State-space representation for CRONE controllers

This article proposes an infinite-dimensitional state-space representation and a finite-dimensional approximation for fractional linear filters with transfer function C-d(s) := C-0 ((1 + s/w(b)) /(1 + s/w(h)))(d). where 0 < w(b) < w(h) and d is a real number. This representation is derived from the Taylor expansion in zero of the function (1 - z)(d), and is made up of an infinite number of first-order ordinary differential equations. Finite-dimensional approximations obtained by truncating this representation are shown to converge towards C-d in H-infinity. The use of Cd in a standard linear feedback loop (CRONE control) is discussed, and closed-loop exponential and input-output stabilities are shown to be equivalent. The example of CRONE car suspension is presented, for which robustness of closed-loop resonance and step response overshoot vis-a-vis a variation in the vehicule mass is achieved. Copyright (C) 1998 IFAC.