**3. Main Results**

Proving the boundedness of the distributed interval observer is equal to proving the convergence of the error dynamics. The Lyapunov function is a great choice for it. For error dynamics *e*˙(*t*) = *Aee*(*t*), the following Lyapunov function is constructed:

$$V(t) = \mathfrak{e}^T(t) P e(t),\tag{18}$$

where *P* is a positive symmetric matrix with suitable dimensions.

In the integer-order system, *V*˙ (*t*) < 0 is the sufficient condition to prove the convergence of error dynamics. *V*˙ (*t*) < 0 is equivalent to

$$A\_{\mathfrak{c}}^{T}\mathcal{P} + \mathcal{P}A\_{\mathfrak{c}} \prec 0.\tag{19}$$

Equation (19) is the strict LMI that can be computed with MATLAB. Formula (19) can yield that *Ae* is a Hurwitz matrix. By configuring matrix *Ae*, the convergence of the system can be reached.

However, the above details are mainly about the integer-order system. There are no corresponding lemmas about the fractional-order system. Therefore, the fractional-order extension of a Lyapunov candidate function is introduced to demonstrate the convergence of the error dynamics by referring [37].

**Lemma 4.** *Consider error dynamics <sup>e</sup>*(*t*) <sup>∈</sup> *<sup>R</sup><sup>n</sup> to be a continuous and derivable function. Then, for any time t* ≥ 0*, the fractional derivative of the Lyapunov function is*

$$D\_t^a V(t) \le \left( D\_t^a e^T(t) \right) P e(t) + e^T(t) P (D\_t^a e(t)),\tag{20}$$

*where V*(*t*) = *eT*(*t*)*Pe*(*t*) *is the Lyapunov function connected with e*(*t*)*.*

**Proof.** Denote that *J* = (*D<sup>α</sup> <sup>t</sup> eT*)*Pe* + *eTP*(*D<sup>α</sup> <sup>t</sup> <sup>e</sup>*) <sup>−</sup> *<sup>D</sup><sup>α</sup> <sup>t</sup> V*.

According to the definition of fractional calculus, *D<sup>α</sup> <sup>t</sup> V*(*t*) is equivalent to

$$\begin{split} D\_t^\mu V(t) &= \frac{1}{\Gamma(1-\alpha)} \int\_0^t \frac{\dot{V}(\tau)}{(t-\tau)^\alpha} d\tau\\ &= \frac{1}{\Gamma(1-\alpha)} \int\_0^t \frac{e^T(\tau) P \dot{e}(\tau) + \dot{e}^T(\tau) P e(\tau)}{(t-\tau)^\alpha} d\tau. \end{split} \tag{21}$$

Considering (21), *J* is rewritten as

$$\begin{split} J &= \frac{1}{\Gamma(1-a)} \int\_{0}^{t} \frac{e^{T}(t)P\dot{e}(\tau) + \dot{e}^{T}(\tau)P\dot{e}(t) - e^{T}(\tau)P\dot{e}(\tau) - \dot{e}^{T}(\tau)P\dot{e}(\tau)}{(t-\tau)^{a}} d\tau \\ &= \frac{1}{\Gamma(1-a)} \int\_{0}^{t} \frac{(e^{T}(t) - e^{T}(\tau))P\dot{e}(\tau) + \dot{e}^{T}(\tau)P(e(t) - e(\tau))}{(t-\tau)^{a}} d\tau \\ &= -\frac{1}{\Gamma(1-a)} \int\_{0}^{t} \frac{\dot{z}^{T}(\tau)Pz(\tau) + z^{T}(\tau)P\dot{z}(\tau)}{(t-\tau)^{a}} d\tau, \end{split} \tag{22}$$

where *z*(*τ*) = *e*(*t*) − *e*(*τ*).

