**4. Robust Sensor Fault Reconstruction**

In this case and without loss of generality, we define a new state *yn*(*t*) <sup>∈</sup> *<sup>R</sup><sup>p</sup>* that converts (1) with *fa*(*t*) = 0 to a similar form presented in the previous section, i.e., (1) with *fs*(*t*) = 0, so that a similar algorithm can be used. To this aim, let us define:

$$
\dot{y}\_n(t) + A\_n y\_n(t) = A\_n y(t) \tag{45}
$$

where *An* is a stable PD matrix. Then, one obtains:

$$
\dot{y}\_{\text{fl}}(t) = -A\_{\text{fl}}y\_{\text{fl}}(t) + A\_{\text{fl}}\mathbf{C}z(t) + A\_{\text{fl}}F\_{\text{f}}f\_{\text{s}}(t). \tag{46}
$$

Now, an augmented system with *n* + *p* states is defined as:

$$
\begin{split}
\underbrace{\begin{bmatrix}
\dot{z}(t) \\
\dot{y}\_n(t)
\end{bmatrix}}\_{A\_N} &= \underbrace{\begin{bmatrix}
A & 0 \\
A\_n \mathbb{C} & -A\_n
\end{bmatrix}}\_{A\_N} \begin{bmatrix}
z(t) \\
y\_n(t)
\end{bmatrix} + \underbrace{\begin{bmatrix}
B \\
0
\end{bmatrix}}\_{B\_N} u(t) \\ &+ \underbrace{\begin{bmatrix}
0 \\
A\_n F\_s
\end{bmatrix}}\_{F\_N} f\_s(t) + \underbrace{\begin{bmatrix}
D \\
0 \\
D\_N
\end{bmatrix}}\_{D\_N} d(t) + \underbrace{\begin{bmatrix}
M \\
0 \\
\end{bmatrix}}\_{M\_N} \partial(t, y, u) \\
y\_n(t) &= \underbrace{\begin{bmatrix}
0 & I\_p \\
0 & I\_p
\end{bmatrix}}\_{C\_N} \begin{bmatrix}
z(t) \\
y\_n(t)
\end{bmatrix}.
\end{split}
\tag{47}
$$

Using this augmented model, it is evident that the sensor fault reconstruction (SFR) can be handled similar to the AFR procedure discussed in the previous section.
