**6. Example and Comparisons**

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 *α* = 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 :

$$\begin{cases} \ \_{t\_0}D\_t^\alpha \hat{\mathbf{x}}(t) = \sum\_{i=1}^M h\_i \left( \mathbf{x}(t) \right) \left[ A\_i \mathbf{x}(t) + B\_i u(t) + F\_i \bar{u}(t) \right], \\\ y(t) = \mathbf{C} \mathbf{x}(t) + G \bar{u}(t), \end{cases} \tag{68}$$

$$\begin{aligned} \text{where: } A\_1 &= \begin{bmatrix} -2 & 1 & 1 \\ 1 & -3 & 0 \\ 2 & 1 & -4 \end{bmatrix}, A\_2 = \begin{bmatrix} -3 & 2 & -2 \\ 5 & -3 & 0 \\ 0.5 & 0.5 & -4 \end{bmatrix}, B\_1 = \begin{bmatrix} 1 \\ 0.3 \\ 0.5 \end{bmatrix}, B\_2 = \begin{bmatrix} 0.5 \\ 1 \\ 0.25 \end{bmatrix}, \\ C &= \begin{bmatrix} 1 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}, F\_1 = \begin{bmatrix} 0.5 \\ -1 \\ 0.25 \end{bmatrix}, F\_2 = \begin{bmatrix} -1 \\ 0.52 \\ 1 \end{bmatrix}, G = \begin{bmatrix} 0.9 \\ 0.9 \end{bmatrix}. \\ \text{The activation functions are chosen in the form:} \end{aligned}$$

The activation functions are chosen in the form:

$$\begin{cases} \ h\_1(\mathbf{x}) = \frac{1 - \tanh(\mathbf{x}\_1)}{2}, \\\ h\_2(\mathbf{x}) = 1 - h\_1(\mathbf{x}) = \frac{1 + \tanh(\mathbf{x}\_1)}{2}. \end{cases} \tag{69}$$

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 (*ys*1, *ys*2), the outputs estimated by

the FOUIO (*yo*1, *yo*2) and the fractional order unknown input observer (FOUIO) (*yfo*1, *yfo*2) in the absence of unknown inputs. Figure 2 shows the outputs estimation error (a and b) in the absence of unknown inputs (*ys*<sup>1</sup> − *yo*1, *ys*<sup>2</sup> − *yo*2) and (*ys*<sup>1</sup> − *yfo*1, *ys*<sup>2</sup> − *yfo*2). The two Figures 1 and 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 1.** Outputs of the fractional-order Takagi–Sugeno (FOTS) system and its estimation by FOUIO and fractional order unknown input observer (FOUIO) in a free of fault case.

**Figure 2.** Output estimation Error in a free of fault case. (**a**) Error output estimation of the FOTS system by FUIO in a free of fault case. (**b**) Error output estimation of the FOTS system by FOUIO in a free of fault case.

Figure 3 presents the states of the FOTS system (*xs*1, *xs*2, *xs*3) and their estimations given by the FUIO (*xo*1, *xo*2, *xo*3) and the FOUIO (*xfo*1, *xfo*2, *xfo*3) in the absence of unknown inputs. Figure 4 shows the state estimation errors (a and b) in the absence of unknown inputs (*xs*<sup>1</sup> − *xo*1, *xs*<sup>2</sup> − *xo*2, *xs*<sup>3</sup> − *xo*3) and (*xs*<sup>1</sup> − *xfo*1, *xs*<sup>2</sup> − *xfo*2, *xs*<sup>3</sup> − *xfo*3). The two Figures 3 and 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.

**Figure 3.** State of the FOTS system and its estimation by FUIO and FOUIO in a free of fault case.

**Figure 4.** State estimation error in free of fault case. (**a**) State estimation error between the FOTS system and FUIO in a free of fault case. (**b**) State estimation error between the FOTS system and FOUIO in a free of fault case.

