Living with Unstable Zero Dynamics:

The Story of Feedback Limitations


Professor Petar Kokotovic
Department of Electrical and

Computer Engineering

University of California – Sta. Barbara


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.

Friday, March 14, 2003

3:30 – 4:30 p.m.

1500 EECS