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Hyperplane method for reachable state estimation for linear time-invariant systems

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Abstract

A numerical algorithm is presented for generating inner and outer approximations for the set of reachable states for linear time-invariant systems. The algorithm is based on analytical results characterizing the solutions to a class of optimization problems which determine supporting hyperplanes for the reachable set. Explicit bounds on the truncation error for the finite-time case yield a set of so-called ε-supporting hyperplanes which can be generated to approximate the infinite-time reachable set within an arbitrary degree of accuracy. At the same time, an inner approximation is generated as the convex hull of points on the boundary of the finite-time reachable set. Numerical results are presented to illustrate the hyperplane method. The concluding section discusses directions for future work and applications of the method to problems in trajectory planning in servo systems.

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References

  1. Vinter, R.,A Characterization of the Reachable Set for Nonlinear Control Systems, SIAM Journal on Control and Optimization, Vol. 18, No. 6, pp. 599–610, 1980.

    Google Scholar 

  2. Khrustalev, M. M.,Exact Description of Reachable Sets and Global Optimality Conditions for Dynamic Systems, Automation and Remote Control, Vol. 49, No. 5, pp. 597–605, 1988 (in Russian).

    Google Scholar 

  3. Gayek, J. E., andVincent, T. L.,On the Asymptotic Stability of Boundary Trajectories, International Journal of Control, Vol. 41, No. 4, pp. 1077–1086, 1985.

    Google Scholar 

  4. Gayek, J. E., andVincent, T. L.,On the Intersection of Controllable and Reachable Sets, Journal of Optimization Theory and Applications, Vol. 50, No. 2, pp. 267–278, 1980.

    Google Scholar 

  5. Gayek, J. E., andVincent, T. L.,Approximating Reachable Sets for a Class of Linear Control Systems, International Journal of Control, Vol. 43, No. 2, pp. 441–453, 1986.

    Google Scholar 

  6. Grantham, W. J.,Estimating Reachable Sets, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 103, No. 2, pp. 420–422, 1980.

    Google Scholar 

  7. Summers, D.,Lyapunov Approximation of Reachable Sets for Uncertain Linear Systems, International Journal of Control, Vol. 41, No. 5, pp. 1235–1243, 1985.

    Google Scholar 

  8. Summers, D.,Lyapunov Estimation of Reachable Sets, Abstracts of Papers to the American Mathematics Society, Vol. 10, p. 132, 1989.

    Google Scholar 

  9. Hermes, H., andLaSalle, J. P.,Functional Analysis and Time-Optimal Control, Academic Press, New York, New York, 1969.

    Google Scholar 

  10. Snow, D. R.,Reachable Regions and Optimal Controls, Advances in Control Systems, Edited by C. T. Leondes, Academic Press, New York, New York, Vol. 5, pp. 133–196, 1967.

    Google Scholar 

  11. Athans, M. A., andFalb, P. L.,Optimal Control, McGraw-Hill, New York, New York, pp. 553–554, 1966.

    Google Scholar 

  12. Leitmann, G.,The Calculus of Variations and Optimal Control, Plenum, New York, New York, 1981.

    Google Scholar 

  13. Director, S., andHatchel, G.,The Simplicial Approximation Approach to Design Centering, IEEE Transactions on Circuits and Systems, Vol. CAS-24, No. 7, pp. 363–372, 1977.

    Google Scholar 

  14. Mozumder, P. K.,Statistical Quality Control for VLSI Fabrication Processes, MS Thesis, Carnegie Mellon University, 1986.

  15. Graettinger, T. J., andKrogh, B. H.,Reference Signal Derivative Constraints for Guaranteed Tracking Performance, Proceedings of the 1989 American Control Conference, Pittsburgh, Pennsylvania, pp. 2702–2707, 1989.

  16. Aggarwal, R., andLeitmann, G.,Avoidance Control, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 99, No. 2, pp. 152–154, 1977.

    Google Scholar 

  17. Sticht, D., Vincent, T., andSchultz, D.,Sufficiency Theorems for Target Capture, Journal of Optimization Theory and Applications, Vol. 17, No. 5, pp. 523–543, 1975.

    Google Scholar 

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Authors and Affiliations

  1. Department of Electrical Engineering and Computer Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania

    T. J. Graettinger (Graduate Student) & B. H. Krogh (Associate Professor)

Authors
  1. T. J. Graettinger
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  2. B. H. Krogh
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Communicated by M. Simaan

This research was supported in part by Digital Equipment Corporation through the American Electronics Association Fellowship Loan Program and by the National Science Foundation under Grant No. ECS-84-04607.

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Graettinger, T.J., Krogh, B.H. Hyperplane method for reachable state estimation for linear time-invariant systems. J Optim Theory Appl 69, 555–588 (1991). https://doi.org/10.1007/BF00940689

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