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Hybrid systems: Chattering approximation to relaxed controls

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Hybrid Systems III (HS 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1066))

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Abstract

We explain a main connection between relaxed control and hybrid systems. We summarize the non-multiple agent aspects of our research. We describe new algorithms for approximating relaxed optimal controls based on a generalized form of linear programming using convex analysis. Finally we use chattering to derive an analogue of the Hamilton-Jacobi-Bellman equation and dynamic programming in our hybrid systems model.

Research supported by U.S. Army Research Office contract DAAL03-91-C-0027.

Research supported by SDIO contract DAA H-04-93-C-0113, Dept. of Commerce Agreement 70-NANB5H1164.

Research supported by U.S. Army Research Office contract DAAL03-91-C-0027, SDIO contract DAA H-04-93-C-0113, Dept. of Commerce Agreement 70-NANB5H1164.

Research supported by U.S. Army Research Office contract DAAL03-91-C-0027, Dept. of Commerce Agreement 70-NANB5H1164, NSF grant DMS-9306427.

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References

  1. P. Antsaklis, W. Kohn, A. Nerode, S. Sastry, eds.: Hybrid Systems II, Lecture Notes in Computer Science vol. 999, Springer-Verlag. (1995)

    Google Scholar 

  2. J. P. Aubin and I. Ekeland: Applied Non-Linear Analysis Wiley. (1984)

    Google Scholar 

  3. A. Bejancu: Finsler geometry and applications, Harwood. (1990)

    Google Scholar 

  4. R. Belleman, Dynamic Programming, Princeton University Press, Princeton, NJ, (1957).

    Google Scholar 

  5. R.E. Bellman and R. Kalaba, Dynamic Programming and Modern Control Theory, Academic Press, New York (1965).

    Google Scholar 

  6. E. Cartan: Les Espaces de Finsler, Actualities scientifiques et industrielle 79, Exposes de geometrie II. (1934)

    Google Scholar 

  7. F. H. Clarke: Methods of Dynamics and Nonsmooth Optimization, SIAM, Philadelphia. (1989)

    Google Scholar 

  8. F. H. Clarke: Optimization and Nonsmooth Analysis, SIAM, Philadelphia. (1990)

    Google Scholar 

  9. J. N. Crossley, J. B. Remmel, R. A. Shore, and Moss E. Sweedler: Logical Methods Birkhauser. (1993)

    Google Scholar 

  10. S. Eilenberg: Automata, Languages, and Machines (vol. A), Academic Press, New York. (1974)

    Google Scholar 

  11. I. Ekeland and R. Temam: Convex analysis and variational problems North-Holland. (1976)

    Google Scholar 

  12. I. Ekeland: Infinite Dimensional Optimization and Convexity, University of Chicago Lecture Notes in Mathematics, University of Chicago Press. (1983)

    Google Scholar 

  13. I. Ekeland: Convexity methods in Hamiltonian mechanics, Springer-Verlag. (1990)

    Google Scholar 

  14. P. Finsler: Über Kurven und Flächen in allegmeinen Räumen. Diss. Göttingen (1918) (republished by Verlag Birkhäuser Basel (1951))

    Google Scholar 

  15. X. Ge, W. Kohn, A. Nerode, and J. B. Remmel: Algorithms for Chattering Approximations to Relaxed Optimal Control. MSI Tech. Report 95-1, Cornell University. (1995)

    Google Scholar 

  16. R. L. Grossman, A. Nerode, A. Ravn, and H.Rischel, eds., Hybrid Systems, Lecture Notes in Computer Science 736, Springer-Verlag. (1993)

    Google Scholar 

  17. W. Kohn: A Declarative Theory for Rational Controllers, Proc. 27th IEEE CDC. (1988) 130–136.

    Google Scholar 

  18. W. Kohn and T. Skillman: Hierarchical Control Systems for Autonomous Space Robots, Proc. AIAA. (1988) 382–390.

    Google Scholar 

  19. W. Kohn: Application of Declarative Hierarchical Methodology for the Flight Telerobotic Servicer, Boeing Document G-6630-061, Final Report of NASA-Ames Research Service Request 2072, Job Order T1988. Jan 15 (1988)

    Google Scholar 

  20. W. Kohn: Autonomous Space Robots, Proc. AIAA Conf. on Guidance, Navigation, and Control, v. 1. (1988), 382–390.

    Google Scholar 

  21. W. Kohn: The Rational Tree Machine: Technical Description and Mathematical Foundations, IR and D BE-499, Technical Document 905-10107-1, Boeing Computer Services. July 7 (1989)

