Relaxing Convergence Constraints in Local Priority Hysteresis Switching Logic

This paper addresses certain inherent limitations of
local priority hysteresis switching logic. Our main result establishes
that under persistent excitation assumption, it is possible to
relax constraints requiring strict positivity of local priority and
hysteresis switching constants. Relaxing these constraints allows the
adaptive system to reach optimality which implies the performance
improvement. The unconstrained local priority hysteresis switching
logic is examined and conditions for global convergence are derived.

Authors:

• Mubarak Alhajri

Keywords:

• Adaptive control
• convergence
• hysteresis constant
• hysteresis switching.

References:

[1] M. Alhajri and M. Safonov. Setting the hysteresis constant to zero in
adaptive switching control. In American Control Conference (ACC),
pages 3542–3546. IEEE, 2011.
[2] M. Alharashani. Relaxing convergence assumptions for continuous
adaptive control. PhD thesis, University of Southern California, 2010.
[3] B. D. O. Anderson. Exponential Stability of Linear Equations Arising
in Adaptive Identification. IEEE Transactions on Automatic Control,
22:83–88, 1977.
[4] K. J. Astrom and T. Bohlin. Numerical identification of linear dynamic
systems from normal operating records. In Theory of Self-Adaptive
Control Systems, page 96, 1966.
[5] K. J. A˚ stro¨m and B. Wittenmark. Adaptive control. Courier Corporation,
2013.
[6] G. Battistelli, J. A. Hespanha, E. Mosca, and P. Tesi. Unfalsified adaptive
switching supervisory control of time varying systems. In Decision
and Control, 2009 held jointly with the 2009 28th Chinese Control
Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference
on, pages 805–810. IEEE, 2009.
[7] D. Bertsekas. Nonlinear programming. 1999. Athena Scientific, Belmont,
MA, 1999.
[8] D. P. Bertsekas. Nonlinear Programming. Athena Scientific Belmont,
MA, 1999.
[9] R. Bitmead. Persistence of excitation conditions and the convergence
of adaptive schemes. IEEE Transactions on Information Theory,
30(2):183–191, 1984.
[10] S. Boyd and S. S. Sastry. Necessary and Sufficient Conditions
for Parameter Convergence in Adaptive Control. Automatica,
22(6):629–639, 1986.
[11] Z. Han and K. S. Narendra. New concepts in adaptive control
using multiple models. Automatic Control, IEEE Transactions on,
57(1):78–89, 2012.
[12] J. Hespanha, D. Liberzon, and A. Morse. Hysteresis-based switching
algorithms for supervisory control of uncertain systems* 1. Automatica,
39(2):263–272, 2003.
[13] J. Hespanha, D. Liberzon, A. S. Morse, B. D. O. Anderson, T. S.
Brinsmead, and F. D. Bruyne. Multiple model adaptive control. Part
2: Switching. International Journal of Robust and Nonlinear Control,
11(5):479–496, 2001.
[14] A. Morse. Towards a unified theory of parameter adaptive control.
II. Certainty equivalence and implicit tuning. IEEE Transactions on
Automatic Control, 37(1):15–29, 1992.
[15] A. S. Morse. Supervisory control of families of linear set-point
controllers — Part I: Exact matching. Automatic Control, IEEE
Transactions on, 41(10):1413–1431, Oct 1996.
[16] A. S. Morse. Supervisory control of families of linear set-point
controllers — Part II: Robustness. IEEE Transactions on Automatic
Control, 42(11):1500–1515, Nov 1997.
[17] A. S. Morse, D. Q. Mayne, and G. C. Goodwin. Applications of
hysteresis switching in parameter adaptive control. IEEE Transactions
on Automatic Control, 37(9):1343–1354, Sep 1992.
[18] J. R. Munkres. Topology: A First Course. Prentice-Hall, Englewood
Cliffs, NJ, 1975.
[19] K. Narendra and A. Annaswamy. Persistent excitation in adaptive
systems. International Journal of Control, 45(1):127–160, 1987.
[20] S. V. Patil, Y.-C. Sung, and M. G. Safonov. Unfalsified adaptive control
with reset and bumpless transfer. In Decision and Control (CDC), 2014
IEEE 53rd Annual Conference on, pages 1264–1270. IEEE, 2014.
[21] P. Rosa and C. Silvestre. Multiple-model adaptive control using
set-valued observers. International Journal of Robust and Nonlinear
Control, 24(16):2490–2511, 2014.
[22] K. S. Sajjanshetty and M. G. Safonov. Unfalsified adaptive control:
Multi-objective cost-detectable cost functions. In Decision and Control
(CDC), 2014 IEEE 53rd Annual Conference on, pages 1283–1288.
IEEE, 2014.
[23] M. Stefanovic and M. Safonov. Safe adaptive switching control:
Stability and convergence. Automatic Control, IEEE Transactions on,
53(9):2012–2021, Oct. 2008.
[24] M. Vaezi, A. Izadian, and M. Deldar. Adaptive control of a hydraulic
wind power system using multiple models. In Industrial Electronics
Society, IECON 2014-40th Annual Conference of the IEEE, 2014.