Identification in the presence of symmetry: oscillator networks

Obtaining an accurate quantitative model for a given dynamical system is of central importance in many areas of systems theory, including control theory. In many applications, the basic structural features of the system model can be determined from a consideration of the physical laws which govern the system behavior, so that what remains is parametric system identification, that is, determining the values of the parameters in the model using measurements of inputs and outputs. We focus in particular on adaptive parameter identification (G. Goodwin and K. Sin, 1984; S. Sastry and M. Bodson, 1989), since this approach has proven to be compatible with a great many system theoretic objectives (e.g., model based prediction and control). We begin an investigation of the role that structural properties such as symmetry play in the problem of system identification. For definiteness, we concentrate on determining the values of coupling parameters from observed outputs of systems of coupled oscillators. The techniques are of more general applicability, but by keeping the physical system simple and definite, the effects of symmetry in the analysis can be clearly seen.