Finite Rank Criteria for ${H^\infty}$ Control of Infinite-Dimensional Systems

This paper studies the ${H^\infty}$ control of infinite-dimensional systems whose transfer matrices are expressible as a series connection of a rational transfer matrix and a scalar (possibly irrational) inner function. This class of systems is adequate for describing many control problems in practice, when weighting functions are rational and plants have at most finitely many unstable modes. We show that this problem can be reduced to solving two matrix-valued Riccati equations and an additional rank condition. Furthermore, the obtained controller structure is characterized by the inner function and controllers for the finite-dimensional part. This result provides us with a solution that is easily implementable without much prior knowledge on infinite-dimensional control theory. A numerical example is given to illustrate the result.