**3. Robust Actuator Faults Reconstruction**

In this part, assuming that the proposed SMO gains in (7) are well-designed, an efficient approach is proposed for a robust AFR procedure. Relying on the results of Theorem 2.3, one obtains that *ey*˜ = *e*˙*y*˜ = 0 as *t* → ∞. Then:

$$\dot{e}\_1(t) = \mathcal{A}\_1 e\_1(t) - \mathcal{M}\_1 \partial(t, y, u) - \mathcal{D}\_1 d(t) \tag{33}$$

$$0 = \mathcal{A}\_3 e\_1(t) - \mathcal{M}\_2 \partial(t, y, u) - \mathcal{D}\_2 d(t) - \mathcal{F}\_2 f\_a(t) + P\_0^{-1} v\_{eq}$$

where *veq* is obtained by approximating *v* in (8):

$$
\sigma\_{\text{eff}} = -\rho(t, y, \mu) c\_{\tilde{y}} \left( \varepsilon + ||c\_{\tilde{y}}|| \right)^{-1} \tag{34}
$$

where *ε* > 0. From (34), one obtains:

$$\begin{aligned} \dot{e}\_1(t) &= (LA\_3 + A\_1)e\_1(t) - (M\_1 + LM\_2)\partial(t, y, u) \\ &- (D\_1 + LD\_2)d(t) \\ 0 &= \mathcal{T} \begin{pmatrix} A\_3e\_1(t) - M\_2\partial(t, y, u) - \\ D\_2d(t) - F\_2f\_a(t) \end{pmatrix} + P\_0^{-1}v\_{\varepsilon\eta}. \end{aligned} \tag{35}$$

This implies:

$$P\_0^{-1} \upsilon\_{\epsilon \eta} = \mathcal{T} \begin{pmatrix} -A\_3 \varepsilon\_1(t) + M\_2 \partial(t, y, u) + \\ D\_2 d(t) + F\_2 f\_a(t) \end{pmatrix} . \tag{36}$$

Now, the goal is to minimize or eliminate the effects of disturbance and uncertainty signals on the AFR. To this end, the reconstruction signal is defined as:

$$\hat{f}\_i = \mathcal{W}\mathcal{T}^T P\_0^{-1} \upsilon\_{eq} \tag{37}$$

where *<sup>W</sup>* = [*W*1, *<sup>F</sup>*−<sup>1</sup> <sup>22</sup> ]. Multiplication of both sides in (37) by *<sup>W</sup>*<sup>T</sup> *<sup>T</sup>* implies:

$$f\_i(t) = \,\_+f\_a(t) - WA\_3e\_1(t) + [WD\_2, WM\_2] \left[ \begin{array}{c} d(t) \\ \partial(t, y, u) \end{array} \right].\tag{38}$$

From (36), we have:

$$\begin{aligned} e\_1(s) &= -\left(sI - \left(LA\_3 + A\_1\right)\right)^{-1} \times \\ \left[LD\_2 + D\_1, LM\_2 + M\_1\right] \left[\begin{array}{c} d(t) \\ \partial(t, y, u) \end{array}\right]. \end{aligned} \tag{39}$$

Substitution of (40) in (39) results:

$$\begin{aligned} \hat{f}\_i(t) &= f\_d(t) + G(s) \begin{bmatrix} d(t) \\ \partial(t, y, u) \end{bmatrix} \\ G(s) &= \begin{bmatrix} \ W D\_2 & \ W M\_2 \end{bmatrix} + \\ &\quad \begin{aligned} &W A\_3 (sI - (LA\_3 + A\_1))^{-1} \times \\ &\quad \begin{bmatrix} LD\_2 + D\_1 & LM\_2 + M\_1 \end{bmatrix}. \end{aligned} \tag{40}$$

Therefore, the effect of *<sup>d</sup>*(*t*) *∂*(*t*, *y*, *u*) on the fault reconstruction signal will be minimized or bounded if:

$$\|\|G(s)\|\|\_{\infty} < \zeta$$

where *ξ* is a small constant. Let define *P*˜ in (31) as:

$$
\vec{P} = \begin{bmatrix} P\_{11} & P\_{12} \\ \vec{P}\_{12}^T & \vec{P}\_{22} \end{bmatrix} > 0 \tag{42}
$$

where *P*˜ <sup>22</sup> <sup>∈</sup> *<sup>R</sup>p*×*<sup>p</sup>* and *<sup>P</sup>*˜ <sup>11</sup> <sup>∈</sup> *<sup>R</sup>*(*n*−*p*)×(*n*−*p*). By applying the Bounded Real Lemma (BRL) [38], the inequality (42) is converted to:

$$
\begin{bmatrix}
\Phi\_{11} & \Phi\_{12} & -\left(WA\_3\right)^T \\
\Phi\_{12}^T & -\frac{\kappa}{3}I & \left(\mathcal{W}\left[\begin{array}{cc} D\_2 & M\_2 \end{array}\right]\right)^T \\
\Phi\_{11} = \mathcal{P}\_{11}A\_1 + A\_1^T\mathcal{P}\_{11} + \mathcal{P}\_{12}A\_3 + A\_3^T\mathcal{P}\_{12}^T \\
\Phi\_{12} = -\left(\mathcal{P}\_{11}\left[\begin{array}{cc} D\_1 & M\_1 \end{array}\right] + \mathcal{P}\_{12}\left[\begin{array}{cc} D\_2 & M\_2 \end{array}\right]\right).
\end{bmatrix}
\tag{43}
$$

By solving (44), one obtains *W* and *P*˜. Then, by substituting *W* in (38) results:

$$f\_i(t) \simeq f\_a(t). \tag{44}$$
