Characterization of a Topological Obstruction to Reach Control by Continuous State Feedback

Abstract

This paper studies a topological obstruction to solving the reach control problem (RCP) by continuous state feedback. Given a simplex and given an affine control system defined on the simplex, the RCP is to find a state feedback to drive closed-loop trajectories initiated in the simplex through an exit facet, without first exiting through other facets. We distill the problem as one of continuously extending a function that maps into a sphere from the boundary of a simplex to its interior. As such, we employ techniques from the extension problem of algebraic topology. Unlike previous work on the same problem, in this paper we remove unnecessary restrictions on the dimension of the simplex, the number of inputs of the system, and the particular geometry of the subset of the state space where the obstruction arises. Thus, the results of this paper represent the culmination of our efforts to characterize the topological obstruction. The conditions obtained in the paper are easily checkable and fully characterize the obstruction.

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Notes

  1. Up to a degenerate case where \({{\mathcal {H}}}= {{\mathbb {S}}}^{m-1}\), but \({{\mathcal {H}}}^* \ne {{\mathbb {S}}}^{m-1}\).

References

  1. Andres J (2006) Topological principles for ordinary differential equations. In: Cañada A, Drábek P Fonda A (eds) Handbook of differential equations: ordinary differential equations, vol 3. Elsevier, Amsterdam, pp 1–101

  2. Ashford G, Broucke ME (2013) Time-varying affine feedback for each control on simplices. Automatica 49(5):1365–1369

    Article  MathSciNet  MATH  Google Scholar

  3. Border KC (1989) Fixed point theorems with applications to economics and game theory. Cambridge University Press, Cambridge

    MATH  Google Scholar

  4. Borsuk K (1967) Theory of retracts. Państwowe Wydawnictwo Naukowe, Warsaw

  5. Bredon GE (1997) Topology and geometry. Springer, Berlin

    MATH  Google Scholar

  6. Broucke ME (2010) Reach control on simplices by continuous state feedback. SIAM J Control Optim 48(5):3482–3500

    Article  MathSciNet  MATH  Google Scholar

  7. Broucke ME, Ganness M (2014) Reach control on simplices by piecewise affine feedback. SIAM J Control Optim 52(5):3261–3286

    Article  MathSciNet  MATH  Google Scholar

  8. Broucke ME, Ornik M, Mansouri A (2015) A topological obstruction in a control problem. Systems and control letters. Submitted

  9. Danzer L, Grünbaum B, Klee V (1963) Hellys theorem and its relatives. In: Klee V (ed) Convexity. American Mathematical Society, Providence, RI, pp 101–180

  10. Davis JF, Kirk P (2001) Lecture notes in algebraic topology. American Mathematical Society, Providence

    Book  MATH  Google Scholar

  11. tom Dieck T, (2008) Algebraic topology. European Mathematical Society, Zürich

  12. Eilenberg S (1940) Cohomology and continuous mappings. Ann Math 41(1):231–251

    Article  MathSciNet  MATH  Google Scholar

  13. Elbassioni K, Tiwary HR (2011) On a cone covering problem. Comput Geom 44(3):129–134

    Article  MathSciNet  MATH  Google Scholar

  14. Habets LCGJM, Collins PJ, van Schuppen JH (2006) Reachability and control synthesis for piecewise-affine hybrid systems on simplices. IEEE Trans Autom Control 51:938–948

    Article  MathSciNet  Google Scholar

  15. Habets LCGJM, van Schuppen JH (2004) A control problem for affine dynamical systems on a full-dimensional polytope. Automatica 40:21–35

    Article  MathSciNet  MATH  Google Scholar

  16. Helwa MK, Broucke ME (2014) Reach control of single-input systems on simplices using multi-affine feedback. Math Theory Networks Syst. pp 748–755

  17. Hopf H (1931) Ãœber die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math Ann 104(1):637–665

    Article  MathSciNet  MATH  Google Scholar

  18. Li H, Hestenes D, Rockwood (2001) A Spherical conformal geometry with geometric algebra. In: Sommer G (ed) Geometric computing with clifford algebras: theoretical foundations and applications in computer vision and robotics, pp. 61–75