Then, the proof for Lemma 1 is equivalent to proving that

$$\frac{1}{\Gamma(1-\alpha)} \int\_{0}^{t} \frac{(\dot{z}^{T}(\tau))Pz(\tau) + z^{T}(\tau)P\dot{z}(\tau)}{(t-\tau)^{\alpha}} d\tau \le 0. \tag{23}$$

For *u* = *zT*(*t*)*Pz*(*t*), one can deduce that

$$\frac{d\mathbf{u}}{dt} = \dot{z}^T P z(t) + z^T(t) P \dot{z}(t). \tag{24}$$

Then, *<sup>v</sup>*(*τ*) = (*t*−*τ*)−*<sup>α</sup>* <sup>Γ</sup>(1−*α*) is denoted, which yields

$$dv(\tau) = \frac{a(t-\tau)^{-\alpha-1}}{\Gamma(1-\alpha)}d\tau. \tag{25}$$

On the basis of (24) and (25), (23) is equal to

$$\frac{z^T(\tau)Pz(\tau)}{\Gamma(1-a)(t-\tau)^a}|\_0^t - \frac{a}{\Gamma(1-a)} \int\_0^t \frac{z^T(\tau)Pz(\tau)}{(t-\tau)^a} d\tau \le 0,\tag{26}$$

with 
$$\frac{z^T(\tau)Pz(\tau)}{\Gamma(1-a)(t-\tau)^a}|\_0^t = \frac{z^T(\tau)Pz(\tau)}{\Gamma(1-a)(t-\tau)^a}|\_{\tau=t} - \frac{z^T(0)Pz(0)}{\Gamma(1-a)(t)^a}.\tag{27}$$

When *τ* = *t*, the first term of (27), has nondeterminacy, and the analysis of its limitation is necessary:

$$\begin{split} \lim\_{\tau \to \tau} \frac{z^T(\tau)Pz(\tau)}{\Gamma(1-a)(t-\tau)^a} &= \frac{-e^T(\tau)Pe(t) + \dot{e}^T(\tau)Pe(\tau) - e^T(t)P\dot{e}(\tau) + e^T(\tau)P\dot{e}(\tau)}{-\Gamma(1-a)a(t-\tau)^{a-1}} \\ &= \frac{-\dot{e}^T(\tau)Pe(t) + \dot{e}^T(\tau)Pe(\tau) - e^T(t)P\dot{e}(\tau) + e^T(\tau)P\dot{e}(\tau)}{-\Gamma(1-a)a}(t-\tau)^{1-a}. \end{split} \tag{28}$$

Due to *<sup>α</sup>* <sup>∈</sup> (0, 1), it is obvious that *<sup>z</sup>T*(*τ*)*Pz*(*τ*) <sup>Γ</sup>(1−*α*)(*t*−*τ*)*<sup>α</sup>* <sup>→</sup> 0 when *<sup>τ</sup>* <sup>=</sup> *<sup>t</sup>*. Expression (26) can be simplified to

$$-\frac{z^T(0)Pz(0)}{\Gamma(1-a)t^a} - \frac{a}{\Gamma(1-a)} \int\_0^t \frac{z^T(\tau)Pz(\tau)}{(t-\tau)^a} d\tau \le 0. \tag{29}$$

For *P* 0 and Γ(·) > 0, it is certain to stand.

**Theorem 1.** *For System* (15)*, there exist bounds x and x, such that x*(*t*) ≤ *x*(*t*) ≤ *x*(*t*) *if the following conditions are satisfied:*


**Proof.** From Properties 3 and 4, the upper bound of the nonlinear function is equal to

$$
\tilde{f}(\overline{\mathbf{x}}\_i, \underline{\mathbf{x}}\_i) = a(\overline{\mathbf{x}}\_i) - b(\underline{\mathbf{x}}\_i). \tag{30}
$$

Then, it follows from (30) that

$$
\tilde{f}(\overline{\mathbf{x}}\_{i\prime}\underline{\mathbf{x}}\_{i}) - \tilde{f}(\mathbf{x}\_{i\prime}\mathbf{x}\_{i\prime}) = (a(\overline{\mathbf{x}}\_{i\prime}) - a(\mathbf{x}\_{i\prime})) - (b(\underline{\mathbf{x}}\_{i\prime}) - b(\mathbf{x}\_{i\prime})).\tag{31}
$$

Functions *a*(·) and *b*(·) are all increasing, which yields

$$
\tilde{f}(\overline{x}\_{i\prime}\underline{x}\_{i}) - \tilde{f}(x\_{i\prime}x\_{i}) \ge 0. \tag{32}
$$

Similarly, one can deduce that

$$
\tilde{f}(\mathbf{x}\_i, \mathbf{x}\_i) - \tilde{f}(\underline{\mathbf{x}}\_i, \overline{\mathbf{x}}\_i) \ge 0. \tag{33}
$$

From *x*(0) ≤ *x*(0) ≤ *x*(0), the initial value of the error dynamics satisfies *e*(0) ≥ 0 and *e*(0) ≥ 0. If matrix Ω is Metzler, and (32) and (33) are true, it follows that *e*(*t*) ≥ 0 and *e*(*t*) ≥ 0, i.e., *x*(*t*) ≤ *x*(*t*) ≤ *x*(*t*), for any *t* ≥ 0.

