Some adaptive control problems which convert to a "classical"problem in several complex variables
Helton, J.W.
Automatic Control, IEEE Transactions on
Volume 46, Issue 12, Dec 2001 Page(s):2038 - 2043
Digital Object Identifier 10.1109/9.975517
Summary:We discuss the equivalence of bi-H∞ control
problems to certain problems of approximation and interpolation by
analytic functions in several complex variables. In bi-H∞
control, the goal is to perform H∞ control
design for a plant where part of it is known and a stable subsystem
δ is not known, i.e. the response at "frequency" s is P(s,
δ(s)). We assume that once our system is running, we can identify
δ online. Thus the problem is to design a function K off-line that
uses this information to produce a H∞ controller via
the formula K(s, δ(s)). The controller should yield a closed loop
system with H∞ gain at most γ no matter which
δ occurs. This is a frequency domain problem. The article shows
how several bi-H∞ control problems convert to two
complex variable interpolation problems. These precisely generalize the
classical (one complex variable) interpolation (AAK-commutant lifting)
problems which lay at the core of H∞ control. These
problems are hard, but the last decade has seen substantial success on
them in the operator theory community. In the most ideal of
bi-H∞ cases these lead to a necessary and sufficient
treatment of the control problem
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