This line of research has been dedicated to putting into use the arsenal of mathematical tools provided by combinatorial, algebraic, and differential topology in robust control problems. The boundary behavior of the Nyquist return difference map, as it was introduced by Horowitz in the Quantitative Feedback Theory, is an issue relevant to the celebrated Brouwer domain invariance theorem, the Caratheodory prime end theorem, and the simplicial approximation theorem, the latter being a foundational result of combinatorial, piecewise-linear topology. Existence of fast robust stability tests on a subspace of uncertainty is an issue relevant to homotopy theory, obstruction theory, and the Fredholm index approach to K-theory as developed by Michael Atiyah. Crucial properties of the complex mu-function are corollaries of the open mapping property of holomorphic functions of several complex variables. Continuity of the mu-function and other robustness margin measures relative to data perturbation is an issue relevant to the differential topology of the Nyquist map.

Our SimplicialVIEW software (developed by Murilo G. Coutinho) has demonstrated that the theoretical constructions of combinatorial, piecewise-linear topology can be implemented in practise using computational geometry to produce an approximate stability boundary as an assembly of simplexes. Visualization of the stability boundary has been made "user friendly" using state of the art computer graphics. The issue of numerical stability and conditioning of the geometric construction have been shown to be related to the differential topology of the map.

Since the core of this unified topological approach has been built, the above research effort is currently being phased out,  and our attention is curently being redirected towards such specialized ramifications as

Selected Publications:

Book

Journal Papers

  1. Jonathan Bar-on and E. A. Jonckheere, ``Phase margins for multivariable control systems," International Journal of Control, vol. 52, pp. 485-498, August 1990.

    2.  
  2. Jonathan R. Bar-on and E. A. Jonckheere, ``Multivariable gain margin," International Journal of Control, Vol. 54, No. 2, pp. 337-365, August 1991.

    3.  
  3. Jonathan R. Bar-on and E. A. Jonckheere, ``The geometry of the multivariable phase margin," IEEE Transactions on Automatic Control, vol. 37, pp. 798-800, June 1992.

    4.  
  4. E. A. Jonckheere and C.-K. Chu, ``Bounded flatness of Q-triangulated regular N-simplexes,'' Applied Mathematics and Computation, vol. 88, pp. 177-198, 1997.

    5.  
  5. E. A. Jonckheere, C.-K. Chu, and C.-Y. Cheng, ``Stochastic complexity of Hex models of robust stability -- A Monte Carlo simulation approach,'' Applied Mathematics and Computation, to appear, 1998.

    6.  
  6. E. A. Jonckheere, C.-K. Chu, and C.-Y. Cheng, ``Simplicial algorithms for computing stationary probabilities of stochastic matrices,'' Applied Mathematics and Computation, vol. 93, pp. 207-217, 1998.

    7.  
  7. R. D. Colgren and E. A. Jonckheere, "H-infinity control of a class of nonlinear systems using describing functions and simplicial algorithms," IEEE Transactions on Automatic Control, vol. 42, No. 5, pp. 707-712, May 1997.

    8.  
  8. M. G. Coutinho and E. A. Jonckheere, "Simplicial computation of neutral stability region in uncertainty space," IEEE Transactions on Automatic Control, to appear, 1998.

    9.  
  9. E. A. Jonckheere and N.-P. Ke, "Complex-analytic theory of the Mu-function," Journal of Mathematical Analysis and its Applications,  vol. 237, pp. 201-239, 1999.

    10.  
  10. E. A. Jonckheere and N.-P. Ke, "Real versus complex stability margin continuity as a smooth versus holomorphic singularity problem," Journal of Mathematical Analysis and its Applications, vol. 237, pp. 541-572, 1999.

    11.  
  11. N. Fathpour and E. A. Jonckheere, "A Brouwer domain invariance approach to boundary behavior of Nyquist maps for uncertain systems," Mathematics of control, Signals, and Systems, vol. 11, pp. 357-371, 1998.

    12.  
  12. E. A. Jonckheere and N.-P. Ke, "Topological theory of 0/0 ambiguities in robust stability," Mathematics of control, Signals, and Systems, submitted, 1997.

    13.  
  13. E. A. Jonckheere, F. Ahamd, and E. Gutkin, "Differential topology of numerical range," Linear Algebra and Its Applications, vol. 279, pp. 227-254, 1998.

    14.  

Conference Papers

  1. E. A. Jonckheere and Chih-Yung Cheng, ``Robust stability and game of Hex,'' 1993 IEEE Regional Conference on Aerospace Control Systems, Westlake Village, California, May 25-27, 1993, pp. 406-410.

    2.  
  2. E. A. Jonckheere, C.-Y. Cheng, and M. G. Coutinho Neto, ``A survey of simplicial algorithms and topology in robust stability,'' 1993 IEEE Regional Conference on Aerospace Control Systems, Westlake Village, California, May 25-27, 1993, pp. 400-405.

    3.  
  3. R. D. Colgren, E. A. Jonckheere, and C.-Y. Cheng, ``H-infinity optimal control of a simple nonlinear system analyzed using simplicial algorithms,'' 1993 IEEE Regional Conference on Aerospace Control Systems, Westlake Village, California, May 25-27, 1993, pp. 411-415.

    4.  
  4. E. A. Jonckheere and Chih-Yung Cheng, ``Robust stability, Morse theory and singularity'' in Proceedings of the 32nd IEEE Conference on Decision and Control, San Antonio, Texas, Dec. 15-17, 1993, FP-2, pp. 3453-3458.

    5.  
  5. E. A. Jonckheere and C.-Y. Cheng, ``Simplicial algorithms on V-triangulation of uncertainty space,'' in Proceedings of the American Control Conference, ACC'94, Baltimore, MD, June 29-July 01, 1994, TM10, pp. 1814-1815.

    6.  
  6. E. A. Jonckheere, M. G. Coutinho, and C.-Y. Cheng, ``A computational geometry approach to simplicial Nyquist maps in robust stability,'' in Proceedings of the American Control Conference, ACC'94, Baltimore, MD, June 29-July 01, 1994, TM14, pp. 1901-1905.

    7.  
  7. E. A. Jonckheere and C.-Y. Cheng, ``Structural stability of robustness margin,'' in Proceedings of the IFAC Symposium on Robust Control Design, Rio de Janeiro, Brazil, September, 14-16, 1994, pp. 423-428.

    8.  
  8. E. A. Jonckheere and N.-P. Ke, ``Stability margin relative to stratified uncertainty space,'' IEEE Conference on Decision and Control, New Orleans, Louisiana, Dec. 1995, WP11-1, pp. 1322-1323.

    9.  
  9. M. G. Coutinho and E. A. Jonckheere, ``SimplicialVIEW -- A package for robust stability analysis,'' Proceedings of the IEEE Conference on Decision and Control, New Orleans, Louisiana, Dec. 1995, WA05-5, pp. 136-141.

    10.  
  10. E. A. Jonckheere and N.-P. Ke, ``Complex-analytic theory of the mu-function,'' American Control Conference, Albuquerque, NM, June 04-06, 1997, pp. 3321-3325.

    11.  
  11. E. A. Jonckheere and N.-P. Ke, ``Topological theory of 0/0 ambiguities in robust stability,'' IEEE Conference on Decision and Control, San Diego, CA, Dec. 1997, pp 4354-4359.

    12.  
  12. E. A. Jonckheere and N.-P. Ke, ``Real versus complex robustness margin as a smooth versus holomorphic singularity problem,'' IEEE Conference on Decision and Control, San Diego, CA, Dec. 1997, pp 3266-3271.

    13.  
Back to main page