*2.3. Fractional-Order Systems*

Consider the following fractional-order system for the *i*-th agent.

$$\begin{cases} D\_t^\mathbf{x} \mathbf{x}\_i(t) = A \mathbf{x}\_i(t) + f(\mathbf{x}\_i(t)),\\ y\_i(t) = \mathbf{C} \mathbf{x}\_i(t), \end{cases} \tag{7}$$

where *xi* <sup>∈</sup> *<sup>R</sup><sup>n</sup>* is the system state, *yi* <sup>∈</sup> *<sup>R</sup><sup>m</sup>* is the output, and *<sup>f</sup>*(*xi*) <sup>∈</sup> *<sup>R</sup><sup>n</sup>* is the Lipschitz function. *<sup>A</sup>* <sup>∈</sup> *<sup>R</sup>n*×*<sup>n</sup>* and *<sup>C</sup>* <sup>∈</sup> *<sup>R</sup>m*×*<sup>n</sup>* are matrices with suitable dimensions. Communication topology G for (7) is a strongly connected balanced graph.

**Property 1.** *Given a matrix <sup>M</sup>* <sup>∈</sup> *<sup>R</sup>n*×*n, <sup>M</sup> is a Metzler matrix if all its elements outside the main diagonal are non-negative. For example, the following matrix is Metzler:*

$$M = \begin{bmatrix} -1 & 1\\ 2 & 0.5 \end{bmatrix}.$$

**Property 2.** *Given a matrix <sup>N</sup>* <sup>∈</sup> *<sup>R</sup>n*×*n, <sup>M</sup> is a Hurwitz matrix if its all real parts of the eigenvalues are negative. For example, the following matrix is Hurwitz:*

$$N = \begin{bmatrix} -3 & -2 \\ -2 & -3 \end{bmatrix}$$

.

For nonlinearity *f*(*xi*), we began the analysis with the following properties.

**Property 3** ([36])**.** *the differentiable global Lipschitz function f*(*xi*) *can be divided into two increasing functions a*(*xi*) *and b*(*xi*)*; their relationship is*

$$f(\mathbf{x}\_i) = a(\mathbf{x}\_i) - b(\mathbf{x}\_i). \tag{8}$$

**Property 4** ([36])**.** *for a Lipschitz function f*(*xi*)*, there exists a differentiable global Lipschitz function* ˜ *f*(*xk*<sup>1</sup> *<sup>i</sup>* , *<sup>x</sup>k*<sup>2</sup> *<sup>i</sup>* )*, such that*


**Remark 1.** *In Property 3, a Lipschitz function is transformed into two increasing functions, which is just for us to introduce the function* ˜ *f*(·, ·)*. Because a*(*xi*) *and b*(*xi*) *are both increasing functions, one can deduce that <sup>∂</sup>* ˜ *f <sup>∂</sup>a*(*xi*) <sup>≥</sup> <sup>0</sup> *and <sup>∂</sup>* ˜ *f <sup>∂</sup>b*(*xi*) ≤ <sup>0</sup> *easily. For <sup>x</sup>* ≤ *<sup>x</sup>* ≤ *x, we have* ˜ *<sup>f</sup>*(*x*, *<sup>x</sup>*) <sup>≤</sup> ˜ *<sup>f</sup>*(*x*, *<sup>x</sup>*) <sup>≤</sup> ˜ *f*(*x*, *x*)*, which are the upper and lower bounds of the Lipschitz function f*(*x*) *in the structure of distributed interval observers.*

Assuming that the bounds of the system state satisfy *xi* ≤ *xi* ≤ *xi*, on the basis of Properties 3 and 4, one can deduce that

$$f(\underline{\mathbf{x}}\_{i}, \mathbf{\overline{x}}\_{i}) \le f(\mathbf{x}\_{i}, \mathbf{x}\_{i}) = f(\mathbf{x}\_{i}) \le f(\mathbf{\overline{x}}\_{i}, \underline{\mathbf{x}}\_{i}). \tag{9}$$

