The approaches review of the framework application in identification problems is fulfilled. It is showed that this concept can have different interpretations of identification problems. In particular, the framework is understood as a frame, structure, system, platform, concept, and basis. Two directions of this concept application are allocated: 1) the framework integrating the number of methods, approaches or procedures; b) the mapping describing in the generalized view processes and properties in a system. We give the review of approaches that are the basis of the second direction. They are based on the analysis of virtual geometric structures. These mappings (frameworks) differ in the theory of chaos, accidents, and the qualitative theory of dynamic systems. Introduced mappings (frameworks) are not set a priori, and they are determined based of the experimental data processing. The main directions analysis of geometrical frameworks application is fulfilled in structural identification problems of systems. The review includes following directions: i) structural identification of nonlinear systems; ii) an estimation of Lyapunov exponents; iii) structural identifiability of nonlinear systems; iv) the system structure choice with lag variables; v) system attractor reconstruction.

Approach to the analysis of nonlinear dynamic systems structural identifiability (SI) under uncertainty proposed. This approach has a difference from methods applied to SI estimation of dynamic systems in the parametrical space. Structural identifiability interpreted as of the structural identification possibility a nonlinear system part. We show that the input has S-synchronization property for the solution of the SI task. The identifiability method based on the analysis of structures. The input parameter effect on the possibility of the system SI estimation is studied.

Lyapunov exponents (LE) identification prob-lem of dynamic systems with periodic coefficients is con-sidered under uncertainty. LE identification is based on the analysis of framework special class describing dy-namics of their change. Upper bound for the smallest LE and mobility limit for the large LE are obtained and the indicator set of the system is determined. The graphics criteria based on the analysis of framework special class features are proposed for an adequacy estimation of obtained LE estimations. The histogram method is applied to check for obtained estimation set. We show that the dynamic system can have the LE set.

The structural-parametrical method for design of adaptive observers (AO) for nonlinear dynamic sys-tems under uncertainty is proposed. The design of AO is consisting of two stages. The structural stage allowed identifying a class of nonlinearity and its structural pa-rameters. The solution of this task is based on an estima-tion of the system structural identifiability (SI). The method and criteria of the system structural identi-fiability are proposed. Effect of an input on the SI is showed. We believe that the excitation constancy condition is satisfied for system variables. Requirements to the input at stages of structural and parametrical design of AO differ. The parametrical design stage AO uses the results obtained at the first stage of the adaptive observer construction. Two cases of the structural information application are considered. The main attention is focused on the case of the insufficient structural information. Adaptive algorithms for tuning of parameters AO are proposed. The uncertainty estimation procedure is proposed. Stability of the adaptive system is proved. Simulation results confirmed the performance of the proposed approach.

The method for construction adaptive observ-ers (AO) time-varying linear dynamic objects at non-fulfillment of condition excitation constancy (EC) is pro-posed. Synthesis of the adaptive observer is given as the solution of two tasks. The solution first a problem is a choice of the constant matrix decreasing the effect of EC condition. Procedures for obtaining of this matrix are proposed. The matrix specifies restrictions for a vector of parameters AO. The solution of the second problem gives a method of design adaptive multiplicative algorithms in the presence of the obtained restrictions. Procedures for an estimation uncertainty in an object are proposed. They are based on obtaining of static models giving the forecast change of uncertainty. Optimum estimations of the uncertainty are obtained which minimize an error between outputs of the object and AO. An exponential dissipativity of adaptive system is proved. The results of the modeling confirming the effectiveness of designed methods and procedures are presented.

The method of construction adaptive observers for linear time-varying dynamical systems with one input and an output is offered. Adaptive algorithms for identification are designed. Adaptive algorithms not realized as an adaptive system contains parametric uncertainty (PU). Realized adaptive algorithms of identification parameters system are offered. They on the procedure of the estimation PU and algorithm of signal adaptation are based. The algorithm of velocity change system parameters estimation is proposed. Estimations PU and its misalignments are obtained. Boundedness of trajectories an adaptive system is proved. Exponential stability conditions of the adaptive system are obtained. Iterative procedure of construction a parametric restrictions area is proposed. Simulation results have confirmed the efficiency of the method construction an adaptive observer.

The new approach to structural identification of nonlinear dynamic systems under uncertainty is pro-posed. It is based on the analysis of virtual frameworks (VF), reflecting a state of a nonlinear part system. Con-struction VF is based on obtaining special an informa-tional set describing a steady state of a nonlinear dynamic system. Introduction VF demands an estimation of structural identifiability of a system. This concept is associated with nonlinearity of system and properties VF. The method of an estimation of structural identifiability is proposed. The appearance of the insignificant virtual frameworks, not satisfying to the condition of structural identifiability, is considered. Algorithms for an estimation of a nonlinearity class on the basis of the analysis of sector sets are proposed. Methods and procedures of the estimation of framework single-valued and multiple-valued nonlinearities are proposed. The method of the structurally-frequency analysis is proposed and applied to validate the obtained solutions. VF is proposed for identification of an order and a spectrum of eigenvalues of a linear dynamic system. The possibility of application VF for the problem solving of identification static systems is shown.