To validate the advantages of the proposed fractional order unknown input observer, the system (

68) represented by the FOTS model with UPV is considered with

$\alpha =0.8$. The state estimation is carried out by means of two fuzzy unknown input observers, the first with integer order and the second one with fractional order. The unknown inputs considered may be noise, faults or modeling uncertainties.

#### Example and Simulation Results

Consider the FOTS model (

8), which is defined as follows:

where:

${A}_{1}=\left[\begin{array}{ccc}-2& 1& 1\\ 1& -3& 0\\ 2& 1& -4\end{array}\right]$,

${A}_{2}=\left[\begin{array}{ccc}-3& 2& -2\\ 5& -3& 0\\ 0.5& 0.5& -4\end{array}\right]$,

${B}_{1}=\left[\begin{array}{c}1\\ 0.3\\ 0.5\end{array}\right]$,

${B}_{2}=\left[\begin{array}{c}0.5\\ 1\\ 0.25\end{array}\right]$,

$C=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 1\end{array}\right]$,

${F}_{1}=\left[\begin{array}{c}0.5\\ -1\\ 0.25\end{array}\right]$,

${F}_{2}=\left[\begin{array}{c}-1\\ 0.52\\ 1\end{array}\right]$,

$G=\left[\begin{array}{c}0.9\\ 0.9\end{array}\right].$The activation functions are chosen in the form:

Two cases are considered for simulation, the first one in the absence of unknown inputs (unknown inputs are null), and the second one in the presence of unknown inputs. The unknown input considered is accompanied by an additive noise. To have a treatment close to reality, the initial values of the system are chosen non-null, but the initial values of the two unknown input observers are chosen equal to zero.

The outputs and the states of the FOTS system with their estimations given by the fuzzy integer and fractional order unknown input observers, and the unknown inputs and their estimates will be compared and analyzed.

Case 1: Absence of unknown input.

At first, the case of the absence of unknown inputs (unknown inputs are null) will be evaluated.

Figure 1 shows the two outputs of the considered FOTS system

$(y{s}_{1},y{s}_{2})$, the outputs estimated by the FOUIO

$(y{o}_{1},y{o}_{2})$ and the fractional order unknown input observer (FOUIO)

$(yf{o}_{1},yf{o}_{2})$ in the absence of unknown inputs.

Figure 2 shows the outputs estimation error (a and b) in the absence of unknown inputs

$(y{s}_{1}-y{o}_{1},y{s}_{2}-y{o}_{2})$ and

$(y{s}_{1}-yf{o}_{1},y{s}_{2}-yf{o}_{2})$. The two

Figure 1 and

Figure 2 show that the FOUIO gives better output estimation for the considered system. The decreased quality of the output estimation at the moment

$t=0$ is because of the choice of the initial values.

Figure 3 presents the states of the FOTS system

$(x{s}_{1},x{s}_{2},x{s}_{3})$ and their estimations given by the FUIO

$(x{o}_{1},x{o}_{2},x{o}_{3})$ and the FOUIO

$(xf{o}_{1},xf{o}_{2},xf{o}_{3})$ in the absence of unknown inputs.

Figure 4 shows the state estimation errors (a and b) in the absence of unknown inputs

$(x{s}_{1}-x{o}_{1},x{s}_{2}-x{o}_{2},x{s}_{3}-x{o}_{3})$ and

$(x{s}_{1}-xf{o}_{1},x{s}_{2}-xf{o}_{2},x{s}_{3}-xf{o}_{3})$. The two

Figure 3 and

Figure 4 show that the fuzzy observer with unknown inputs gives a better state estimation of the FOTS system. The decreased quality of the state estimation at the moment

$t=0$ is because of the choice of the initial values.

Now, the case of the presence of unknown inputs will be evaluated.

Case 2: Presence of unknown input and measurement noise simultaneously.

Figure 5 shows the outputs of the FOTS system

$(y{s}_{1},y{s}_{2})$, and their estimations given by the FUIO

$(y{o}_{1},y{o}_{2})$ and the fuzzy FOUIO

$(yf{o}_{1},yf{o}_{2})$ in the presence of unknown inputs.

Figure 6 shows the outputs estimation error (a and b) in the presence of unknown inputs

$(y{s}_{1}-y{o}_{1},y{s}_{2}-y{o}_{2})$ and

$(y{s}_{1}-yf{o}_{1},y{s}_{2}-yf{o}_{2})$. The two

Figure 5 and

Figure 6 show that the FOUIO gives a better output estimation for the FOTS system. The decreased quality of the outputs estimation at the moment

$t=0$ is because of the choice of the initial values.

Figure 7 presents the states of the FOTS system

$(x{s}_{1},x{s}_{2},x{s}_{3})$ and their estimations given by the FUIO

$(x{o}_{1},x{o}_{2},x{o}_{3})$ and the FOUIO

$(xf{o}_{1},xf{o}_{2},xf{o}_{3})$ in the presence of unknown inputs.

Figure 8 shows the state estimation errors (a and b) in the presence of unknown inputs

$(x{s}_{1}-x{o}_{1},$$x{s}_{2}-x{o}_{2},$$x{s}_{3}-x{o}_{3})$ and

$(x{s}_{1}-xf{o}_{1},$$x{s}_{2}-xf{o}_{2},$$x{s}_{3}-xf{o}_{3})$. The two

Figure 7 and

Figure 8 show that the FOUIO gives a better state estimation for the FOTS system. The decreased quality of the state estimation at the moment

$t=0$ is because of the choice of the initial values.

Analyzing the convergence conditions of the proposed FOUIO, if the condition (

23) on the term

$\omega \left(t\right)$ is not satisfied or the value of the constant

$\delta $ is very important (impossibility of finding a solution with Theorem 1, Theorem 2) offers the possibility of designing the observer with unknown input.

Figure 9 shows the considered unknown input with normal noise (ubar), their estimations given by the FUIO (ubar FUIO), FOUIO (ubar FOUIO) and the unknown input without noise (ubar without noise).

Figure 10 shows the unknown input estimation errors (ubar-ubar FUIO, ubar-ubar FOUIO).

The two

Figure 9 and

Figure 10 show that the FOUIO gives a better unknown input estimation of the FOTS system, but it cannot be decoupled from the noise. The decreased quality of the unknown input estimation at the moment

$t=0$ is because of the choice of the initial values.

In the presence of adding random measurement noises bounded by 0.01, the unknown input estimated based on the proposed observer is noisy. Indeed, the presence of measurement noise, at high frequency, decreases the quality of reconstruction of the unknown input.