**4. Fractional Order Takagi–Sugeno Unknown Input Observer**

The proposed fractional order Takagi–Sugeno fuzzy unknown input observer is given by the following equations:

$$\begin{cases} \ \_{\, \, \, \, \, \, \, \, \, \, \Gamma} \mathbf{1}\_{t}^{a} \mathbf{x}(t) = \sum\_{i=0}^{M} h\_{i} \left( \hat{\mathbf{x}}(t) \right) \left[ \mathbf{N}\_{i} \mathbf{z}(t) + \mathbf{G}\_{i} \boldsymbol{u}(t) + L\_{i} \mathbf{y}(t) \right], \\\ \quad \mathbf{\hat{x}}(t) = \mathbf{z}(t) + \mathbf{H} \mathbf{y}(t). \end{cases} \tag{13}$$

The state and the output estimation can be defined as:

$$\begin{aligned} \dot{\hat{x}}(t) &= \mathbf{x}(t) - \hat{\mathbf{x}}(t), \\ &= \mathbf{x}(t) - z(t) + H \mathbf{C} \mathbf{x}(t) + H \mathbf{E} \mathbf{\bar{u}}(t), \\ &= P \mathbf{x}(t) - z(t) + H \mathbf{E} \mathbf{\bar{u}}(t), \end{aligned} \tag{14}$$

where

$$P = I + HC.\tag{15}$$

Hence, the dynamics of the state estimation error is

$$\begin{split} \, \_{t \, \_0} D\_t^a \mathfrak{X}(t) &= P\_{t \, \_0} D\_t^a \mathfrak{x}(t) - \_{t \, \_0} D\_t^a \mathfrak{z}(t) + H E\_{t \, \_0} D\_t^a \mathfrak{z}(t) \\ &= \sum\_{i=1}^M h\_i \left( \mathfrak{x}(t) \right) \left[ P A\_i \mathfrak{x}(t) + P B\_i \mathfrak{u}(t) + P E\_i \mathfrak{z}(t) \right. \\ &\quad + P \omega(t) - N\_l \mathfrak{z}(t) - G\_l \mathfrak{u}(t) - L\_l \mathfrak{y}(t) \right] + H E\_{t \, \_0} D\_t^a \mathfrak{z}(t), \end{split} \tag{16}$$

replacing *y*(*t*) and *z*(*t*) by their respective expressions given by (11) and (13), the state error is given as follows:

$$\begin{aligned} \mathbf{^\_{L}}D\_{\mathbf{i}}^{\mathbf{a}}\hat{\mathbf{x}}(t) &= \sum\_{i=1}^{M} h\_{i} \left( \hat{\mathbf{x}}(t) \right) \left[ \left( \mathbf{P} \mathbf{A}\_{i} - \mathbf{N}\_{i} - \mathbf{K}\_{i} \mathbf{C} \right) \mathbf{x}(t) + \left( \mathbf{P} \mathbf{B}\_{i} - \mathbf{G}\_{i} \right) \mathbf{u}(t) + \left( \mathbf{P} \mathbf{E}\_{i} - \mathbf{K}\_{i} \mathbf{E} \right) \mathbf{d}(t) + \mathbf{P} \boldsymbol{\omega}(t) \right. \\ &\left. + \mathbf{N}\_{i} \mathbf{e}(t) \right] + H \mathbf{E}\_{l\_{0}} \mathbf{D}\_{l}^{\mathbf{a}} \hat{\mathbf{u}}(t), \end{aligned} \tag{17}$$

with *Ki* = *NiH* + *Li*.

If the next conditions are satisfied:

$$HE = 0,\tag{18}$$

$$N\_{\bar{i}} = PA\_{\bar{i}} - K\_{\bar{i}} \mathbf{C}\_{\prime} \tag{19}$$

$$PB\_{\bar{l}} = G\_{\bar{l}\nu} \tag{20}$$

$$PE\_i = K\_i E \tag{21}$$

and

$$L\_i = K\_i - N\_i H. \tag{22}$$

Then, the dynamics of the state estimation error become:

$$\mathbf{u}\_{t\_0} D\_t^n \vec{\mathbf{x}}(t) = \sum\_{i=1}^{M} h\_i \left( \vec{\mathbf{x}}(t) \right) \left[ N\_i \vec{\mathbf{x}}(t) + P \omega(t) \right],\tag{23}$$

thus showing that the dynamics of the state estimation error is disturbed by *ω*(*t*).

To synthesize the matrices of the observer (13), two techniques are proposed.

## *4.1. First Approach*

It is assumed that the term *ω*(*t*) defined in (12) satisfies the following Lipschitz condition:

$$\left|\omega(t)\right| \le \delta \left|\tilde{\mathbf{x}}(t)\right|,\tag{24}$$

where *δ* is a positive constant.

**Lemma 1** (see [55])**.** *Let M and N be matrices of the appropriate sizes, then the following property holds:*

$$M^T N + N^T M \le \eta M^T M + \eta^{-1} N^T N, \qquad \eta > 0. \tag{25}$$

**Theorem 1.** *A fractional order unknown input observer (13) for system (11) exists if there exists a positive definite matrix X, matrices Mi, S, positive scalars η and δ satisfying the following conditions for all i* = 1, ··· , *M:*

$$
\begin{bmatrix}
\Theta\_l & \left(X + \text{SC}\right) \\
\left(X + \text{SC}\right)^T & -\lambda I
\end{bmatrix} < 0,\tag{26}
$$

$$SE = 0,\tag{27}$$

$$(X + S\mathbb{C})\,E\_i = M\_i E\_\prime \tag{28}$$

*where*

$$\boldsymbol{\Theta}\_{i} = \boldsymbol{A}\_{i}^{T} \left( \boldsymbol{X} + \boldsymbol{\mathsf{C}}^{T} \boldsymbol{S} \right) + \left( \boldsymbol{X} + \boldsymbol{\mathsf{SC}} \right) \boldsymbol{A}\_{i} - \boldsymbol{\mathsf{C}}^{T} \boldsymbol{M}\_{i}^{T} - \boldsymbol{M}\_{i} \boldsymbol{\mathsf{C}} + \eta \boldsymbol{\delta}^{2} \boldsymbol{I}. \tag{29}$$

*Then, the fractional order observer (13) is completely defined by:*

$$H = X^{-1}S\_{\prime} \tag{30}$$

$$K\_i = X^{-1} M\_{i\nu} \tag{31}$$

$$N\_{\bar{i}} = \left(I + H\mathbb{C}\right)A\_{\bar{i}} - K\_{\bar{i}}\mathbb{C}\_{\prime} \tag{32}$$

$$L\_i = K\_i - N\_i H\_\prime \tag{33}$$

*and*

$$G\_{\bar{l}} = \left(I + HC\right)B\_{\bar{l}}.\tag{34}$$

**Proof of Theorem 1.** In order to found the existence conditions of the fractional order observer in Theorem 1, Lemma 1 can be introduced:

Considering the following quadratic Lyapunov function:

$$V(t) = \mathfrak{x}(t)^T X \mathfrak{x}(t), \qquad X = X^T > 0,\tag{35}$$

its derivative with regard to time is given by:

$$D\_{t0}D\_t^\mathbf{u}V(t) \le\_{t0} D\_t^\mathbf{u} \tilde{\mathbf{x}}(t)^T \mathbf{X} \tilde{\mathbf{x}}(t)(t) + \tilde{\mathbf{x}}(t)^T \mathbf{X}\_{t0} D\_t^\mathbf{u} \tilde{\mathbf{x}}(t). \tag{36}$$

