Adaptive observer design for time-varying nonlinear systems with unknown polynomial parameters

Many control methods involve the use of real-time values of the vector of state variables or its estimates. The article considers the problem of state variables observer design for a nonlinear non-stationary plant of a wider class compared to the known analogs. To solve the problem, some assumptions are introduced and assume that the plant parameters are partially unknown functions of time that have a polynomial form. Each unknown parameter is polynomial functions of time with unknown coefficients. The problem of observer design is solved in a class of identification methods that involve the transformation of the original nonlinear mathematical model of the plant to a linear static regression. In this problem, instead of the usual unknown constant parameters, there are unknown functions of time which are estimated. To recover variables of unknown parameters, the method of dynamic regressor extension and mixing (DREM) is used. The method allows getting monotone estimates, as well as accelerating the convergence of estimates to true values. The proposed approach allows obtaining accurate parametrizations of a nonlinear nonstationary system, including exponentially decaying terms associated with using dynamic filters. The resulting regression equations explicitly depend on the tuning parameters and changing the values of these parameters yields a system of linearly independent regression equations, which can be decomposed then into scalar regression equations. An observer of the parameters and state variables of the system is designed on the basis of scalar regression equations and considered assumptions about models of non-stationary parameters. The application of the proposed approach allows solving the problems of restoring unmeasured variables and signals of real control systems and also makes it possible to identify unknown time-varying parameters, which in turn is an actual self-contained problem. The approach can be applied in control of chemical processes, electrical converters, as well as in a number of other technical applications.