D-optimal input design for identification of a continuous system using sum of squares polynomial

This paper presents a paradigm for designing inputs to identify linear continuous-time SISO systems. The goal here is to design a band-limited spectrum that is D-optimal and satisfies certain input/output power constraints. The D-optimal criterion with input/output constraints lead to a set of optimal moments. In order to determine the spectrum from these moments, it is first approximated by a finite order sum of squares (s.o.s.) polynomial. It is shown that the coefficients of the s.o.s. polynomial and its moments are linearly related. These coefficients can be determined by minimizing the distance between the optimal moments and the moments of the finite order approximation under the constraint that the spectrum is non-negative. Alternatively, the linear relationship between the moments and coefficients of the spectrum can also be enforced as additional constraints along with the power constraints and non-negativity constraints in the D-optimal framework. A simulation example, with a 2^nd order system, is presented to illustrate the proposed method of determining the power spectrum and the input.