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 (*ys*1, *ys*2), and their estimations given by the FUIO (*yo*1, *yo*2) and the fuzzy FOUIO (*yfo*1, *yfo*2) in the presence of unknown inputs. Figure 6 shows the outputs estimation error (a and b) in the presence of unknown inputs (*ys*<sup>1</sup> − *yo*1, *ys*<sup>2</sup> − *yo*2) and (*ys*<sup>1</sup> − *yfo*1, *ys*<sup>2</sup> − *yfo*2). The two Figures 5 and 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 5.** Outputs for the FOTS system and its estimation by FUIO and FOUIO in a faulty case.

**Figure 6.** Output for error estimation faulty case. (**a**) Outputs for error estimation between the FOTS system and FUIO in a faulty case. (**b**) Outputs error estimation between the FOTS system and FOUIO in a faulty case.

Figure 7 presents the states of the FOTS system (*xs*1, *xs*2, *xs*3) and their estimations given by the FUIO (*xo*1, *xo*2, *xo*3) and the FOUIO (*xfo*1, *xfo*2, *xfo*3) in the presence of unknown inputs. Figure 8 shows the state estimation errors (a and b) in the presence of unknown inputs (*xs*<sup>1</sup> − *xo*1, *xs*<sup>2</sup> − *xo*2, *xs*<sup>3</sup> − *xo*3) and (*xs*<sup>1</sup> − *xfo*1, *xs*<sup>2</sup> − *xfo*2, *xs*<sup>3</sup> − *xfo*3). The two Figures 7 and 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 *ω*(*t*) is not satisfied or the value of the constant *δ* is very important (impossibility of finding a solution with Theorem 1, Theorem 2) offers the possibility of designing the observer with unknown input.

**Figure 7.** State estimation for the FOTS system and its estimations by FUIO and FOUIO in a faulty case.

**Figure 8.** Output estimation error in faulty case. (**a**) State estimation error between the FOTS system and FUIO in a faulty case. (**b**) State estimation error between the FOTS system and FOUIO in a faulty case.

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 9.** Unknown input and their estimations.

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

**Figure 10.** Unknown input errors estimation.

The two Figures 9 and 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.

#### **7. Conclusions**

In this paper, a new approach is proposed for designing a fractional order Takagi–Sugeno unknown input observer for a nonlinear system described by FOTS models with UPV. The first step is to rewrite the FOTS system in the form of a disturbed equivalent FOTS and with measurable premise variables. After that, two cases are considered; the first one uses the hypothesis that the perturbation, which appears after rewriting the FOTS model, verifies a Lipschitz condition, while the second one does not use this hypothesis. In this second case, another method is developed and based on an *L*<sup>2</sup> approach. The convergence conditions of the proposed observers are given in the form of linear matrix inequalities (LMI) that can be easily solved with conventional digital tools.

The obtained results show that a good convergence of the outputs and the state estimation errors is observed using the new proposed FOUIO. The state of the system can be estimated even in the presence of an unknown input varying rapidly since it is totally decoupled from the state. An improvement in the dynamics of the proposed observer is possible by placing the poles.

In future work, it would be interesting to study the decoupling of the noise and the estimation of the unknown inputs using the augmented systems.

**Author Contributions:** Conceptualization, A.D. and D.D.; methodology, D.D. and A.T.A.; software, A.D., D.D. and S.A.; validation, A.D. and A.T.A.; formal analysis, A.T.A.; investigation, S.A.; resources, S.A.; writing–original draft preparation, A.D. and S.A.; writing–review and editing, A.D., D.D. and A.A.; visualization, D.D. and S.A.; funding acquisition, A.T.A.

**Funding:** This research was funded by Prince Sultan University, Riyadh, Saudi Arabia.

**Acknowledgments:** The authors would like to thank Prince Sultan University, Riyadh, Saudi Arabia for supporting and funding this work. Special acknowledgment to Robotics and Internet-of-Things Lab (RIOTU) at Prince Sultan University, Riyadh, SA. Also, the authors wish to acknowledge the editor and anonymous reviewers for their insightful comments, which have improved the quality of this publication.

**Conflicts of Interest:** The authors declare no conflict of interest.
