**1. Introduction**

Fractional calculus is the definition of the differential and integral of arbitrary order systems. During the last 20 years, fractional calculus grew into a hot topic and attracted increasing interest [1,2]. It plays an important role when people are faced with natural dynamics problems. It works well when it is applied in describing the memory and hereditary properties of manifold materials. In addition, the applications of fractional calculus are rapidly expanding, such as non-Fickian dynamics [3], fractional boundary value problems [4], and variable-order thermostat models [5].

Due to the inaccessibility of the system state in many physical backgrounds and the availability of output information, research on observation problems is very significant. With the indepth study of observation problems, interval observers are widely accepted as an efficient tool in reconstructing the system state with nonlinearity or bounded uncertainty [6,7]. The concept of interval observers was first proposed by Gouzé, and it was successfully applied to uncertain biological systems [8]. After that, research on interval observers can be divided into two parts. The first is based on monotone system theory, and the second was developed from set-membership estimation. The monotone-systemtheory-based method requires researchers to design suitable observer gains in order to ensure that the error dynamics is positive and bounded. In this case, the bounds from the interval observer converge to the original system. Inspired by this idea, there are many new techniques for various systems emerging. For linear time-invariant systems, the timevarying coordinate transformation technique was used in interval observer design [9]. Efimov et al. also extended the interval estimation technique to nonlinear time-varying systems [10]. An interval observer for switched systems was also introduced in [11]. On the other hand, a set-membership estimation-based method combined robust observer design with reachability analysis [12]. Interval observers in this framework obviously improved estimation accuracy. Additionally, scholars presented a new integrated version of interval observers that achieved an expected tradeoff between robust estimation conservatism and computational complexity [13]. In general, research on interval observers for integer-order

**Citation:** Zhang, H.; Huang, J.; He, S. Fractional-Order Interval Observer for Multiagent Nonlinear Systems. *Fractal Fract.* **2022**, *6*, 355. https:// doi.org/10.3390/fractalfract6070355

Academic Editors: Thach Ngoc Dinh, Shyam Kamal and Rajesh Kumar Pandey

Received: 31 May 2022 Accepted: 23 June 2022 Published: 25 June 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

systems is fruitful, which is instructive for us to design interval observers of fractional-order systems.

A tremendous expansion for multiagent systems (MASs) has been witnessed, which mainly regards output regulation [14], observer design [15] and consensus control [16]. It started from simple two-order systems and attracted more attention on various complex MASs, such as switched [17], nonlinear [18], and time-delaying [19] systems. The observers designed for MASs are called distributed observers if the communication topology is taken into account. A distributed observer was designed for leader-following MASs, and an actual vehicle model was used to verify the function of the distributed observer [20]. In [21], the existence condition of the distributed observer was established to recover the state of nonlinear MASs. There are some works for fractional-order MASs in [22] and [23]. The consensus control problem for fractional-order MASs under a fixed topology was studied by stability theory and the Lyapunov method [22]. For unknown nonlinear dynamics, an adaptive control protocol was introduced that was used in a trajectory tracking problem [23]. There were also some interesting results about the observation problem in [24–27]. Traditional observers and distributed controllers were both used in fractional-order MASs [24]. Compared with [24], the time-varying formation control was investigated, and the consensus protocol was designed to match a switching topology [25]. For the fractional-order heterogeneous nonlinear MASs, the consensus control problem was converted into the stabilisation problem using the distributed control technique [26]. In [27], the object was the fractional-order system with unknown orders, and a robust observer-based controller was formulated to solve the consensus problem.

For these works [24–27], the introduced observers were all Luenberger-like. When the original system suffers from uncertain disturbance or nonlinearity, Luenberger-like observers are unable to recover the system state. It is challenging to recover the system state when the general fractional-order system has a complex communication with its neighborhood. Therefore, a distributed interval observer is a better choice for fractionalorder MASs. In many physical problems such as trajectory tracking or output regulation, distributed interval observers provide a more accurate state information, which is a great step forward for the consensus control protocol design. However, there is not any paper focusing on distributed interval observers for fractional-order MASs. In the field of fractional calculus, there are some interesting works about fractional-order systems, including state reconstruction and algorithm programming [28–30]. Motivated by the interval observer mentioned above, we developed a framework of distributed interval observers for the general fractional-order MASs. The specific contributions are outlined as follows:


The rest of the paper is structured as follows. Section 2 mainly presents some preparation work, including the fractional differential, graph theory and fractional-order systems. Section 3 proposes the distributed interval observer design method for fractional-order MASs. An illustrative example is given in Section 4 to verify the validity of the designed observer. Lastly, conclusions and future work are drawn in Section 5.

Notations: For two vectors *<sup>x</sup>* <sup>∈</sup> *<sup>R</sup><sup>n</sup>* and *<sup>y</sup>* <sup>∈</sup> *<sup>R</sup>n*, *<sup>x</sup>* <sup>&</sup>lt; (≤)*<sup>y</sup>* are understood with *xi* < (≤)*yi*, *i* ∈ {1, ... , *n*}. For *A* ≺ 0 and *A* 0, symbol ≺ () means that matrix *A* is positive(negative) definite. For a matrix *B*, there exist three properties: *B*<sup>+</sup> = *max*(*B*, 0), *<sup>B</sup>*<sup>−</sup> <sup>=</sup> *<sup>B</sup>*<sup>+</sup> <sup>−</sup> *<sup>B</sup>* and <sup>|</sup>*B*<sup>|</sup> <sup>=</sup> *<sup>B</sup>*<sup>+</sup> <sup>+</sup> *<sup>B</sup>*−. Matrix *<sup>B</sup><sup>T</sup>* denotes the transpose of *<sup>B</sup>*, and *He*(*B*) is defined as *He*(*B*) = *<sup>B</sup>* <sup>+</sup> *<sup>B</sup>T*. *<sup>λ</sup>min*(*A*) represents the minimal nonzero eigenvalue of *<sup>A</sup>*. <sup>⊗</sup> is

the Kronecker product used in MASs. 1*<sup>N</sup>* means a *N*-order column vector, and all elements are 1.
