On Problem of Parameters Identification of Dynamic Object

In this paper, some problem formulations of dynamic object parameters recovery described by non-autonomous system of ordinary differential equations with multipoint unshared edge conditions are investigated. Depending on the number of additional conditions the problem is reduced to an algebraic equations system or to a problem of quadratic programming. With this purpose the paper offers a new scheme of the edge conditions transfer method called by conditions shift. The method permits to get rid from differential links and multipoint unshared initially-edge conditions. The advantage of the proposed approach is concluded by capabilities of reduction of a parametric identification problem to essential simple problems of the solution of an algebraic system or quadratic programming. Keywords—dynamic objects, ordinary differential equations, multipoint unshared edge conditions, quadratic programming, conditions shift I. PROBLEM FORMULATION ET’s consider a problem of parameters identification of a linear non-autonomous dynamic system: [ ], , ), ( ) ( ) ( ) ( ) ( 0 k t t t t C p t B t x t A t x ∈ + + = & (1) where n E t x ∈ ) ( phase state of system; l E p∈ required parameters; ) ( ), ( ), ( t C t B t A given matrixes with dimensions ) 1 ( ), ( ), ( × × × n l n n n respectively, moreover const t A ≡/ ) ( . There are m initially-edge conditions of a system that can also depend on unknown parameters: β ξ α ν ν ν ˆ ˆ ) ˆ ( ˆ 0 = + ∑ = p t x k , (2) where , ,..., 1 , 0 , ˆ k t = ν ν -times moments from ], , [ 0 k t t k k t t t t = = ˆ , ˆ 0 0 , the matrixes β ξ α ν ˆ , ˆ , ˆ with respective dimensions ) 1 ( ), ( ), ( × × × m l m n m are given. Let's mark a general problem of linear systems of differential equations with the multipoint unshared edge conditions. The problem (1), (2), generally speaking, concerns to this class of problems at fixed values of parameter p . The problem is connected to the complexity of obtaining of constructive necessary and sufficient conditions of the solution existence of a boundary value problem such as (1), (2), that is studied by many scientists, starting from activities of Tamarkin, Valle-Poussin and other scientists ([1], [3]). Let ], , [ ), , max( ) ( , ) ( 0 k t t t l n t B rang n t A rang ∈ = = m rang rang k k = = ] ˆ , , ˆ ,..., ˆ [ ] , ˆ ,..., ˆ [ 0 0 β ξ α α ξ α α . Kamil Aida-zade is with the State Oil Academy of Azerbaijan, Department of Applied Mathematics, Baku, Azerbaijan. Cemal Ardil is with the Azerbaijan National Academy of Aviation, Baku, Azerbaijan. Depending on a ratio between values of matrixes ranks, participating in (1), (2), the following cases, corresponding to the different problem formulations, are possible. Case А: l n m + = . Then there is a single vector of parameters p and corresponding solution of a boundary value problem (1), (2) (problem A). Case В: l n m + < . Then the system vector of parameters, satisfying (1), (2), is not unique and there are additional conditions on system parameters and status in the form of equality with the number no more than m l n − + and inequality