**Abdelghani Djeddi 1,†, Djalel Dib 1,†, Ahmad Taher Azar 2,3,\*,† and Salem Abdelmalek 4,†**


Received: 21 August 2019; Accepted: 11 October 2019; Published: 16 October 2019

**Abstract:** This paper presents a new procedure for designing a fractional order unknown input observer (FOUIO) for nonlinear systems represented by a fractional-order Takagi–Sugeno (FOTS) model with unmeasurable premise variables (UPV). Most of the current research on fractional order systems considers models using measurable premise variables (MPV) and therefore cannot be utilized when premise variables are not measurable. The concept of the proposed is to model the FOTS with UPV into an uncertain FOTS model by presenting the estimated state in the model. First, the fractional-order extension of Lyapunov theory is used to investigate the convergence conditions of the FOUIO, and the linear matrix inequalities (LMIs) provide the stability condition. Secondly, performances of the proposed FOUIO are improved by the reduction of bounded external disturbances. Finally, an example is provided to clarify the proposed method. The obtained results show that a good convergence of the outputs and the state estimation errors were observed using the new proposed FOUIO.

**Keywords:** fractional order unknown input fuzzy observer; fractional order Takagi–Sugeno models; *L*<sup>2</sup> optimization; linear matrix inequalities; unmeasurable premise variables
