New Approach To The Formation Of The Adequate Diagnostic Matrix Of The Gas Turbine Engine

This paper presents the new approach to the formation of the gas turbine engine diagnostic matrix employing A.Tikhonov’s regularization method and taking into account the compressor properties shift under the condition of engine air-gas channel alteration. This method allows eliminating the certain inadequacy of the diagnostic matrices in some cases and removes the restrictions on their implementation for gas turbine engines diagnostics. The elaborated regularization algorithm of the calculation-identification matrix reversion permits to determine the diagnostic matrix persistently. The suggested method of registration of the compressor properties shift allows providing the adequacy of the engine mathematical model taking into consideration the depreciation of the engine and air-gas channel and consequently obtaining the adequate diagnostic matrix. It is offered to employ the obtained diagnostic model in the on-board systems of the gas turbine engine control and diagnostics. INTRODUCTION Gas turbine engine is a complex technical system, operation of which is determined by the whole range of the complicated physical processes, interconnected and interacting tightly. The contemporary gas turbine engines comprise a lot of components and have a complicated structure; these components interact continuously with the external environment in the process of operation, and cooperate with other subsystems of the aircraft as well. The accurate forecast cannot be done for the further technical state of such complicated system as aircraft engine since it is impossible to present the full comprehensive description of the impact different factors can have on the engine in different situations. Accordingly, there is a necessity to take the decision on the engine state under the circumstance of indeterminacy. The task solved by the diagnostics system in the operation process concerns receiving the data necessary for exact decision taking the further engine running as an object of diagnostics, and for possible maximal decreasing the existing indeterminacy. Nowadays there are different methods and diagnostic models for control and diagnostics of the gas turbine engines, but it is worth mentioning that there is no universal method or model. That is the reason why the diagnostics system comprises as a rule the whole complex of models and methods employed for determining and forecasting the engine technical state. Mathematical models as a means of forecasting the object state possess an important advantage – it is the possibility of examining the engine state under the influence of different factors under different conditions. The issue of developing the general mathematical model is identification and discovering connections between the fact of malfunction appearance and the engine parameters alteration as an object of diagnostics. The parameters under control are chosen on this purpose, then the algorithms of parameters transformation are worked out, the accuracy of measurement and the diagnostics regularity are specified, the complete association between the diagnostics results and the engine technical state are established. One of the methods of the engine air gas channel diagnostics is the method of diagnostic matrices. The essence of the method is the following: the equations in minor deviations describing the gas turbine engine become the basis for constructing and forming the diagnostic matrix 1 C A B − = , where the matrix A components comprise the coefficients of the calculated parameters, and the matrix B components comprise the coefficients of measured parameters. As it was said above the task of determination of the gas turbine engine diagnostic matrix is an equivalent of the task of the stable reversion of the calculation identification matrix which as a rule occurred to be the ill-conditioned one (here is the matrix of the first type) or the matrix, the components of which fully or partly are specified approximately (here is the matrix of the first type also) or it is the underdetermined matrix since the values of certain components are not identified (here is the matrix of the second type). Consequently, for its stable reversion it is necessary to employ the special methods implementing the apparatus of the poorly conditioned problems theory (Morozov 1984, Engl et al 1985). In case when the calculation identification matrix is the matrix of the first type, the general theoretical fundamental of constructing the regulating reversed Proceedings 25th European Conference on Modelling and Simulation ©ECMS Tadeusz Burczynski, Joanna Kolodziej Aleksander Byrski, Marco Carvalho (Editors) ISBN: 978-0-9564944-2-9 / ISBN: 978-0-9564944-3-6 (CD) calculation identification matrix is the following (Tikhonov et al 1986): the element/vector 0 u U ≡ ∉ is given, where U is Hilbert space (in this case it is finite dimensional) and the equation Az u, = (1) is considered, where A : Z U → is a linear operator (in this case it is finite dimensional, in other words matrix), Z is Hilbert space, and z Z ∉ is the unknown desired element/vector. Then finding the element z Z ∉ which is to be continuously dependent on the right part u of the equation (1) will result in determining the stable reversed operator 1 A . − For fulfilling this, instead of the operator equation (1) the optimization problem U z Z Az u min ∉ − → (2) is considered { } R marks the class of these positive definite self-adjoint operators Q that the quadric form ( ) 2 U Z Az Qz,z + is not less than 2 2 Z z , β ⋅ in other words ( ) 2 2 2 U Z Z Az qz,z z , β + ≥ ⋅ where ( ) 0 A,Q β β = > is a constant, not dependent on ( ) z D Q . ∉ The composed function is introduced: [ ] ( ) ( ) 2 def U Z M Q; z,u Az u Qz,z , z D Q . ≡ − + ∉ Then the element ( ) ex z D Q , ∉ minimizing the composed function [ ] M Q,z,u , satisfies the equation ( ) * * Q A A z A u, + = having the unique solution ( ) z R Q u, = where ( ) ( ) 1 def * * R Q Q A A A . − ≡ + Then the element u is given approximately, in other words it is assumed that u u , δ = + % where δ is a certain random process with the values in U, for which the probabilistic average is zero, in other words { } 0 E . δ = It is marked ( ) ( ) ( ) { } 2 2 def z Q;z u ; E R Q z z u δ   ∆ ≡ −   % and ( ) { } { } ( ) ( ) { } { } ( ) 2 2