On the robust stability of plant families with poles crossing the imaginary axis: A new small gain condition

A standard assumption made when employing the small gain theorem for robust stabilization of an uncertain plant family is that the plant family cannot have any poles arbitrarily crossing the imaginary axis or that the number of unstable poles must remain fixed. This assumption can with structured uncertainties. Considered in this paper is the problem of robust stabilization of single-input single-output plant families with structured uncertainties, possibly with poles crossing the imaginary axis. A necessary and sufficient condition for robust stabilization of such a family is derived by using the zero exclusion condition. Also given is a small gain like sufficient condition for robust stability formulated as a mixed sensitivity problem in the H_∞ setting. This new condition is less conservative than the well known standard small gain condition. With the new small gain condition in hand the typical robust performance problem for plants with structured uncertainties in the sense of quantitative feedback theory (QFT) can also be posed as a mixed sensitivity problem in the H_∞ setting.