**Remark 2.** *For Theorem 1, we constructed a frame for* (7)*. The positivity of the error dynamics is guaranteed, which implies x*(*t*) ≤ *x*(*t*) ≤ *x*(*t*)*. However, an interval observer not only requires the error dynamics be positive, but also that the upper and lower errors are in a convergence of zero. Then, we give the following theorem.*

**Theorem 2.** *On the basis of the result in Theorem 1, given constant τ* > 0 *and positive matrix P* = *PT, if there exists a solution such that*

$$\hat{\Pi} = \begin{bmatrix} He(PA - lIC + PN\_1) - 2\tau I & PN\_2 + N\_3^T P \\ PN\_3 + N\_2^T P & He(PA - lIC + PN\_4) - 2\tau I \end{bmatrix} \prec 0,$$

$$\gamma > \frac{\tau}{a(\mathcal{L})'} $$

*where γ is the coupling strength, L* = *P*−1*U and M* = *P*−<sup>1</sup> *are the observer gains, then System* (15) *is a distributed interval observer.*

**Proof.** The Lyapunov function is chosen as follows:

$$V(t) = \sum\_{i=1}^{N} r\_i \overline{e}\_i^T(t) P \overline{e}\_i(t) + \sum\_{i=1}^{N} r\_i \underline{e}\_i^T(t) P \underline{e}\_i(t). \tag{34}$$

From Lemma 1, the fractional derivative of *V*(*t*) is

$$\begin{split} D\_t^u V(t) \le \sum\_{i=1}^N r\_i (D\_t^u \overline{e}\_i^T)(t) P \overline{e}\_i(t) + \sum\_{i=1}^N r\_i \underline{e}\_i^T(t) P (D\_t^u \underline{e}\_i(t)) \\ + \sum\_{i=1}^N r\_i (D\_t^u \underline{e}\_i^T)(t) P \underline{e}\_i(t) + \sum\_{i=1}^N r\_i \overline{e}\_i^T(t) P (D\_t^u \overline{e}\_i(t)). \end{split} \tag{35}$$

Substituting the error dynamics into (35)

