**1. Introduction**

Recently, the interest in fractional derivatives and integral applications, as well as in theoretical and practical works, has grown immensely, see for example [1–6]. The main aspects, concept and several applications of fractional calculus are outlined, for example, in [7–14]. This is essentially due to the fact that various physical systems are well described by a fractional order state equation [15–17].

Growing applications have attracted interest in studying the state estimation of fractional differential equations in a linear case [18–21] and in a nonlinear case [22–25]. It is well known that the study of the problem of stabilization of the fractional order system is particularly important for the synthesis of the observer [26–36].

Takagi–Sugeno (TS) fuzzy models have also attracted attention in recent years. The main feature of this class of nonlinear models is to represent the local dynamics of each fuzzy implication (rule) by linear system models. It has been effectively employed in the implementation of nonlinear systems [37–41]. Takagi–Sugeno models have been broadly utilized to represent nonlinear integer-order systems. However, the fuzzy Takagi–Sugeno scheme remains very efficient for nonlinear fractional

order systems (FOS) [42–44]. Therefore, the use of fractional-order Takagi–Sugeno (FOTS) models to represent nonlinear FOS will be introduced in this paper. Several approaches confirm that the validity functions of Takagi–Sugeno representation rely on measurable premise variables, whereas various applications, like diagnosis, consider that those variables rely on the input and state variables of the system that are usually immeasurable [45–48].

Takagi–Sugeno uses premise variables for computing weighting functions. Premise variables can be known (inputs or outputs of the system), or unknown variables taken as the state of the system to be estimated. State variables are usually unmeasurable, but they can be measured by the introduction of sensors, with an additional cost, but the right choice is to estimate the state variables in order to avoid the effects of sensor and shareholder faults that may have appeared on the inputs or outputs of the system considered. This justifies the research works on the state estimation of systems [48,49]. In order to use the state of the system as premise variables, then the states must be estimated, hence the need to synthesize an adequate observer able to estimate the state of the system despite the presence of unknown inputs and disturbances. Hence, it is motivating to deem the common state of unmeasurable variables such as system states. The problem appears especially in the structure of the state of the TS observer.

To implement a fuzzy observer for TS systems with unmeasurable premise variables (UPV), several methods have been evolved, comprising those which take account of analytical advances of an estimation error [50–52], and those which use the error description by a TS model with uncertainty or unorganized disruption [49]. The present work presents the Takagi–Sugeno unknown input fractional order observers design for FOTS models with UPV.

The main objective of the current paper is to found new stability and stabilization conditions using FOTS systems with UPV in the continuous case, to implement observers for nonlinear systems. The case where the weighting functions rely on premise variables depending on unmeasurable system states is considered. First, the representation of FOTS systems with UPV and their observers will be considered, which are given under the linear matrix inequalitie (LMI) formulation. Then, an analysis of the stability of the state estimation error studied by using the minimization of the *L*<sup>2</sup> norm of the transfer from bounded unknown exogenous disturbances to the state estimation error will be established. An application example is designed to demonstrate the performance of the suggested approach.

This paper is organized in the following way. The next section provides some background on the fractional calculus. The FOTS model is presented in Section 3. The main results of the paper, namely the synthesis of the fractional fuzzy observer based unmeasurable premise variables, are presented in Section 4. A new proposed method for unknown input estimation of the fractional order Takagi–Sugeno unknown input observer is given in Section 5. A numerical example is given in Section 6 to demonstrate the efficiency and validity of the proposed approach. Finally, the paper ends with concluding remarks and future perspectives in Section 7.
