State observers in the design of eigenstructures for enhanced sensitivity

The problem of closed-loop enhanced sensitivity design is as follows: Given a linear time invariant system, find a (realizable) feedback gain such that: (1) the closed-loop is stable in the reference and the potentially damaged states, and (2) the eigenstructure includes a subset of poles, with desirable derivatives, that lie in a part of the plane where identification is feasible. This paper shows that pole derivatives with respect to system parameters for a controller/observer system, contrary to the assumption often made, depend on both the controller and the observer gains, i.e. the separation principle holds for placing the poles but does not extend to the pole derivatives. Closed-form expressions for the derivatives with due consideration to both gains are presented. Examination shows that the sum of these derivatives is independent of both gains, is constant along the nonlinear paths traced by the poles as damage increases and, provided the damage affects only the stiffness, is nearly zero.