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International Hybrid Systems Workshop

HS 1994: Hybrid Systems II pp 294–321Cite as

Hybrid systems as Finsler manifolds: Finite state control as approximation to connections

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 999))

Abstract

Hybrid systems are networks of interacting digital devices and continuous plants reacting to a changing environment. Our multiple agent hybrid control architecture ([KN93b], [KN93c]) is based on the notion of hybrid system state. The latter incorporates evolution models using differential or difference equations, logic constraints, and geometric constraints. The set of hybrid states of a hybrid system can be construed in a variety of ways as a differentiable (or a C ∞) manifold which we have called the carrier manifold ([KNRG95]). We have suggested that for control problems the coordinates of points of the carrier manifolds should be selected to incorporate all information about system state, control state, and environment needed to choose new values of control parameters.

This paper provides an outline of our ongoing work which expresses a wide class of control problems as relaxed, or convexified, calculus of variations problems on carrier manifolds. Weierstrass pointed out that many calculus of variations problems with time as an independent variable can be rewritten as parametric problems, that is, as problems where time is not an explicit variable, by moving to a larger manifold where time is an extra variable, x n+1=t, with the extra constraint x n+1=1. Then Finsler (1918) pointed out that each such parametric calculus of variations problem induces a metric ground form ds 2 associated with the manifold. We reduce a wide class of optimal control problems to calculating the Cartan affine connection associated with the Finsler metric ground form induced by the corresponding convexified calculus of variations problem. For any prescribed ε of deviation from the variational minimum, we then use chatterings of control flows on the manifold to approximate to a connection and use this to compute a finite state control program which enforces ε-optimal behavior. Thus we find that Finsler manifolds and their associated differential geometry and control flows form a computational basis for extraction and verification of hybrid systems and intelligent control systems in general. We remark that other connections, not arising from metrics, can be usefully employed in control too.

Here we discuss only the simplest problem of the calculus of variations and optimal control: find an “optimal” path x(t) among admissible paths on the manifold M extending from initial point x(t 0 )=x0 to endpoint x(t 1 )=x1 with a given end direction. x(t 1 )=y 1. The theory of Finsler manifolds is not familiar to most electrical engineers and certainly not to computer scientists. The goal of this paper is to give some historical background and a brief description of how one interprets connections as optimal controls and extraction of finite state control programs as extraction of approximations to connections. One can also unwind and use connections similarly for non-parametric problems. A book length mathematical treatment is in preparation.

Sponsored by Army Research Office contract DAAL03-91-C-0027, DARPA-US ARMY AMCCOM (Picatinny Arsenal, N. J.) contract DAAA21-92-C-0013 to ORA Corp., SDIO contract DAAH04-93-C-O113

Sponsored by Army Research Office contract DAAL03-91-C-0027, DARPA-US ARMY AMCCOM (Picatinny Arsenal, N. J.) contract DAAA21-92-C-0013 to ORA Corp., SDIO contract DAAH04-93-C-O113

Sponsored by Army Research Office contract DAAL03-91-C-0027, DARPA-US ARMY AMCCOM (Picatinny Arsenal, N. J.) contract DAAA21-92-C-0013 to ORA Corp., SDIO contract DAAH04-93-C-O113 and NSF grant DMS-9306427

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Author information

Authors and Affiliations

  1. Sagent Corporation, Bellevue, Washington

    Wolf Kohn

  2. Mathematical Sciences Institute, Cornell University, 14853, Ithaca, NY

    Anil Nerode

  3. Department of Mathematics and Department of Computer Science, Universityof California at San Diego, 92093, San Diego, CA

    Jeffrey B. Remmel

Authors
  1. Wolf Kohn
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  2. Anil Nerode
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  3. Jeffrey B. Remmel
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Panos Antsaklis Wolf Kohn Anil Nerode Shankar Sastry

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Kohn, W., Nerode, A., Remmel, J.B. (1995). Hybrid systems as Finsler manifolds: Finite state control as approximation to connections. In: Antsaklis, P., Kohn, W., Nerode, A., Sastry, S. (eds) Hybrid Systems II. HS 1994. Lecture Notes in Computer Science, vol 999. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60472-3_15

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  • DOI: https://doi.org/10.1007/3-540-60472-3_15

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