*Dα <sup>t</sup> V*(*t*) = *N* ∑ *i*=1 *ri*(Ω*iei* + Δ*f*(*xi*))*TPei* + *N* ∑ *i*=1 *rie<sup>T</sup> <sup>i</sup> P*(Ω*iei* + Δ*f*(*xi*)) + *N* ∑ *i*=1 *ri*(Ω*iei* + Δ*f*(*xi*))*TPei* + *N* ∑ *i*=1 *rie<sup>T</sup> <sup>i</sup> P*(Ω*iei* + Δ*f*(*xi*)) ≤ *N* ∑ *i*=1 *ri*(Ω*iei* + *F*1*ei* + *F*2*ei*)*TPei* + *N* ∑ *i*=1 *rie<sup>T</sup> <sup>i</sup> P*(Ω*iei* + *F*1*ei* + *F*2*ei*) + *N* ∑ *i*=1 *rie<sup>T</sup> <sup>i</sup> P*(Ω*iei* + *F*3*ei* + *NFei*) + *N* ∑ *i*=1 *ri*(Ω*iei* + *F*3*ei* + *F*4*ei*)*TPei* = ((*<sup>R</sup>* <sup>⊗</sup> (*<sup>A</sup>* <sup>−</sup> *LC*) <sup>−</sup> *<sup>γ</sup>R*L ⊗ *<sup>M</sup>*)*<sup>e</sup>* <sup>+</sup> *<sup>R</sup>* <sup>⊗</sup> *<sup>F</sup>*1*<sup>e</sup>* <sup>+</sup> *<sup>R</sup>* <sup>⊗</sup> *<sup>F</sup>*2*e*)*T*(*<sup>I</sup>* <sup>⊗</sup> *<sup>P</sup>*)*<sup>e</sup>* <sup>+</sup> *<sup>e</sup>T*(*<sup>I</sup>* <sup>⊗</sup> *<sup>P</sup>*)((*<sup>R</sup>* <sup>⊗</sup> (*<sup>A</sup>* <sup>−</sup> *LC*) <sup>−</sup> *<sup>γ</sup>R*L ⊗ *<sup>M</sup>*)*<sup>e</sup>* <sup>+</sup> *<sup>R</sup>* <sup>⊗</sup> *<sup>F</sup>*1*<sup>e</sup>* <sup>+</sup> *<sup>R</sup>* <sup>⊗</sup> *<sup>F</sup>*2*<sup>e</sup>* + ((*<sup>R</sup>* <sup>⊗</sup> (*<sup>A</sup>* <sup>−</sup> *LC*) <sup>−</sup> *<sup>γ</sup>R*L ⊗ *<sup>M</sup>*)*<sup>e</sup>* <sup>+</sup> *<sup>R</sup>* <sup>⊗</sup> *<sup>N</sup>*3*<sup>e</sup>* <sup>+</sup> *<sup>R</sup>* <sup>⊗</sup> *<sup>F</sup>*4*e*)*T*(*<sup>I</sup>* <sup>⊗</sup> *<sup>P</sup>*)*<sup>e</sup>* <sup>+</sup> *<sup>e</sup>T*(*<sup>I</sup>* <sup>⊗</sup> *<sup>P</sup>*)((*<sup>R</sup>* <sup>⊗</sup> (*<sup>A</sup>* <sup>−</sup> *LC*) <sup>−</sup> *<sup>γ</sup>R*L ⊗ *<sup>M</sup>*)*<sup>e</sup>* <sup>+</sup> *<sup>R</sup>* <sup>⊗</sup> *<sup>F</sup>*3*<sup>e</sup>* <sup>+</sup> *<sup>R</sup>* <sup>⊗</sup> *<sup>N</sup>*4*e*) <sup>=</sup> *<sup>e</sup>T*(*<sup>R</sup>* <sup>⊗</sup> (*He*(*PA* <sup>−</sup> *PLC* <sup>+</sup> *PF*1)) <sup>−</sup> *<sup>γ</sup>*(*R*<sup>L</sup> <sup>+</sup> <sup>L</sup>*TR*) <sup>⊗</sup> *PM*)*<sup>e</sup>* <sup>+</sup> *<sup>e</sup>T*(*<sup>R</sup>* <sup>⊗</sup> (*He*(*PA* <sup>−</sup> *PLC* <sup>+</sup> *PF*4)) <sup>−</sup> *<sup>γ</sup>*(*R*<sup>L</sup> <sup>+</sup> <sup>L</sup>*TR*) <sup>⊗</sup> *PM*)*<sup>e</sup>* + *eT*(*PF*<sup>2</sup> + *F<sup>T</sup>* <sup>3</sup> *<sup>P</sup>*)*<sup>e</sup>* + *<sup>e</sup>T*(*PF*<sup>3</sup> + *<sup>F</sup><sup>T</sup>* <sup>2</sup> *P*)*e*, (36)

where <sup>Ω</sup>*<sup>i</sup>* <sup>=</sup> *<sup>A</sup>* <sup>−</sup> *LC* <sup>−</sup> *<sup>γ</sup>M*∑*<sup>N</sup> <sup>j</sup>*=<sup>1</sup> <sup>L</sup>*ij*, <sup>Δ</sup>*f*(*xi*) = ˜ *<sup>f</sup>*(*xi*, *xi*) <sup>−</sup> ˜ *f*(*xi*, *xi*) and Δ*f*(*xi*) = ˜ *f*(*xi*, *xi*) − ˜ *f*(*xi*, *xi*).

