Circle Criterion and Boundary Control Systems in Factor Form: Input–output Approach

A circle criterion is obtained for a SISO Lur’e feedback control system consisting of a nonlinear static sector-type controller and a linear boundary control system in factor form on an infinite-dimensional Hilbert state space H previously introduced by the authors (Grabowski and Callier, 1999). It is assumed for the latter that (a) the observation functional is infinite-time admissible, (b) the factor control vector satisfies a compatibility condition, and (c) the transfer function belongs to H(Π) and satisfies a frequency-domain inequality of the circle criterion type. We also require that the closed-loop system be wellposed, i.e. for any initial state x0 ∈ H the truncated input and output signals uT , yT belong to L (0, T ) for any T > 0. The technique of the proof adapts Desoer-Vidyasagar’s circle criterion method (Desoer and Vidyasagar, 1975, Ch. 3, Secs. 1 and 2, pp. 37–43, Ch. 5, Sec. 2, pp. 139–142 and Ch. 6, Secs. 3 and 4, pp. 172–174), and uses the input-output map developed by the authors (Grabowski and Callier, 2001). The results are illustrated by two transmission line examples: (a) that of the loaded distortionless RLCG type, and (b) that of the unloaded RC type. The conclusion contains a discussion on improving the results by the loop-transformation technique.