On the Shape of Finite Dimensional Approximations of Feedback Functional Gains

An important technique in design of controllers for uid ow is that of compensator design. The use of compensators allows for measurement of the state at a small number of places in the uid and estimation of the state based on those measurements. However, questions arise regarding what states to measure, what locations to place sensors and how to design low order (practical) controllers. One approach to low order compensator design has been discussed in BK2] and uses the spatial characteristic of feedback gains to design reduced order controllers. That work relies on approximating the gains for the innnite dimensional system by a small number of low order polynomials (constants or linear elements). As we show through numerical examples for the simply supported Euler Bernoulli beam, the spatial structure of the approximating feedback gains can depend on the inner product chosen for the Hilbert space. The reason for this is that the inner product shapes the Galerkin approximations and projections. Recently F] it was shown that stability behavior for approximations could be improved by using an inner product diierent than the standard energy inner product. In this paper we explore the eeects of inner product choice on the structure of approximate feedback functional gains.