Living with Unstable Zero Dynamics:

The Story of Feedback Limitations

**Computer Engineering **

Like a person, an engineering technique becomes wise and effective
only when its limits are fully understood. For classical feedback theory, such
limits were described already in its birth certificate issued by Bode in the
1940's. Along with its many abilities, feedback control is hindered by the
presence of unstable poles and/or zeros of the transfer function of the plant.
Bode, a native of Urbana, Illinois, quantified these limits with his celebrated
integrals. Some 40 years later, Freuenberg and Looze working in Urbana,
generalized Bode's integrals to linear multivariable feedback systems. A
further step was made in 1993 by Qiu and Davison who employed cheap optimal
control to translate the frequency domain limits into the time domain and
extend a 1972 optimal regulator result of Kwakernaak and Sivan.

The train of thought was thus set to move to Santa Barbara where
Seron, Braslavsky, Mayne and this
speaker decided to analyze feedback limitations in nonlinear systems for which
frequency domain methods don't work. Since the most intriguing linear
limitations are due to unstable ('nonminimum phase') zeros, their nonlinear
counterpart has been the focus of recent reasearch, and is the main topic of
this talk. Using nonlinear cheap optimal control and the concept on nonlinear
zero dynamics, a clear characterization of feedback limitations in nonlinear
systems has been derived, which also sheds new light on the limitations in
linear systems. Even if the
control effort is free, but zero dynamics are unstable, perfect tracking performance cannot be
achieved. A well defined amount of control energy must be diverted to stabilize
the zero dynamics, causing a tracking error which exactly reflects this amount.

This talk will give structual and qualitative interpretations of
these nonlinear results in a narrative form, understandable to nonexperts, but
it will also include a few analytical expressions for students familiar with
nonlinear control theory.

**3:30 – 4:30 p.m.**