Theoretical Bases of Diagnostic Type Functional Observers

Abstract

The given paper considers the problem of design of a full-scale vector restorer, that is invariant to indefinite inputs. It is considered that a diagnostic restorer and full-scale vector restorer are concept relatives, however not completely equivalent. It is possible to be convinced of it, considering primary differential signal which is formed by comparison of exits of the real operating object and its mathematical model. The initial signal of mathematical model represents only an assessment of an exit of real object the formations of which are defined by quantity and quality of available a priori information. Ideally, at lack of the destabilizing factors, absolute adequacy of mathematical model to real object guarantees a zero signal of a error. Actually, it is impossible to consider in mathematical model all destabilizing factors, and there is no requirement as the model received thus will be too complex and unproductive. Therefore in practice, the actual differential signal is quite slow physical oscillatory process around zero level. All factors which unaccounted in mathematical model are reflected in it, including: unforeseen perturbations and malfunctions, effects of modeling errors, use of models of the underestimated order, fluctuations of parameters of system, instability of a working point, unaccounted or linear nonlinear dependences, organized or natural hindrances, noise, etc.

It is intuitively clear that if the problem of identification and localization of malfunctions is solved, then first, the differential signal has to be sensitive concerning the set list like malfunctions (signatures), and secondly, it is necessary to get rid of the background created by the collateral destabilizing factors which are not of interest. Thus, the differential signal has to be previously processed.The spectral structure of a differential signal is considered. Its components are analyzed. On the basis of the analysis conclusions are drawn: the full-scale state vector restorer filtering of noise in an implicit form as the problem of its mathematical model in state variables automatically creates the corresponding frequency characteristic; selectivity of a full-scale state vector restorer to a certain type of malfunctions is provided with mathematical model of a channel of distribution of fault from an indefinite input to an exit of control object in the assumption that the matrix of distribution of malfunctions is set a priori; the state vector restorer in the Luyenberger’s form will meet the requirements to systems of identification and localization of malfunctions if primary differential signal becomes possible to make independent of cumulative influence result of the destabilizing factor. The mathematical apparatus of a full order restorer for stationary linear dynamic systems, in which system uncertainty is interpreted as indefinite perturbation and in mathematical model are represented as additional uncontrollable inputs. This process is limited to systems of continuous time, however it does not limit generality of the received results as they can be easily transferred to systems of discrete time. Necessary and sufficient conditions for the existence of such restorer are formulated and proved. Recommendations, regarding the design sequence are suggested, special cases are analyzed.