**4. Simulation**

Considering a fractional-order MAS with nonlinearity, the state-space model is similar to (7), where

$$A = \begin{bmatrix} -0.5 & 2\\ 0 & -1 \end{bmatrix}, \quad \mathbb{C} = \begin{bmatrix} 0 & 1 \end{bmatrix}, \quad f(\mathbf{x}\_i) = \sin(\mathbf{x}\_i).$$

For nonlinear function *f*(*xi*), corresponding function ˜ *f*(*xi*, *xi*) is defined as

$$\vec{f}(\overline{x}\_i, \underline{x}\_i) = \sin(\overline{x}\_i) + \overline{x}\_i - \underline{x}\_i.$$

For functions *a*(*xi*) = *sin*(*xi*) + *xi* and *b*(*xi*) = *xi*, it is obvious that *a*(*xi*) and *b*(*xi*) are increasing functions. *a*(*xi*) = *sin*(*xi*) + *xi* and *b*(*xi*) = *xi* are substituted into (13); then,

$$\begin{aligned} &\sin(\overline{\mathfrak{x}}\_{i}) + \overline{\mathfrak{x}}\_{i} - \underline{\mathfrak{x}}\_{i} - \sin(\underline{\mathfrak{x}}\_{i}) \\ &= (\sin(\overline{\mathfrak{x}}\_{i}) - \sin(\underline{\mathfrak{x}}\_{i})) + (\overline{\mathfrak{x}}\_{i} - \underline{\mathfrak{x}}\_{i}) + (\underline{\mathfrak{x}}\_{i} - \underline{\mathfrak{x}}\_{i}) \\ &\leq 2(\overline{\mathfrak{x}}\_{i} - \underline{\mathfrak{x}}\_{i}) + (\underline{\mathfrak{x}}\_{i} - \underline{\mathfrak{x}}\_{i}) \\ &= F\_{1}\overline{\mathfrak{x}}\_{i} + F\_{2}\underline{\mathfrak{x}}\_{i}. \end{aligned}$$

We chose *F*<sup>1</sup> = 2*I* and *F*<sup>2</sup> = *I*. Similarly, we have *F*<sup>3</sup> = *I* and *F*<sup>4</sup> = 2*I*. Laplacian matrix L is


By using Lemma 3, we obtained generalized algebraic connectivity *a*(L) = 1. By solving the LMI in Theorem 2, the observer gains can be computed:

$$L = \begin{bmatrix} -1.1771 \\ -0.2282 \end{bmatrix}, \ M = \begin{bmatrix} 0.1906 & 0.0757 \\ 0.0757 & 0.1470 \end{bmatrix}, \tau = 2.9151.$$

Then we chose *γ* = 3. The initial value of the original system state is defined as *x*(0)=[12352 − 1 − 1 4] *<sup>T</sup>*. The initial value of the distributed interval observer is defined as *x*(0)=[6 7 8 10 7 4 4 9] *<sup>T</sup>* and *<sup>x</sup>*(0)=[−<sup>4</sup> <sup>−</sup> <sup>3</sup> <sup>−</sup> 2 0 <sup>−</sup> <sup>4</sup> <sup>−</sup> <sup>6</sup> <sup>−</sup> <sup>6</sup> <sup>−</sup> <sup>1</sup>] *T*.

By performing Steps 5–8 in Algorithm 1, we can obtain Figures 1–6. Then, Figures 1 and 2 show the original system state of the four agents. Figures 3 and 4 display the bounds from a distributed interval observer. *vij* means the upper bound of the *j*th state of the *i*th agent, while *uij* means the lower bound of the *j*th state of the *i*th agent. From Figures 3 and 4, the bounds of the distributed observer trap the state of the original system. Define that *e*+ *ij* <sup>=</sup> *xij* <sup>−</sup> *xij* and *<sup>e</sup>*<sup>+</sup> *ij* = *xij* − *xij*. For Figures 5 and 6, the error between the observer and the original system could be reduced to a bounded value, which implies that the distributed interval observer is feasible.

**Algorithm 1** Distributed interval estimation for fractional-order MASs.

Step 1: Given the constant matrix L, compute the generalized algebraic connectivity *a*(L);


**Figure 1.** The first state of the four-agents.

**Figure 2.** The second state of the four-agents.

**Figure 3.** The bounds of the first state for the four-agents.

**Figure 4.** The bounds of the second state for the four-agents.

**Figure 5.** The error of the first state for the four-agents.

**Figure 6.** The error of the second state for the four-agents.
