ABSTRACT
Mathematical models as described by equations (2.3) and (2.9) above are of the “forward” type; that is, the parameters m, c, and k are assumed to be known, as well as the initial conditions. The mathematical model then predicts the resultant model behavior y(t) at any time t from the solution formulas (2.4) or (2.10). This is typically the approach taken in sensitivity investigations, which is quite useful, and can provide important features of the model as functions of parameters (see [2, 6, 9, 12] and the references therein). However, in reality, not all parameters are directly measurable (e.g., most springs in mechanical devices come without specification of the spring constant k). Instead, we may have sparse and noisy measurements of displacements (using proximity sensors) and/or accelerations (using accelerometers). From this information, we need to find the unknown parameters. Problems of this type are called inverse or parameter estimation problems and are ubiquitous in modeling. Finding the solutions to an inverse problem is, in general, nontrivial because of non-uniqueness difficulties that arise. This undesirable feature is often due to noisy data and insufficient number of observations. For a discussion on the non-uniqueness as well as other issues such as stability in inverse problems we refer the interested reader to [1, 3].
