Control of linear dynamic systems with set constrained disturbances. IEEE Trans. Autom. Control 16(5), 411–423 (1971)

    Article  MathSciNet  Google Scholar 

  13. E.C. Kerrigan, Robust constraint satisfaction: invariant sets and predictive control, Ph.D. thesis. University of Cambridge, 2001

    Google Scholar 

  14. E. Kofman, H. Haimovich, M.M. Seron, A systematic method to obtain ultimate bounds for perturbed systems. Int. J. Control 80(2), 167–178 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. I. Kolmanovsky, E.G. Gilbert, Theory and computation of disturbance invariant sets for discrete-time linear systems. Math. Probl. Eng. 4(4), 317–367 (1998)

    Article  MATH  Google Scholar 

  16. A.V. Kuntsevich, V.M. Kuntsevich, Invariant sets for families of linear and nonlinear discrete systems with bounded disturbances. Autom. Remote Control 73(1), 83–96 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. V.M. Kuntsevich, B.N. Pshenichnyi, Minimal invariant sets of dynamic systems with bounded disturbances. Cybern. Syst. Anal. 32(1), 58–64 (1996)

    Article  MATH  Google Scholar 

  18. G. Leitmann, Guaranteed asymptotic stability for a class of uncertain linear dynamical systems. J. Optim. Theory Appl. 27(1), 99–106 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Leitmann, On the efficacy of nonlinear control in uncertain linear systems. J. Dyn. Syst. Meas. Control 103(2), 95–102 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. P. McLane, Optimal stochastic control of linear systems with state- and control-dependent disturbances. IEEE Trans. Autom. Control 16(6), 793–798 (1971)

    Article  Google Scholar 

  21. S. Olaru, J.A. De Doná, M.M. Seron, F. Stoican, Positive invariant sets for fault tolerant multisensor control schemes. Int. J. Control 83(12), 2622–2640 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. S.V. Raković, E.C. Kerrigan, D.Q. Mayne, Reachability computations for constrained discrete-time systems with state-and input-dependent disturbances, in Proceedings of the 42nd IEEE Conference on Decision and Control, vol. 4 (2003), pp. 3905–3910

    Google Scholar 

  23. S.V. Raković, E.C. Kerrigan, K.I. Kouramas, D.Q. Mayne, Invariant approximations of the minimal robust positively invariant set. IEEE Trans. Autom. Control 50(3), 406–410 (2005)

    Article  MathSciNet  Google Scholar 

  24. V. Reppa, S. Olaru, M.M. Polycarpou, Structural detectability analysis of a distributed sensor fault diagnosis scheme for a class of nonlinear systems, in Proceedings of the 9th IFAC SAFEPROCESS (Paris, France, 2015), pp. 1485–1490

    Google Scholar 

  25. R.M. Schaich, M. Cannon, Robust positively invariant sets for state dependent and scaled disturbances, in Proceedings of the 54th IEEE Conference on Decision and Control (2015)

    Google Scholar 

  26. M.M. Seron, X.W. Zhuo, J.A. De Doná, J.J. Martínez, Multisensor switching control strategy with fault tolerance guarantees. Automatica 44(1), 88–97 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. M.M. Seron, J.A. De Doná, Robust fault estimation and compensation for LPV systems under actuator and sensor faults. Automatica 52, 294–301 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. F. Stoican, S. Olaru, Set-theoretic Fault-tolerant Control in Multisensor Systems Wiley-ISTE, (2013)

    Google Scholar 

  29. J.L. Willems, J.C. Willems, Feedback stabilizability for stochastic systems with state and control dependent noise. Automatica 12(3), 277–283 (1976)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The results presented in this chapter were initiated in the framework of the French-Italian collaborative research project Galileo 2014. The authors would like to thank Prof. F. Blanchini, Prof. D. Casagrane, Prof. S. Miani, and G. Giordano for the fruitful discussion on the topic. This work was supported by the People Programme (Marie Curie Actions) of the European FP7 (2007–2013) under REA grant agreement n\(^{\circ }\) 626891 (FUTuRISM).

Author information

Authors and Affiliations

  1. Laboratory of Signals and Systems (UMR CNRS 8506), CentraleSupélec-CNRS-Univ. Paris-Sud, Université Paris-Saclay, 91192, Gif-sur-Yvette, France

    Sorin Olaru & Vasso Reppa

Authors
  1. Sorin Olaru
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vasso Reppa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sorin Olaru .

Editor information

Editors and Affiliations

  1. Laboratory of Signals and Systems, CentraleSupélec-CNRS-Université Paris-Sud, Université Paris-Saclay, Gif sur Yvette, France

    Sorin Olaru

  2. Department of Industrial Automation, Univ. of Chem.Tech. and Metallurgy Department of Industrial Automation, Sofia, Bulgaria

    Alexandra Grancharova

  3. Faculty of Engineering, Porto University Faculty of Engineering, Porto, Portugal

    Fernando Lobo Pereira

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Olaru, S., Reppa, V. (2015). Ultimate Bounds and Robust Invariant Sets for Linear Systems with State-Dependent Disturbances. In: Olaru, S., Grancharova, A., Lobo Pereira, F. (eds) Developments in Model-Based Optimization and Control. Lecture Notes in Control and Information Sciences, vol 464. Springer, Cham. https://doi.org/10.1007/978-3-319-26687-9_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-26687-9_16

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26685-5

  • Online ISBN: 978-3-319-26687-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation