**3. Fractional Order Takagi–Sugeno Model**

Consider the following nonlinear system given by [54]:

$$\begin{cases} \ \_dD\_t^\alpha \mathbf{x}(t) = f\left(\mathbf{x}, \boldsymbol{\mu}\right),\\ \ \_y(t) = \mathbf{g}\left(\mathbf{x}, \boldsymbol{\mu}\right), \end{cases} \tag{6}$$

where *<sup>x</sup>*(*t*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* and *<sup>α</sup>* is the fractional order derivative. *<sup>f</sup>* and *<sup>g</sup>* are nonlinear functions.

Using the well-known transformation by nonlinear sector, the following TS fuzzy system is given [49]:

$$\begin{cases} \ \_{a}D\_{t}^{\mu}\mathbf{x}(t) = \sum\_{i=0}^{M} h\_{i} \left( \zeta(t) \right) \left[ A\_{i}\mathbf{x}(t) + B\_{i}\mathbf{u}(t) \right], \\\ \mathbf{y}(t) = \sum\_{i=0}^{M} h\_{i} \left( \zeta(t) \right) \left[ \mathbf{C}\_{i}\mathbf{x}(t) + D\_{i}\mathbf{u}(t) \right], \end{cases} \tag{7}$$

where *<sup>u</sup>*(*t*) <sup>∈</sup> <sup>R</sup>*<sup>m</sup>* is the input vector, *<sup>x</sup>*(*t*) <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is the state vector, and *<sup>y</sup>*(*t*) <sup>∈</sup> <sup>R</sup>*<sup>p</sup>* represents the output vector. *Ai* <sup>∈</sup> <sup>R</sup>*n*×*n*, *Bi* <sup>∈</sup> <sup>R</sup>*n*×*nu* , *Ci* <sup>∈</sup> <sup>R</sup>*n*×*ny* and *Di* <sup>∈</sup> <sup>R</sup>*ny*×*nu* are known matrices. *hi* (*ξ*(*t*)) are the weighting functions relying on the premise variables *ξ*(*t*) which can be measurable (input or output of the system) or unmeasurable variables (state of the system). It can also be an external signal. These functions confirm the following so-called convex sum property:

$$\begin{cases} \displaystyle \sum\_{i=0}^{M} h\_i \left( \xi(t) \right) = 1, \\\displaystyle 0 \le l\_i \left( \xi(t) \right) \le 1, \quad \forall i \in \{1, 2, \ldots, M\}. \end{cases} \tag{8}$$

In this paper, the target is to model a fractional order fuzzy TS observer with UPV. Thus, the work is dedicated to the problem of state estimation for nonlinear fractional order systems characterized by continuous time TS models, with unknown input *u*¯(*t*).

Under the hypothesis *C*<sup>1</sup> = *C*<sup>2</sup> = ··· = *C* and *Di* = 0, the FOTS model in the presence of unknown inputs and a measurable premise variable can be defined as follows:

$$\begin{cases} \ \_dD\_t^\mathbf{u} \mathbf{x}(t) = \sum\_{i=0}^M h\_i \left( \mathbf{x}(t) \right) \left[ A\_i \mathbf{x}(t) + B\_i \mathbf{u}(t) + E\_i \mathbf{u}(t) \right], \\\ y(t) = \mathbf{C} \mathbf{x}(t) + E \vec{u}(t), \end{cases} \tag{9}$$

where *<sup>u</sup>*¯(*t*) <sup>∈</sup> <sup>R</sup>*<sup>q</sup>* (*<sup>q</sup>* <sup>&</sup>lt; *<sup>p</sup>*) is the input vector, and *Ei* <sup>∈</sup> <sup>R</sup>*n*×*nu*¯ and *<sup>E</sup>* <sup>∈</sup> <sup>R</sup>*ny*×*nu*¯ are known matrices. This can be rewritten as:

$$\begin{cases} \ \_4D\_t^a \mathbf{x}(t) = \sum\_{l=0}^M h\_l \left( \hat{\mathbf{x}}(t) \right) \left[ A \mathbf{x}(t) + B \boldsymbol{u}(t) + E \boldsymbol{u}(t) + \left( h\_l \left( \mathbf{x}(t) \right) - h\_l \left( \hat{\mathbf{x}}(t) \right) \right) \left( A \mathbf{x}(t) + B \boldsymbol{u}(t) + E \boldsymbol{u}(t) \right) \right] \\\ \quad \mathbf{y}(t) = \mathbf{C} \mathbf{x}(t) + E \boldsymbol{u}(t). \end{cases} \tag{10}$$

After rewriting the model (9), we obtain:

$$\begin{cases} \ \_{a}D\_{t}^{a}\mathbf{x}(t) = \sum\_{i=0}^{M} h\_{i} \left( \dot{\mathbf{x}}(t) \right) \left[ A\_{i}\mathbf{x}(t) + B\_{i}u(t) + E\_{i}\vec{u}(t) + \omega(t) \right],\\ \ y(t) = \mathbf{C}\mathbf{x}(t) + E\vec{u}(t). \end{cases} \tag{11}$$

This form corresponds to a perturbed FOTS model with measurable premise variables (estimated state of the system), where:

$$
\omega(t) = \sum\_{i=1}^{M} \left( h\_i \left( x(t) - \hat{x}(t) \right) \right) \left[ A\_i x(t) + B\_i u(t) + E\_i d(t) \right]. \tag{12}
$$

This term is considered a global bounded and asymptotically vanishing perturbation.