  19. Mehta K (2012) A Topological obstruction in a control problem. M.A.Sc. Thesis, Department of Electrical and Computer Engineering. University of Toronto

  20. Moarref M, Ornik M, Broucke ME (2016) An obstruction to solvability of the reach control problem using affine feedback. Automatica 71:229–236

    Article  MathSciNet  MATH  Google Scholar

  21. Müger M (2016) Topology for the working mathematician. http://www.math.ru.nl/~mueger/topology.pdf. Accessed 13 Mar 2017

  22. Naber GL (2000) Topological methods in euclidean spaces. Dover, Mineola

    MATH  Google Scholar

  23. Ornik M, Broucke ME (2015) Some results on an affine obstruction to reach control. arXiv:1507.02957 [math.OC]

  24. Ornik M, Broucke ME (2015) A topological obstruction to reach control by continuous state feedback. In: IEEE conference on decision and control. pp. 2258–2263

  25. Ornik M, Broucke ME (2016) On a topological obstruction in the reach control problem. Mathematical and computational approaches in advancing modern science and engineering. In: Proceedings of AMMCS-CAIMS 2015, Springer

  26. PrzymusiÅ„ski T (1978) Collectionwise normality and absolute retracts. Fundam Math 98(1):61–73

    MathSciNet  MATH  Google Scholar

  27. Repovs D, Semenov PV (2013) Continuous selections of multivalued mappings. Springer, Berlin

    MATH  Google Scholar

  28. Robinson CV (1942) Spherical theorems of Helly type and congruence indices of spherical caps. Am J Math 64(1):260–272

    Article  MathSciNet  MATH  Google Scholar

  29. Roszak B, Broucke ME (2006) Necessary and sufficient conditions for reachability on a simplex. Automatica 42(11):1913–1918

    Article  MathSciNet  MATH  Google Scholar

  30. Semsar-Kazerooni E, Broucke ME (2014) Reach controllability of single input affine systems. IEEE Trans Autom Control. 59(3):738–744

    Article  MathSciNet  MATH  Google Scholar

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Correspondence to Melkior Ornik.

Additional information

This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Appendix

Appendix

Lemma 11

([18]) Suppose (A3) holds. Let \(y\in {{\mathbb {S}}}^{m-1}\). Suppose \(y\in {{\mathcal {C}}}_j\). The stereographic projection of \({{\mathcal {C}}}_j\backslash \{-y\}\) centered at \(-y\) equals:

  1. (i)

    a closed half-space in \({{\mathbb {R}}}^{m-1}\) if \(h_j\cdot y=0\),

  2. (ii)

    an \(m-1\)-dimensional closed ball in \({{\mathbb {R}}}^{m-1}\) if \(h_j\cdot y<0\).

Lemma 12

Suppose (A3) and (A4) hold. \({{\mathcal {H}}}\) is locally contractible.

Proof

Since local contractibility is a local property, we need only study a neighborhood of any point in \({{\mathcal {H}}}\). To that end, let \(x \in {{\mathcal {H}}}\) and suppose without loss of generality \(x \in {{\mathcal {H}}}_1 \cap \cdots \cap {{\mathcal {H}}}_r\), and \(x \notin {{\mathcal {H}}}_{r+1},\ldots ,{{\mathcal {H}}}_{\kappa +1}\). Since all \({{\mathcal {H}}}_j\)'s are closed in \({{\mathbb {S}}}^{m-1}\) there is a neighborhood \({{\mathcal {W}}}\) of x such that \({{\mathcal {W}}}\cap {{\mathcal {H}}}= {{\mathcal {W}}}\cap \bigcup _{j=1}^r {{\mathcal {H}}}_j\). Now consider \(-x\). It is certainly outside some neighborhood of x. We will shrink \({{\mathcal {W}}}\) so that \(-x \not \in {{\mathcal {W}}}\). We will prove that \({{\mathcal {T}}}= \left( \bigcup _{j=1}^r {{\mathcal {H}}}_j\right) \backslash \{-x\}\) is locally contractible, from which it follows \({{\mathcal {W}}}\cap {{\mathcal {H}}}\) is locally contractible.

