Hâˆ‍ and Sliding Mode Observers for Linear Time Invariant Fractional Order Dynamic Systems With Initial Memory Effect

Abstract

The Hâˆ‍ and sliding mode observers are important in integerorder dynamic systems. However, these observers are not well explored in the field of fractionalorder dynamic systems. In this paper, the Hâˆ‍ filter and the fractionalorder sliding mode unknown input observer are developed to estimate state of the linear timeinvariant fractionalorder dynamic systems with consideration of proper initial memory effect. As the first result, the fractionalorder Hâˆ‍ filter is introduced, and it is shown that the gain from the noise to the estimation error is bounded in the sense of the Hâˆ‍ norm. Based on the extended bounded real lemma, the Hâˆ‍ filter design is formulated in a linear matrix inequality form, and it will be seen that numerical methods to solve convex optimization problems are feasible in fractionalorder systems (FOSs). As the second result of this paper, not only state but also unknown input disturbance are estimated by fractionalorder slidingmode unknown input observer, simultaneously. In this paper, it is shown that the design and stability analysis of the two estimation techniques are not related with the initial history. Through two numerical examples, the performance of the fractionalorder Hâˆ‍ filter and the fractionalorder slidingmode observer is illustrated with consideration of the initialization functions.