*6.1. Actuator Fault Reconstruction*

First, the following stabilizing controller is designed:

$$\boldsymbol{\mu}(t) = \begin{bmatrix} -14.34 & -1.26 & -0.01 & 0.33 & -3.06 \\ 21.89 & 0.28 & 0.07 & -0.82 & 9.37 \end{bmatrix} \boldsymbol{\chi}(t).$$

During the simulation, we assume *x*(0) = [0.5, 1, 1, 1.5, 0.5] *<sup>T</sup>*; the disturbance *d*(*t*) = *u*(*t* − 25) and the uncertainty *∂*(*t*, *y*, *u*) = [0, 0.5, 2]*y* are also considered. It is easy to check that Assumption 1 is satisfied for this system, so the proposed method is applicable. Using the results in Theorem 2, the transformation matrix T*<sup>b</sup>* is calculated as:

$$T\_b = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{bmatrix}.$$

Then, the LMI (44) is solved to minimize the effects of disturbances and uncertainties. Consequently, the observer gains and the AFR are obtained using (9), (33), and (38) as:

$$\mathbf{G}\_{l} = \begin{bmatrix} -0.1 & -3.3 & 0 \\ 0.01 & 0.1 & 0 \\ -1.55 & 71.6 & 0 \\ -0.8 & 1.55 & 0 \\ 0 & 0 & -0.1 \end{bmatrix}, \mathbf{G}\_{n} = \begin{bmatrix} -0.001 & 0.07 & 0 \\ 0.001 & -0.07 & 0 \\ 2.75 & 2.68 & 0 \\ 1.37 & 1.3 & 0 \\ 0 & 0 & -0.13 \end{bmatrix}.$$

The associated matrices *L* and *P*<sup>0</sup> are calculated as:

$$L = \begin{bmatrix} 1 & 1 & 0 \\ -1 & -1 & 0 \\ 0 & -1 & 1 \end{bmatrix}, P\_0 = \begin{bmatrix} -13.35 & 14.06 & 0 \\ 14.06 & -14.05 & 0 \\ 0 & 0 & 7.53 \end{bmatrix}.$$

The Lyapunov matrix *P* is also obtained from (30):

$$P = \begin{bmatrix} 8.41 & 0 & 0.1 & -0.8 & 0 \\ 0 & 5.65 & -0.01 & 0 & 0 \\ 0.1 & 0 & 0 & -0.1 & 0 \\ -0.81 & 0 & -0.1 & 0.72 & 0 \\ 0 & 0 & 0 & 0 & 7.53 \end{bmatrix}.$$

The parameters *ε* and *ρ* are selected as 0.5 and 10, respectively. Then, choosing *<sup>ξ</sup>* <sup>=</sup> <sup>1</sup> <sup>×</sup> <sup>10</sup>−3, the matrix *<sup>W</sup>* is calculated as *<sup>W</sup>* <sup>=</sup> [−0.676, <sup>−</sup>0.581, <sup>−</sup>0.28].

Figures 1 and 2 show the effectiveness of the proposed AFR algorithm reconstructing faults simultaneously occurring in both actuators in the presence of the mentioned unknown disturbances/uncertainties.

**Figure 1.** Illustration of robust actuator fault reconstruction (fault on the first actuator).

**Figure 2.** Illustration of robust actuator fault reconstruction (fault on the second actuator).
