The ideal design of an airplane should include built-in sensors that are pre-blended in the perfect aerodynamic shape. Each built-in sensor is expensive to blend in and requires continuous maintenance and data processing, so we would like to use as few sensors as possible. The ideal formulation of the corresponding optimization problem is, e.g., to minimize the average detection error for fault locations. However, there are two obstacles to this ideal formulation:
--First, this ideal formulation requires that we know the probabilities of different fault locations and the probabilities of different aircraft exploitation regimes. In reality, especially for a new aircraft, we do not have those statistics (and for the aging aircraft, the statistics gathered from its earlier usage may not be applicable to its current state). Therefore, instead of a well-defined optimization problem, we face a problem of not so well defined problem of optimization under uncertainty.
--Second, even if we know the probabilities, the corresponding optimization problem is very computation-consuming and difficult to solve.
In this paper, we overcome the first obstacle by using maximum entropy approach (MaxEnt) to select the corresponding probability distributions.
To overcome the second obstacle, we use the symmetry approach. Namely, the basic surface shapes are symmetric (with respect to some geometric transformations such as rotations or shifts). The MaxEnt approach results in distributions that are invariant with respect to these symmetries, and therefore, the resulting optimality criterion (be it the minimum of detection error, or the minimum of fault location error, etc.) is also invariant with respect to these same symmetries. It turns out that for an arbitrary optimality criterion that satisfies the natural symmetry conditions (crudely speaking, that the relative quality of two sensor placements should not change if we simply shift or rotate two placements), the line formed by the optimally placed sensors (optimally with respect to this criterion) can be described as an orbit of the corresponding Lie transformation groups. As a result, we describe the optimal sensor placements.
A similar problem of optimal sensor placement is also discussed for space structures.