Zastosowanie LMI do syntezy obserwatorów liniowych ciągłych układów niecałkowitego rzędu

W pracy rozpatrzono problem syntezy obserwatorów liniowych układów ciągłych niecałkowitego rzędu. Wykorzystując aparat liniowych nierówności macierzowych (LMI) sformułowano warunki oraz podano procedury do wyznaczania macierzy wzmocnień obserwatorów, dla rzędu ? układu: 0 < α < 1 i 1 < α < 2. Rozważania zilustrowano przykładem liczbowym. Obliczenia i symulacje wykonano w środowisku Matlab/Simulink.

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Many sophisticated analytical procedures to control system design are based on the assumption that the full state vector is available for measurement. The example of such control procedure is placement of the unstable system eigenvalues. In many systems of practical importance, however, the entire state vector is not available for measurement. In some systems, for example, measurements may require the use of costly measurement devices and it may be unreasonable to measure all state variables. An auxiliary dynamical system, which reconstructs the state vector, is known as a full-order or an identity observer, and is coupled to the original system through the available system inputs and outputs [14]. The paper presents a problem of synthesis of full-order observers for fractional continuous-time linear systems. It has been shown that this problem can be formulated and solved by the use of linear matrix inequalities (LMI) methods [15]. Two cases of fractional order i.e. 0 < α < 1 and 1 < α < 2 of the system (1) have been considered. Necessary and sufficient conditions (Theorem 1 and 2) for solvability of the problem as well as two procedures (Procedure 1 and 2) for computation of a gain matrix L of asymptotic stable observer (2) have been given. The proposed approach is illustrated with a practical example. Numerical calculations have been performed in the Matlab package and accompanied by public domain software: SeDuMi solver and YALMIP parser. The LMI approach to observer synthesis for fractional continuous-time linear systems has not been considered as yet.