A. Steven Morse

Center for Computational Vision and Control

Department of Electrical Engineering

Yale University

**Scale Independent Hysteresis Switching and Some of Its Applications
**

``Scale-independence'' is a property of certain switching algorithms used in an adaptive
context which is key to proving an algorithm's correctness when operating in the face of noise
and
disturbance inputs. The concept of *dwell-time switching*, exploited in our earlier work
on
supervisory control, has the advantage of being scale-independent. Unfortunately, the
existence of a prescribed dwell-time makes it impossible to rule out the possibility of
finite escape in applications of dwell-time switching to the adaptive control on nonlinear
systems. On the other hand, the popular idea of hysteresis switching, originally devised by
Middleton, Goodwin, Hill and Mayne, does not have the shortcoming. Unfortunately, hysteresis
switching is not a scale-independent algorithm. Moreover, to date it is not known how to
analyze systems employing hysteresis switching except in higly unrealistic [noise free, exact
matching] situations when switching necessarily in finite time. In this talk we describe,
analyse, and compare with dwell-time and hysteresis switching, a new form of chatter-free
switching which does not employ a prescribed dwell-time and which is scale independent. To
demonstrate the switching logic's utility, we consider its use as a component of an
estimator-based supervisor whose purpose is to orchestrate the switching of a sequence of
candidate set-point controllers into feedback with an imprecisely modeled siso process so as
to cause the output of the process to approach and track a constant reference input. The
process is assumed to be modeled by a siso linear system whose transfer function is in the
union of a number of subclasses, each subclass being small enough so that one of the candidate
controllers would solve the set-point tracking problems, were the process's transfer function
to be one of the members of the subclass. It is shown that if the number of candidate
controllers is finite, it is possible to derive in a straight forward manner a reasonably
explicit formula for the exponentially weighted L2 gain induced between the supervisory
control system's disturbance input and its output tracking error.

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