### Optimization and Systems Theory Seminar

Friday, August 20, 1999, 11.00-12.00, Room 3721, Lindstedtsv. 25

** Jorge Gonçalves**

Laboratory for Information and Decision Systems

Massachusetts Institute of Technology

Cambridge, MA 02139-4307

USA

E-mail:
jmg@mit.edu

####
Global quadratic stability of limit cycles is common in relay feedback
systems

For a large class of relay feedback systems (RFS) there will be limit
cycle oscillations. Conditions to check existence and *local*
stability of limit cycles for these systems are well known.
*Global* stability conditions, however, are practically non-existent.
This paper presents conditions in the form of linear matrix
inequalities (LMIs) that guarantee *global* asymptotic stability
of a limit cycle induced by a relay with hysteresis in feedback with
an LTI stable system. The analysis is based on finding global
quadratic Lyapunov functions for a Poincaré maps associated with
the RFS. We found that most Poincaré map induced by an LTI flow
between two hyperplanes can be represented as a linear transformation
analytically parametrized by a scalar function of the state.
Moreover, level sets of this function are convex. The search for
globally quadratic Lyapunov functions is then done by solving a set
of LMIs. Most RFS analyzed by the authors were proven to be globally
stable. Systems analyzed include minimum-phase systems, systems of
relative degree larger than one, and of high dimension. This leads
us to believe that quadratic stability of associated Poincaré maps
is common in RFS.

Calendar of seminars

*Last update: August 6, 1999 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*