**Proof of Theorem 2.** To prove Theorem 2, the real bounded Lemma 1 [56] is used.

The dynamics of the fractional state estimation error are given by:

$$\mathbf{h}\_{l0} D\_t^a \vec{\mathbf{x}}(t) = \sum\_{i=1}^{M} h\_i \left( \mathbf{\hat{x}}(t) \right) \left[ \mathbf{N}\_i \vec{\mathbf{x}}(t) + P \omega(t) \right]. \tag{51}$$

Consider the following Lyapunov quadratic function:

$$V(t) = \tilde{\mathbf{x}}(t)^T X \tilde{\mathbf{x}}(t), \\ X = X^T > 0. \tag{52}$$

Its derivative with regard to time is specified by:

$$D\_{t0}D\_t^aV(t) \le\_{t0} D\_t^a \tilde{\mathbf{x}}(t)^T \mathbf{X} \tilde{\mathbf{x}}(t) + \tilde{\mathbf{x}}(t)^T \mathbf{X}\_{t0} D\_t^a \tilde{\mathbf{x}}(t). \tag{53}$$

By substituting (23), the dynamic of the quadratic Lyapunov function is obtained:

$$\mu\_{t\_0} D\_t^{\pi} V(t) \le \sum\_{i=1}^{M} \mu\_i \left( \dot{\mathfrak{x}}(t) \right) \left[ \ddot{\mathfrak{x}}(t)^T \left( N\_i^T X + X N\_i \right) \ddot{\mathfrak{x}}(t) + \ddot{\mathfrak{x}}(t)^T X P \omega(t) + \omega(t)^T P^T X \ddot{\mathfrak{x}}(t) \right]. \tag{54}$$

In order to mitigate the impact of *ω*(*t*) on the state estimation error, the *L*<sup>2</sup> [52] will be used. It can guarantee:

$$\frac{\|\vec{x}(t)\|\_2}{\|\omega(t)\|\_2} < \delta\_\prime \qquad \delta > 0. \tag{55}$$

The system of the state estimation error is stable and the gain *L*<sup>2</sup> noted *δ* of the transfer from *ω*(*t*) to *x*˜(*t*) is bounded, if:

$$
\partial\_{t0} D\_t^{\alpha} V\left(\ddot{\mathbf{x}}(t)\right) + \ddot{\mathbf{x}}(t)^T \ddot{\mathbf{x}}(t) - \delta^2 \omega(t)^T \omega(t) \, \mathrm{d}\,\mathrm{d}t \tag{56}
$$

By substituting *<sup>t</sup>*0*D<sup>α</sup> <sup>t</sup> V* (*x*˜(*t*)), the inequality (56) becomes:

$$\begin{split} \sum\_{i=1}^{M} \mu\_{i} \left( \mathfrak{x}(t) \right) \left[ \mathfrak{x}(t)^{T} \left( \mathcal{N}\_{i}^{T} \mathcal{X} + \mathcal{X} \mathcal{N}\_{i} \right) \mathfrak{x}(t) + \mathfrak{x}(t)^{T} \mathcal{X} P \omega(t) + \omega(t)^{T} P^{T} \mathcal{X} \mathfrak{x}(t) \\ + \mathfrak{x}(t)^{T} \mathfrak{x}(t) - \delta^{2} \omega(t)^{T} \omega(t) \right]. \end{split} \tag{57}$$

The estimation error converges to zero and the gain *L*<sup>2</sup> of the transfer from *ω*(*t*) to *x*˜(*t*) is bounded by *δ* if the following inequality is verified:

$$
\sum\_{i=1}^{M} \mu\_i \left( \mathfrak{X}(t) \right) \begin{bmatrix} N\_i^T X + X N\_i + I & XP \\ P^T X & -\delta^2 I \end{bmatrix} < 0. \tag{58}
$$

The convex sum property of the activation functions makes it possible to write the following sufficient condition: 

$$
\begin{bmatrix}
N\_i^T X + X N\_i + I & XP \\
P^T X & -\delta^2 I
\end{bmatrix} < 0. \tag{59}
$$

Using the expression (19) of *Ni* and the changes of variables *Mi* = *XKi* and ¯ *δ* = *δ*<sup>2</sup> (48) becomes:

$$\begin{bmatrix} \begin{array}{cc} \Theta\_{\bar{i}} & \text{XP} \\ P^T X & -\bar{\delta}I \end{array} \end{bmatrix} < 0, \forall \bar{i} = 1, \dots, M \tag{60}$$

where

$$\Theta\_{\dot{i}} = A\_{\dot{i}}^T P^T X + X P A\_{\dot{i}} - \mathbb{C}^T M\_{\dot{i}}^T - M\_{\dot{i}} \mathbb{C} + I.c.$$

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

$$XHE = 0.\tag{61}$$

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

$$SE = 0.\tag{62}$$

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

$$(X + S\mathbb{C})\,E\_i = M\_i E.\tag{63}$$

Since *P* = *I* + *HC*, replacing *P* in (60), the matrix inequality of Theorem 2 is obtained. The conditions (47)–(49) of Theorem 2 are thus demonstrated.

## **5. Unknown Inputs Estimation**

In system (11), the unknown input *u*¯(*t*) appears with the influence matrix:

$$\Phi(t) = \begin{bmatrix} \sum\_{i=1}^{M} h\_i \left( \mathfrak{X}(t) \right) E\_i \\ E \end{bmatrix}. \tag{64}$$

For the estimation of the unknown input, it is necessary that the rank of the matrix Φ(*t*) is verified at each time *t* for the following condition:

$$\text{rank}\left(\Phi(t)\right) = q\_\prime \tag{65}$$

where *q* is the dimension of *u*¯(*t*). If this condition is satisfied, Φ(*t*) is full-rank column and its pseudo-inverse left Φ−1(*t*) exists:

$$\boldsymbol{\Phi}^{-}(t) = \left(\boldsymbol{\Phi}^{T}(t)\boldsymbol{\Phi}(t)\right)^{-1}\boldsymbol{\Phi}^{T}(t). \tag{66}$$

The unknown input can then be calculated according to the state estimated as follows:

$$\hat{u}(t) = \Phi^- \left[ \begin{array}{c} \, \_{t\_0}D\_t^a \mathfrak{X}(t) - \sum\_{i=1}^M h\_i \left( \mathfrak{X}(t) \right) \left[ A\_i \mathfrak{X}(t) + B\_i u(t) \right] \\\ y(t) - \mathbb{C} \hat{\mathfrak{X}}(t) \end{array} \right],\tag{67}$$

under condition (65) the asymptotic convergence from *x*ˆ(*t*) to *x*(*t*) results in the asymptotic convergence of *u*ˆ¯(*t*) to *u*¯(*t*).
