Robust multivariable PI-controllers for linear systems in Banach state spaces

In this paper we give a generalization of the PI-controller design theory proposed by Kobayashi [T.A. Kobayashi, Digital PI-controller for distributed parameter systems, SIAM J. Control Optim. 26 (1988) 1399–1414] and later by Xu and Jerbi [C.Z. Xu, H. Jerbi, A robust PI-controller for infinite-dimensional systems, Internat. J. Control 61 (1995) 33–45] to stabilize and regulate a large class of infinite-dimensional linear systems in Banach space. We prove that even if the dynamic operator A of the open loop system does not generate a holomorphic semigroup, the invertibility of C A −1 B (where C is an unbounded observation operator and B is a bounded control operator for the system) is still a necessary and sufficient condition for the existence of a robust PI-controller.