*6.2. Sensor Fault Reconstruction*

First, by choosing *An* = <sup>20</sup>*I*3×3, the matrices of the associated augmented model in (48) are obtained.

$$A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 8.9 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 25.5 & 32.25 & -13 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -20 & 0 & 0 & 0\\ 0 & 20 & 0 & 0 & 0 & -20 & 0 & 0\\ 0 & 0 & 20 & 0 & 0 & 0 & -20 & 0 \end{bmatrix}, B = \begin{bmatrix} 0 & 0\\ 0 & 0\\ 0 & 0\\ 10 & 0\\ 0 & 6.68\\ 0 & 0\\ 0 & 0 \end{bmatrix}$$

$$\mathbf{C} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix}, D = \begin{bmatrix} 1\\ 0\\ 0\\ 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0 \end{bmatrix}, H\_{\mathbf{s}} = \begin{bmatrix} 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 20 \end{bmatrix},$$

Then, using a similar procedure the matrix *Tb* is obtained as:

$$T\_{b} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \end{bmatrix}$$

.

.

Using this coordinate transformation, the equivalent model is obtained. Then, a stabilizing controller is designed as *u*(*t*) = *Kz*(*t*), where:

$$K = \begin{bmatrix} -201.7 & -10.1 & -0.1 & 2.7 & -0.1 & 1.3 & -8.4 & 56 \\ -4214.3 & -6 & -18.7 & 31.5 & 0.3 & 1.2 & -188.2 & 84.4 \end{bmatrix}$$

In this case, we assume *x*(0)=[1, 0.5, 1, 0.5, 1.5, 1, 2, 0.5] *<sup>T</sup>*, *<sup>∂</sup>*(*t*, *<sup>y</sup>*, *<sup>u</sup>*) = [0.3, <sup>−</sup>0.5, 0], *<sup>y</sup>* <sup>=</sup> 0.3*z*<sup>5</sup> − 0.5*z*6, and *d*(*t*) = *u*(*t* − 20). Using a similar procedure, the observer gains are obtained for the augmented system as:

$$\mathbf{G}\_{\mathrm{il}} = \begin{bmatrix} -0.1 & 0 & -0.16 \\ 0 & -0.1 & 0.16 \\ -0.01 & 0 & -0.43 \\ -0.01 & -0.01 & 0.3 \\ 0 & 0 & 0.3 \\ 0 & -0.01 & -1.3 \\ -1.3 & 0 & 1.3 \\ -1.3 & -0.3 & -20.7 \end{bmatrix}, \mathbf{G}\_{\mathrm{l}} = \begin{bmatrix} 0 & 0 & 0.04 \\ -0.01 & -0.01 & 0 \\ -0.45 & -2.15 & 0 \\ -0.69 & -0.97 & 0 \\ -14.26 & -938.31 & 0 \\ 2.77 & 239.93 & 0 \\ -0.74 & -2.77 & 0 \\ 0 & 0 & -0.07 \end{bmatrix}.$$

The parameters *ε* and *ρ* are selected as 0.1 and 15, 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>* = [−0.651, <sup>−</sup>1.923, 0.309]. In Figures 3–5, the performance of the proposed robust SFR is illustrated in the presence of the disturbances/uncertainties.

**Figure 3.** Illustration of robust sensor fault reconstruction (pitch angle sensor fault).

**Figure 4.** Illustration of robust sensor fault reconstruction (rotor speed sensor fault).

**Figure 5.** Illustration of robust sensor fault reconstruction (generator speed sensor fault).

### *6.3. Simultaneous Actuator and Sensor Faults*

First, using the transformation in (49), the system is decomposed as in (50) and (51). Then, using the results in Theorem 5, the LMIs (60) and (61) are solved, and the observer gains are obtained from (64) and (65):

$$\mathbf{G}\_{n} = \begin{bmatrix} 0.41 & -0.43 & -1.25 \\ -0.27 & 0.12 & -0.16 \\ -1.16 & -0.24 & -2.25 \end{bmatrix}, \mathbf{G}\_{l} = \begin{bmatrix} 8.87 & -66.57 & 2.22 \\ 10.45 & -0.6 & 0.52 \\ -2.45 & 7.92 & -0.52 \end{bmatrix}.$$

Finally, the simultaneous actuator and sensor faults are reconstructed using (69) and (74). In this case, the parameters are chosen as given in the previous part. Figures 6 and 7 show the comparison of the simultaneous actuator and sensor fault reconstruction of the proposed method with [32] in the presence of *∂*(*t*, *y*, *u*) = [0.3, −0.5, 0], *y* = 0.3*z*<sup>5</sup> − 0.5*z*6, and *d*(*t*) = *u*(*t* − 20). The results verify that despite the existence of unknown disturbance and uncertainty, the proposed method performs well in the reconstruction of both sensor and actuator faults.

Considering the dynamics of disturbance in the sliding mode observer design, there was a reduced impact of disturbance in fault reconstruction in comparison with the approach presented in [32]. In other words, the proposed approach in Theorem 5 has the

quick response in the fault reconstruction process when disturbance is entered to the system. In order to measure and investigate the performance of the proposed methods, it is required to use quantitative criteria. In Table 1, the norm specifications of the sensor and actuator fault detection errors for the proposed approach and the method represented in [32] are calculated. As can be seen in Table 1, the proposed approach improves the accuracy of the actuator fault reconstruction more than 10% and the accuracy of the sensor fault reconstruction more than 4%.

**Figure 6.** The simultaneous actuator and sensor fault reconstruction using the approach proposed in Theorem 5.

**Figure 7.** The simultaneous actuator and sensor fault reconstruction using the approach presented in [32].


**Table 1.** The comparison of the norm specification of the simultaneous faults reconstruction.
