Skip to main content
SpringerLink
Log in

Ellipsoidal Techniques: Reachability and Control Synthesis

  • Chapter
  • First Online:
Dynamics and Control of Trajectory Tubes

Part of the book series: Systems & Control: Foundations & Applications ((SCFA,volume 85))

  • 1115 Accesses

Abstract

This chapter describes the ellipsoidal techniques for control problems introduced in earlier chapters. We derive formulas for reachability sets using the properties of ellipsoids and relations from convex analysis. The formulas are derived through inductive procedures. They allow calculation of both external and internal ellipsoidal approximations of forward and backward reachability sets with any desired level of accuracy. The approximations are illustrated on examples explained in detail, then followed by ellipsoid-based formulas for problems of reachability and control synthesis.

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

Access this chapter

Log in via an institution
eBook
USD 19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 29.99
Price excludes VAT (USA)
  • Durable hardcover 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

Institutional subscriptions

Notes

  1. 1.

    In the following formulas symbol p[⋅ ] stands for the pair {p 0,  p(s), s ∈ [t 0, t]}.

  2. 2.

    Here we also take into account that \((l^{{\prime}}(t)A(t) + dl(t)/dt)Y _{l}(t)\) is bounded with p 0(t) > 0 and p t (τ) > 0 almost everywhere due to the controllability assumption.

  3. 3.

    Here and in the sequel the terms of type o i (ε) are assumed to be such that o i (ε)ε−1 → 0 with ε → 0.

  4. 4.

    Note that for \(l_{2}^{0}/l_{1}^{0} > 0\) or \(l_{2}^{0}/l_{1}^{0} < -t\) we have \(\upsigma \not\in [-t, 0]\). For such vectors the point of support x l[t] will be at either of the vertices of set \(\mathcal{X}[t]\).

  5. 5.

    Thus example was worked out by P. Gagarinov.

  6. 6.

    If \(\mathcal{W}[t]\) is already calculated, the procedure is simple.

  7. 7.

    Such a proof is indicated in [174, Sect. 3.6, p. 212].

References

  1. Aumann, R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. Studies in Applied Mathematics, vol. 15. SIAM, Philadelphia (1994)

    Google Scholar 

  3. Chernousko, F.L.: Ellipsoidal bounds for sets of attainability and uncertainty in control. Optim. Control Appl. Methods 3(2), 187–202 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chernousko, F.L.: State Estimation for Dynamic Systems. CRC Press, Boca Raton (1994)

    Google Scholar 

  5. Kalman, R.E., Ho, Y.-C., Narendra, K.S.: Controllability of linear dynamical systems. Contrib. Theory Differ. Equ. 1(2), 189–213 (1963)

    MathSciNet  Google Scholar 

  6. Krasovski, N.N.: Motion Control Theory. Nauka, Moscow (1968) [in Russian]

    Google Scholar 

  7. Krasovski, N.N.: Rendezvous Game Problems. National Technical Information Service, Springfield (1971)

    Google Scholar 

  8. Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer Series in Soviet Mathematics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  9. Kurzhanskiy, A.A., Varaiya, P.: Ellipsoidal Toolbox. University of California at Berkeley. http://code.google.com/p/ellipsoids/ (2005)

  10. Kurzhanski, A.B., Filippova, T.F.: On the theory of trajectory tubes: a mathematical formalism for uncertain dynamics, viability and control. In: Advances in Nonlinear Dynamics and Control. Progress in Systems and Control Theory, vol. 17, pp. 122–188. Birkhäuser, Boston (1993)

    Google Scholar 

  11. Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. SCFA. Birkhäuser, Boston (1997)

    Book  MATH  Google Scholar 

  12. Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis. Internal approximation. Syst. Control Lett. 41, 201–211 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis. Part I: external approximations. Optim. Methods Softw. 17(2), 177–206 (2002)

    MathSciNet  MATH  Google Scholar 

  14. Kurzhanski, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis. Part II: internal approximations. Box-valued constraints. Optim. Methods Softw. 17(2), 207–237 (2002)

    MathSciNet  MATH  Google Scholar 

  15. Lee, E.B., Markus, L.: Foundations of Optimal Control Theory. Wiley, New York (1967)

    MATH  Google Scholar 

  16. Pontryagin, L.S., Boltyansky, V.G., Gamkrelidze, P.V., Mischenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley Interscience, New York (1962)

    MATH  Google Scholar 

  17. Skelton, R.E.: Dynamic Systems and Control. Wiley, New York (1988)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State (Lomonosov) University, Moscow, Russia

    Alexander B. Kurzhanski

  2. Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA, USA

    Alexander B. Kurzhanski & Pravin Varaiya

Authors
  1. Alexander B. Kurzhanski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pravin Varaiya
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kurzhanski, A.B., Varaiya, P. (2014). Ellipsoidal Techniques: Reachability and Control Synthesis. In: Dynamics and Control of Trajectory Tubes. Systems & Control: Foundations & Applications, vol 85. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-10277-1_3

Download citation

Publish with us

Policies and ethics

Search

Navigation