Arbitrarily Low Sensitivity (ALS) in Linear Distributed Systems Using Pointwise Linear Feedback

One of the major reasons for using feedback is to reduce the sensitivity of systems to (uncertain) plant parameters. In this note, the sensitivity problem is defined for feedback systems with plants described by linear partial differential operators having constant coefficients in a bounded one-dimensional domain. There are also finitely many observation points (and also finitely many lumped feedback loops) and a finite number of (disturbance) inputs. The sensitivity problem is then studied in detail for the heat equation, and comments are made about the linearized (damped) beam equation and the (damped) wave equation. It will be shown that it is possible to reduce, arbitrarily, the sensitivity over any (temporal) frequency interval, uniformly in the space domain (except for the undamped wave equation, where a limitation in the frequency interval is induced by the plant). This reduction may require high-gain feedback around the points where the disturbances appear. Finally, the negative effect (on ALS) of poor knowledge of the disturbance input points is discussed, and possible solutions are suggested.