Unknown input observer design for a class of fractional order nonlinear systems

Analysis and control of fractional order (FO) nonlinear systems is a challenging problem. In earlier works, as highlighted in literature, stability conditions for the FO LTI systems are analytically derived and these results are extended to formulate LMI conditions to express the stability of the FO LTI systems. In present work, design of full order and reduced order observers for imperfect fractional order nonlinear systems is presented. Imperfections in real system are silent dynamics and can be modeled as unknown input. To design observer for such system, unknown input observer (UIO) design concepts are used and LMI conditions for the existence of observer are analytically derived. For this purpose, Differential Mean Value (DMV) theorem is used and nonlinear term in the error dynamics is alternatively expressed in appropriate equivalent form. As a result, error dynamics evolves as Linear Parameter Varying (LPV) system and then stability results for FO LTI systems are extended to stabilize FO nonlinear error dynamical systems. LMI conditions for the existence of unknown input observer for the two cases 0 < α < 1 and 1 < α < 2 are analytically derived. Feasible solution of LMI gives the observer design matrices directly. Finally, results of simulation are presented to authenticate the proposed approach.