    Google Scholar 

  22. W. Kohn: Rational Algebras: A Constructive Approach, IR andD BE-499, Boeing Computer Service Technical Document D-905-10107-2. July 7 (1989)

    Google Scholar 

  23. W. Kohn: Cruise Missile Mission Planning: A Declarative Control Approach, Boeing Computer Services Technical Report. (1989)

    Google Scholar 

  24. W. Kohn: Declarative Multiplexed Rational Controllers, Proc. 5th IEEE Int. Symp. Intelligent Cont. (1990) 794–803.

    Google Scholar 

  25. W. Kohn and C. Johnson: An Algebraic Approach to Formal Verification of Embedded Systems, IRD Tech. Rpt. D-180-31989-1, Boeing Computer Services. (1990)

    Google Scholar 

  26. W. Kohn: A Knowledge-Based Planning and Declarative Control, Boeing Computer Services, Technical Document IRD BCS-021,in ISMIS 91. (1990)

    Google Scholar 

  27. W. Kohn and K. Carlsen: Symbolic Design and Analysis in Control, Proc. 1988 Grainger Lecture Series, U. of Illinois. (1989) 40–52.

    Google Scholar 

  28. W. Kohn and A. Murphy: Multiple Agent Reactive Shop Control, IS-MIS91.

    Google Scholar 

  29. W. Kohn and A. Nerode: Multiple Agent Autonomous Control Systems, Proc. 31st IEEE CDC (Tucson). (1992) 2956–2966.

    Google Scholar 

  30. W. Kohn and A. Nerode: An autonomous control theory: an overview. IEEE Symposium on Computer Aided Control System Design (March 17–19, 1992, Napa Valley, Ca. (1992) 204–210.

    Google Scholar 

  31. W. Kohn and A. Nerode: A Multiple agent Hybrid Control Arcitecture. In Logical Methods (J. Crossley, J. B. Remmel, R. Shore, M. Sweedler, eds.), Birkhauser. (1993) 593–623

    Google Scholar 

  32. W. Kohn and A. Nerode: Models for hybrid systems: automata, topologies, controllability and observability. In [GNRR93]. (1993) 317–356.

    Google Scholar 

  33. W. Kohn and A. Nerode: Multiple Agent Hybrid Control Architecture. In [GNRR93]. (1993) 297–316.

    Google Scholar 

  34. W. Kohn, A. Nerode, and J. B. Remmel, Hybrid Systems as Finsler Manifolds: Finite State Control as Approximation to Connections. In [AKNS95]. (1995)

    Google Scholar 

  35. W. Kohn, A. Nerode, and J. B. Remmel, Hybrid Systems as Finsler Manifolds II: The Bellman-Hamilton-Jacobi equations and Automata, in prep.

    Google Scholar 

  36. W. Kohn, A. Nerode, J. B. Remmel, and X. Ge: Multiple agent hybrid control: carrier manifolds and chattering approximations to optimal control. CDC94 (1994)

    Google Scholar 

  37. W. Kohn, A. Nerode, and J. B. Remmel, A. Yakhnis: Viability in hybrid systems. J. Theoretical Computer Science. 138 (1995) 141–168.

    Article  Google Scholar 

  38. P. D. Loewen, Optimal Control via Nonsmooth Analysis: AMS, Providence. (1993)

    Google Scholar 

  39. J. Lu, X. Ge, W. Kohn, A. Nerode, and N. Coleman: A semi-autonomous multiagent decision model for a battlefield environment. MSI Technical Report, Oct. 1994, Cornell University. (1994)

    Google Scholar 

  40. M. M. Makela: Nonsmooth Optimization: Analysis of Algorithms with Applications to Optimal Control, World Scientific, Singapore. (1993)

    Google Scholar 

  41. Matsumoto, M.: Foundations of Finsler geometry and special Finsler spaces. Kaiseisha Press, Shigkan. (1986)

    Google Scholar 

  42. A. Nerode and X. Ge: Effective Content of the Calculus of Variations, MSI Tech. Report, Feb. 1995, Cornell University. (1995)

    Google Scholar 

  43. A. Nerode, J. James, and W. Kohn: Multiple agent declarative control architecture: A knowledge based system for reactive planning, scheduling and control in manufacturing systems. Intermetrics Report, Intermetrics, Bellevue, Wash. Nov. (1992)

    Google Scholar 

  44. A. Nerode, J. James, and W. Kohn: Multiple Agent Hybrid Control Architecture: A generic open architecture for incremental construction of reactive planning and scheduling. Intermetrics Report, Intermetrics, Bellevue, Wash. June (1994)

