Identification of parameters of non-linear oscillators

Many applied control problems are characterized by a situation where some or all parameters of the initial dynamic system are unknown. In such cases, the problem of identification arises, which consists in determining the unknown parameters of the system based on information about its output - known information about movement. The ability to solve the problem of identification is an essential property of identifiability depends on the analytical structure of the right-hand sides of the dynamics equations and available information [1]. To solve the identification problem itself, this work uses the method of invariant relations [2], which was developed in analytical mechanics and is intended, in particular, for finding partial solutions (dependencies between variables) in problems of the dynamics of a rigid body with a fixed point. The modification of this method to the problems of the theory of control, observation made it possible to synthesize additional connections between the known and unknown quantities of the original system that arise during the movement of its extended model [3 - 5]. It is worth noting that a some more general approach, which forms a suitable method for solving observation problems for nonlinear dynamic systems due to the synthesis of an invariant manifold in the space of an extended system, was proposed in the works [6], [7] as a certain modification of the method stabilization of nonlinear systems I&I (Input and Invariance). The purpose of this work is to spread the method of synthesis of invariant relations in control problems to the problem of identifying parameters of pendulum systems. A general scheme for constructing asymptotically accurate estimates of the parmeters of a two-dimensional dynamical system is proposed. A relatively simple case of the identification problem will be considered, namely: 1) the output of the original system is the complete phase vector and 2) the system depends linearly on the unknown parameters. Generalizations to more general designs of input-output systems, including with the involvement of information about the output obtained on several trajectories, can be carried out using the approach described below and is the subject of a separate study. The computational experiment on the estimation of the parameters of the mathematical pendulum confirms the efficiency of the proposed identification scheme.