The Spectral-Trajectory Identification of Complex Objects by the Realizations of Measured Signals at Operation Monitoring, Control and Diagnostics

An approach of modeling changes in the signal at the output of a complex object operating in the mode of repeated operation cycles with a constant observation interval is discussed in the article. This type includes signals from systems of active diagnostics, active monitoring, adaptive control with an identification dedicated interval, and others. The coefficients of the spectral decomposition of the signal realizations according to the adaptive orthonormal basis are taken as state variables that determine the changes in the output signal. The trajectories of changes in these coefficients in the state space determine the type of the signal change. The article shows that there are standard types of trajectories corresponding typical signal changes. In particular the variability of the spectral decomposition coefficients can be described by algebraic polynomials of various degrees depending on the complexity of the signal variation. That is there is algebraic-polynomial variability of spectral characteristics. The typical changes in the signal leading to the indicated variability are considered in the article: scaling by amplitude; changing phases of all harmonic components of the signal by the same angle; compression or stretching in time; time shift or signal delay. Each of the listed changes is characterized by its own set of polynomial trajectories for the spectral coefficients. This relationship between changes in the signal and the trajectories of changes in the decomposition coefficients is universal and it is preserved for arbitrary time-finite signals. This property can be considered as a general theoretical one describing the deterministic relationship of changes in spectral characteristics from specific changes in the wave shape. In practice it can be used at modeling and recognize changes in signals in control systems, monitoring, diagnostics, nondestructive testing and other digital signal processing systems.