A type of gain scheduling which converts to a “classical” problem in several complex variables

We treat H^∞ control design for a plant where part of it is known and a subsystem δ is not known, that is, the response of the plant at “frequency” s is P(s, δ(s)). We assume that once our control system is running, we can identify the subsystem δ online. Thus the problem is to design a function K off-line that uses this information to produce an H^∞ controller via the formula K(s, δ(s)). The challenge is to pick K so that the controller yields a closed loop system with H^∞ gain at most γ no matter which δ occurs. While this is entirely a frequency domain problem, it can be considered as H^∞ gain scheduling. However, we call this the linear model varying (LMV) control problem. In this article we show that LMV control problems are equivalent to certain problems of interpolation by analytic functions in several complex variables. These precisely generalize the classical interpolation problems which lay at the core of H^∞ control