    Google Scholar 

  45. A. Nerode, J. James, and W. Kohn: Multiple agent reactive control of distributed interactive simulations. Proc. Army Workshop on Hybrid Systems and Distributed Simulation, Feb. 28–March 1. (1994)

    Google Scholar 

  46. A. Nerode, J. James, and W. Kohn: Multiple agent reactive control of wireless distributed multimedia communications networks for the digital battlefield. Intermetrics Report, Intermetrics, Bellevue, Wash. June (1994)

    Google Scholar 

  47. A. Nerode, J. James, W. Kohn, and N. DeClaris: Intelligent integration of medical models. Proc. IEEE Conference on Systems, Man, and Cybernetics, San Antonio. 1–6 Oct. (1994)

    Google Scholar 

  48. A. Nerode, J. James,W. Kohn, and N. DeClaris: Medical information systems via high performance computing and communications. Proc. IEEE Biomedical Engineering Symposium, Baltimore, MD. Nov. (1994)

    Google Scholar 

  49. A. Nerode, J. James, W. Kohn, K. Harbison, and A. Agrawala: A hybrid systems approach to computer aided control system design. Proc. Joint Symposium on Computer Aided Control System Design. Tucson AZ 7–9 March (1994)

    Google Scholar 

  50. A. Nerode, J. B. Remmel, and A. Yakhnis: Hybrid system games: Extraction of control automata and small topologies, MSI Technical Report 93-102, Cornell University. (1993)

    Google Scholar 

  51. A. Nerode, J. B. Remmel, and A. Yakhnis: Hybrid systems and continuous Sensing Games. 9th IEEE Conference on Intelligent Control, August 25–27. (1993)

    Google Scholar 

  52. A. Nerode, J. B. Remmel, and A. Yakhnis: Controllers as Fixed Points of Set-Valued Operators, in [AKNS95]. (1995)

    Google Scholar 

  53. A. Nerode and A. Yakhnis: Modelling hybrid systems as games, CDC92. (1992) 2947–2952.

    Google Scholar 

  54. A. Nerode and A. Yakhnis: Hybrid games and hybrid systems, MSI Technical Report 93-77, Cornell University. (1993)

    Google Scholar 

  55. A. Nerode and A. Yakhnis: An example of extraction of a finite control automaton and A. Nerode's AD-converter for a discrete sensing hybrid system, MSI Technical Report 93-104, Cornell University (1993)

    Google Scholar 

  56. W. Rudin: Functional Analysis, McGraw Hill (1991)

    Google Scholar 

  57. H. Rund: The Differential Geometry of Finsler Spaces, Springer. (1959)

    Google Scholar 

  58. H. Rund: The Hamilton Jacobi Theory, Van Nostrand. (1966)

    Google Scholar 

  59. C. Tuckey: Non-Standard Methods in the Calculus of Variations, Wiley. (1993)

    Google Scholar 

  60. J. Warga: Optimal Control of Differential and Functional Equations, Academic Press, New York and London. (1972)

    Google Scholar 

  61. L. C. Young: Lectures on the Calculus Of Variations and Optimal Control Theory, Chelsea Pub. Company, New York, NY. (1980)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Sciences Institute, Cornell University, 14853, Ithaca, New York

    Xiaolin Ge

  2. Sagent Corporation, 11201 SE 8th Street #J140, 98004, Bellevue, Washington

    Wolf Kohn

  3. Mathematical Sciences Institute, Cornell University, 14853, Ithaca, New York

    Anil Nerode

  4. Department of Mathematics, University of California at San Diego, 92093, La Jolla, California

    Jeffrey B. Remmel

Authors
  1. Xiaolin Ge
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  2. Wolf Kohn
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  3. Anil Nerode
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  4. Jeffrey B. Remmel
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Editor information

Rajeev Alur Thomas A. Henzinger Eduardo D. Sontag

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© 1996 Springer-Verlag Berlin Heidelberg

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Ge, X., Kohn, W., Nerode, A., Remmel, J.B. (1996). Hybrid systems: Chattering approximation to relaxed controls. In: Alur, R., Henzinger, T.A., Sontag, E.D. (eds) Hybrid Systems III. HS 1995. Lecture Notes in Computer Science, vol 1066. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0020937

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  • DOI: https://doi.org/10.1007/BFb0020937

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  • Print ISBN: 978-3-540-61155-4

  • Online ISBN: 978-3-540-68334-6

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