A problem of on-line continuous-time estimation of parameters of polyharmonic function

A problem of on-line continuous-time estimation of parameters (frequencies and amplitudes) of a polyharmonic function is considered. Given problem is a classical problem of the systems theory. Nonlinear dependency of the function from its parameters prevents using standard well-known methods of adaptive control and identification, since the presentation of periodic function as output of a dynamic system leads to the problem of simultaneous estimation of state variables and unknown parameters, that presents the special difficulty in continuous-time. In the recent papers dedicated to the given problem the algorithms of estimation of frequency of the measured sinusoidal signal J=Asin(ωt+φ) were proposed. However these algorithms can not be developed for the estimation of parameters of a polyharmonic signal. A solution of considered problem can find wide practical application in self-learning of robotic systems and adaptive noise damping systems. For example, the problem of overcoming of function uncertainties in control laws arises from the design of mobile robot control systems for a motion along physically detectable, but analytically unknown paths (border of physical object). Solution of this problem is based an using a strategy of self-learning and an algorithm of the approximation of the unknown periodic functional dependencies. Analytical conditions of existence of the solution are presented for the case of on-line continuous-time estimation of the parameters of a polyharmonic function. The design procedure of the estimation algorithm is proposed