Previous chapters have emphasized use of models developed from first principles, or models motivated by physical considerations. Unfortunately physical information is unavailable or only partially available for some systems. Examples include systems that have no physical basis (e.g., economic), very complex systems that are difficult to model (e.g., chemical production processes, power plans), complex systems affected by many poorly understood inputs (e.g., weather, biological), and systems that are partially well modeled but the driving inputs are stochastic with unknown correlation characteristics (e.g., the maneuvering tank problem of Section 3.2.1). In these cases it is often necessary to develop estimation models using empirical methods.
There are two primary methods used for empirical model development. The first approach computes the power spectrum (power spectral density or PSD) of a measured output signal and then attempts to infer the differential or difference equations that will produce that spectrum when driven by white noise inputs. Since white noise has a uniform spectrum, the output spectrum is the magnitude squared of the system transfer function (Fourier transform of the system impulse response). The power spectrum defines the magnitude but not the phase response, so the computed impulse response function is non-unique, That is not a problem for real-valued signals because system poles and zeroes must occur in complex conjugate ...