We treat H{sup}∞ control design for a plant where part of it is known and a subsystemδ is not known, that is, the response of the plant at "frequency" s is P(s, δ(s)). We assume that once our control (closed loop) system is running, we canidentify the subsystem 6 on line. Thus the problem is to design a function K offline that uses this information to produce a H{sup}∞ controller via the formula K(s, δ(s)). The challenge is to pick K so that the controller yields a closed loop systemwith H{sup}∞ gain at mostγ no matter which 6 occurs. While this is entirely a frequency domain problem, it has the flavor of gain scheduling and one might think of it as H{sup}∞ gain scheduling. However, we call this the Linear Model Varying LMVcontrol problem, since it is a strict analog to Linear Parameter Varying control LPV or LFT based control, approaches currently meeting with great success.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 (one complex variable ) interpolation (AAK -commutantlifting) problems which lay at the core of H{sup}∞ control. These problems are hard, but the last decade has seen substantial success on them in the operator theory community, since it has been a focus of efforts by the generation of mathematicians whofollowed AAK-Nagy-Foias-Sarason.
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