By substituting (24), the dynamic of the quadratic Lyapunov function becomes:

$$\mathbb{E}\_{t\_0} D\_t^{\mathbb{E}} V(t) \le \sum\_{i=1}^{M} \mu\_i \left( \hat{\mathbf{x}}(t) \right) \left[ \hat{\mathbf{x}}(t)^T \left( N\_i^T \mathbf{X} + \mathbf{X} N\_i \right) \hat{\mathbf{x}}(t) + \hat{\mathbf{x}}(t)^T \mathbf{X} P \omega(t) + \omega(t)^T \mathbf{P}^T \mathbf{X} \tilde{\mathbf{x}}(t) \right], \tag{37}$$

and when using Lemma 1 and (25), this allows for the following:

$$\begin{split} \|\mathbf{\tilde{x}}^T X P \omega + \omega^T P^T X \mathbf{\tilde{x}} \le \eta \omega^T \omega + \eta^{-1} \mathbf{\tilde{x}}^T X P P^T X \mathbf{\tilde{x}} \\ \le \eta \gamma^2 \mathbf{\tilde{x}}^T \mathbf{\tilde{x}} + \eta^{-1} \mathbf{\tilde{x}}^T X P P^T X \mathbf{\tilde{x}}. \end{split} \tag{38}$$

Substituting (38) in the fractional derivative of the Lyapunov function (36) yields:

$$\mu\_{t\_0} D\_t^{\mathfrak{x}} V = \sum\_{i=1}^{M} \mu\_i \left( \mathfrak{X} \right) \mathfrak{x}^T \left( N\_i^T X + X N\_i + \eta \gamma^2 I + \eta^{-1} X P P^T X \right) \mathfrak{x}. \tag{39}$$

Since the activation functions satisfy condition (8), the fractional derivative of the Lyapunov function is negative if:

$$N\_i^T X + X N\_i + \eta \gamma^2 I + \eta^{-1} X P P^T X < 0. \tag{40}$$

According to (19), Equation (40) becomes:

$$\left(\left(PA\_i - K\_i\mathbb{C}\right)^T X + X\left(PA\_i - K\_i\mathbb{C}\right) + \eta\delta^2 I + \eta^{-1}XPP^T X < 0. \tag{41}$$

It is noted unfortunately that the matrix inequality (41) gives a disadvantage since it is nonlinear with respect to the variables *Ki*, *X* and *η* (more precisely bilinear). A numerical procedure of resolution by linearization is obtained in the following section.

In order to convert these conditions into an LMI formulation, the following change of variables is considered:

$$M\_{\bar{l}} = XK\_{\bar{l}} \tag{42}$$

and by using the Schur complement [16], the linear matrix inequality is obtained:

$$
\begin{bmatrix}
A\_i^T P^T X + X P A\_i - C^T M\_i^T + \eta \delta^2 I & XP \\
P^T X & -\eta I
\end{bmatrix} < 0. \tag{43}
$$

To satisfy condition (18), the equality can be solved:

$$XHE = 0.\tag{44}$$

Using the change of variable *S* = *XH*, linear matrix equality is obtained:

$$SE = 0.\tag{45}$$

The conditions (21) must be satisfied simultaneously, and using the change of variable (42) gives:

$$(X + \mathcal{SC})\, E\_l = M\_l E. \tag{46}$$

Since *P* = *I* + *HC*, replacing *P* in (43), the matrix inequality of Theorem 1 can be obtained. The conditions (26)–(28) of Theorem 1 are thus demonstrated.

## *4.2. Second Approach*

In the case where hypothesis (24) is not satisfied, meaning that the information on its bounded *δ* is not available, the method established in the previous section cannot be applied.

In this section, another method based on the use of the *L*<sup>2</sup> approach is proposed.

**Theorem 2.** *A fractional order unknown input observer (13) for system (11) exists if there exists a positive definite matrix X, matrices Mi, S and positive scalars* ¯ *δ satisfying the following conditions for all i* = 1, ··· , *M:*

$$
\begin{bmatrix}
\Theta\_i & X + \mathcal{SC} \\
\left(X + \mathcal{SC}\right)^T & -\gamma I
\end{bmatrix} < 0,
\tag{47}
$$

$$SE = 0\tag{48}$$

*and*

$$(X + SC)\,E\_l = M\_l E\_r \tag{49}$$

*where*

$$
\Theta\_i = A\_i^T \left( X + \mathbb{C}^T \mathbb{S} \right) + \left( X + \mathbb{S} \mathbb{C} \right) A\_i - \mathbb{C}^T M\_i^T - M\_i \mathbb{C} + I. \tag{50}
$$

*Then, the fractional order UI observer (13) is completely defined by (30)–(34).*