According to Lemma 3, *<sup>R</sup>*<sup>L</sup> <sup>+</sup> <sup>L</sup>*TR* could be simplified as

$$2a(\mathcal{L})\mathbb{E}^{\mathsf{T}}\mathbb{R}\mathbb{E}\leq \mathbb{E}^{\mathsf{T}}(\mathbb{R}\mathcal{L}+\mathcal{L}^{\mathsf{T}}\mathbb{R})\mathbb{E},\tag{37}$$

$$2a(\mathcal{L})\underline{\mathfrak{e}}^{T}\underline{\mathcal{R}}\underline{\mathfrak{e}} \le \underline{\mathfrak{e}}^{T}(\mathcal{R}\mathcal{L} + \mathcal{L}^{T}\mathcal{R})\underline{\mathfrak{e}}.\tag{38}$$

Taking *M* = *P*−1, (37) and (38) into account, it follows from (36) that

$$\begin{split} D\_t^a V(t) \le & \tilde{\varepsilon}^T (R \otimes (H\varepsilon (PA - PLC + PF\_1)) - 2\gamma a(\mathcal{L})R \otimes I\_n)\tilde{\varepsilon} \\ &+ \underline{\mathfrak{e}}^T (R \otimes (H\varepsilon (PA - PLC + PF\_4)) - 2\gamma a(\mathcal{L})R \otimes I\_{\text{fl}} \mathbb{E} \\ &+ \tilde{\varepsilon}^T (PF\_2 + F\_3^T P)\underline{\mathfrak{e}} + \underline{\mathfrak{e}}^T (PF\_3 + F\_2^T P)\tilde{\varepsilon}. \end{split} \tag{39}$$

Considering *γ* > *<sup>τ</sup> <sup>a</sup>*(L), we have

$$\begin{split} D\_t^R V(t) &\leq \overline{\mathfrak{e}}^T (\mathcal{R} \otimes (\textit{He}(\textit{PA} - \textit{PLC} + \textit{PF}\_1)) - 2\tau \mathcal{R} \otimes I\_n \overline{\mathfrak{e}} \\ &+ \underline{\mathfrak{e}}^T (\mathcal{R} \otimes (\textit{He}(\textit{PA} - \textit{PLC} + \textit{PF}\_4)) - 2\tau \mathcal{R} \otimes I\_n \underline{\mathfrak{e}}) \\ &+ \overline{\mathfrak{e}}^T (\mathcal{P} \mathcal{P}\_2 + \mathcal{F}\_3^T \mathcal{P}) \underline{\mathfrak{e}} + \underline{\mathfrak{e}}^T (\mathcal{P} \mathcal{F}\_3 + \mathcal{F}\_2^T \mathcal{P}) \overline{\mathfrak{e}} \\ &= \underline{\mathfrak{e}}^T (t) \mathcal{R} \otimes \Pi \underline{\mathfrak{e}} (t), \end{split}$$

where (*t*)=[*eT*(*t*), *eT*(*t*)]*<sup>T</sup>* and

$$
\Pi = \begin{bmatrix}
He(PA - PLC + PF\_1) - 2\tau I\_n & PF\_2 + F\_3^T P \\
PF\_3 + F\_2^T P & He(PA - PLC + PF\_4) - 2\tau I\_n
\end{bmatrix}.
$$

To satisfy the LMI toolbox, *U* = *PL* is applied in Π, which leads to

$$\hat{\Pi} = \begin{bmatrix} H\varepsilon (PA - UC + PF\_1) - 2\tau I\_n & PF\_2 + F\_3^T P\\ PF\_3 + F\_2^T P & H\varepsilon (PA - UC + PF\_4) - 2\tau I\_n \end{bmatrix}.$$

Then, matrix <sup>Π</sup><sup>ˆ</sup> <sup>≺</sup> 0 is equal to *<sup>D</sup><sup>α</sup> <sup>t</sup> V*(*t*) < 0, which implies that lim*t*→<sup>∞</sup> *e*(*t*) = 0 and lim*t*→<sup>∞</sup> *e*(*t*) = 0. The boundedness of the error dynamics could be guaranteed.

**Remark 3.** *Constant τ is simple without an actual effect. However, it is a parameter connected with γ. If we just compute γ, the LMI toolbox just gives one feasible solution. Nevertheless, if we compute τ, γ* > *<sup>τ</sup> <sup>a</sup>*(L) *would have more regions to select.*

On the basis of Theorem 2, an algorithm was constructed to design distributed interval observers for fractional-order MASs.