We use a stereographic projection centered at \(-x\) of \({{\mathbb {S}}}^{m-1}\backslash \{-x\}\) into \({{\mathbb {R}}}^{m-1}\). By Lemma 11, this projection homeomorphically maps \({{\mathcal {C}}}_i \backslash \{-x\}\) to either a closed half-space in \({{\mathbb {R}}}^{m-1}\) (if \(h_i \cdot x = 0\)), or to a closed ball in \({{\mathbb {R}}}^{m-1}\), if \(h_i\cdot x<0\). Since \({{\mathcal {H}}}_j = {{\mathcal {C}}}(x)\) for \(x \in int ({{\mathcal {F}}}^{{\mathcal {O}}}_j)\), \(j \in I_{{{\mathcal {O}}}_{{\mathcal {S}}}}\), each \({{\mathcal {H}}}_j \backslash \{-x\}\) is the intersection of sets \({{\mathcal {C}}}_i \backslash \{-x\}\), so \({{\mathcal {T}}}\) is the union of intersections of sets \({{\mathcal {C}}}_i\backslash \{-x\}\). By Lemma 11, each \({{\mathcal {C}}}_i \backslash \{-x\}\) is mapped by the same homeomorphism into a convex set: either a half-space or a closed ball. Thus, each \({{\mathcal {H}}}_j \backslash \{-x\}\) is mapped into a convex set. Finally, \({{\mathcal {T}}}\) is homeomorphically deformed into a finite union of convex sets. By Lemma 1, it is locally contractible. \(\square \)

Lemma 13

Suppose (A3) and (A4) hold. Also suppose \({{\mathcal {H}}}\ne {{\mathbb {S}}}^{m-1}\) and \(\bigcap _{{{\mathcal {H}}}_j \ne {{\mathbb {S}}}^{m-1}} {{\mathcal {H}}}_j \ne \emptyset \). Then, \({{\mathcal {H}}}\) is contractible.

Proof

Let \({{\mathcal {Y}}}:= \cap _{{{\mathcal {H}}}_j \ne {{\mathbb {S}}}^{m-1}} {{\mathcal {H}}}_j\). Since each \({{\mathcal {H}}}_j\) is itself an intersection of \({{\mathcal {C}}}_j\)'s, \({{\mathcal {Y}}}\) satisfies Lemma 5. Let \(I' \subset I\) be the index set of \({{\mathcal {C}}}_j\)'s whose intersection forms \({{\mathcal {Y}}}\). By Lemma 5 there exists \(x \in {{\mathcal {Y}}}\subseteq {{\mathcal {H}}}\) such that \(h_k \cdot x < 0\) for all \(k \in I'\). Since \(h_j \cdot (-x) > 0\) for all \(k \in I'\), we know \(-x \notin {{\mathcal {H}}}_j\) for any \({{\mathcal {H}}}_j \subset {{\mathcal {H}}}\). Thus, \(-x \notin {{\mathcal {H}}}\). Consider geodesics on \({{\mathbb {S}}}^{m-1}\) coming out of x. Because the antipodal point \(-x\) is not in \({{\mathcal {H}}}\), there exists a unique geodesic \(f_{x'}\) between x and any point \(x' \in {{\mathcal {H}}}_j\) for any \({{\mathcal {H}}}_j \subset {{\mathcal {H}}}\). Since each \({{\mathcal {H}}}_j\) is Robinson convex (see [9, 28]), the entire path of geodesic \(f_{x'}\) lies inside some \({{\mathcal {H}}}_j \subseteq {{\mathcal {H}}}\), as both x and \(x'\) are in \({{\mathcal {H}}}_j\). Thus, \({{\mathcal {H}}}\) is a star-shaped set with respect to geodesics on a sphere. By a repetition of the standard proof for star-shaped sets in Euclidean spaces, \({{\mathcal {H}}}\) is contractible. \(\square \)

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Ornik, M., Broucke, M.E. Characterization of a topological obstruction to reach control by continuous state feedback. Math. Control Signals Syst. 29, 7 (2017). https://doi.org/10.1007/s00498-017-0192-y

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Keywords

  • Reach control problem
  • Topological obstruction
  • Extension problem
  • Continuous state feedback

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