By using the generalized Taylor formula, ˜ *<sup>f</sup>*(*xi*, *xi*) <sup>−</sup> ˜ *f*(*xi*, *xi*) is written as

$$
\tilde{f}(\overline{\mathbf{x}}\_{i\prime}\underline{\mathbf{x}}\_{i}) - \tilde{f}(\mathbf{x}\_{i\prime}\underline{\mathbf{x}}\_{i}) = \int\_{0}^{1} \frac{\partial \tilde{f}}{\partial \delta\_{1}} (\pi \delta\_{1} + (1 - \pi)\delta\_{2}) d\tau (\delta\_{1} - \delta\_{2}),
\tag{10}
$$

where *δ*<sup>1</sup> = *x x* and *δ*<sup>2</sup> = *x x* . It follows from Property 4 that

$$\begin{split} \bar{f}(\overline{\mathbf{x}}\_{i}, \underline{\mathbf{x}}\_{i}) - \bar{f}(\mathbf{x}\_{i}, \mathbf{x}\_{i}) &= \left[ \int\_{0}^{1} \frac{\partial a}{\partial \mathbf{x}} (\tau \overline{\mathbf{x}} + (1 - \tau)\mathbf{x}) d\tau \right. \\ &\quad \left. - \int\_{0}^{1} \frac{\partial b}{\partial \mathbf{x}} (\tau \mathbf{x} + (1 - \tau)\underline{\mathbf{x}}) d\tau \right] (\delta\_{1} - \delta\_{2}) \\ &= \left[ \mathcal{F}\_{1}(\overline{\mathbf{x}}, \mathbf{x}) - \mathcal{F}\_{2}(\mathbf{x}, \underline{\mathbf{x}}) \right] (\delta\_{1} - \delta\_{2}) . \end{split} \tag{11}$$

Similarly, we have

$$
\tilde{f}(\mathbf{x}\_{i},\mathbf{x}\_{i}) - \tilde{f}(\overline{\mathbf{x}}\_{i},\underline{\mathbf{x}}\_{i}) = [\mathcal{F}\_{\mathbf{3}}(\mathbf{x},\underline{\mathbf{x}}) - \mathcal{F}\_{\mathbf{4}}(\overline{\mathbf{x}},\mathbf{x})](\delta\_{1} - \delta\_{2}),\tag{12}
$$

where matrices F*i*, *i* ∈ {1, ... , 4} are non-negative and can be derived from (11). On the basis of the above discussion, the following property is presented.

**Property 5** ([36])**.** *for f*(*xi*) *and* ˜ *<sup>f</sup>*(·, ·) *defined in Property 4, if Jacobian matrix <sup>∂</sup>* ˜ *f ∂δ*<sup>1</sup> *is bounded, there exist matrices Fi*, *i* ∈ {1, . . . , 4} *such that*

$$\begin{cases} \tilde{f}(\overline{\mathbf{x}}\_{i\prime}\underline{\mathbf{x}}\_{i}) - \tilde{f}(\mathbf{x}\_{i\prime}\mathbf{x}\_{i}) \le F\_{1}\overline{\mathbf{c}}\_{i} + F\_{2}\underline{\mathbf{c}}\_{i\prime} \\ \tilde{f}(\mathbf{x}\_{i\prime}\mathbf{x}\_{i}) - \tilde{f}(\underline{\mathbf{x}}\_{i\prime}\overline{\mathbf{x}}\_{i}) \le F\_{3}\overline{\mathbf{c}}\_{i} + F\_{4}\underline{\mathbf{c}}\_{i\prime} \end{cases} \tag{13}$$

*where ei* = *xi* − *xi and ei* = *xi* − *xi.*

**Example 1.** *For nonlinear function f*(*x*) = *sinx, corresponding functions* ˜ *f*(*x*, *x*) *and* ˜ *f*(*x*, *x*) *are defined as*

$$\begin{cases} f(\overline{x}, \underline{x}) = \sin(\overline{x}) + \overline{x} - \underline{x}, \\ f(\underline{x}, \overline{x}) = \sin(\underline{x}) + \underline{x} - \overline{x}, \end{cases}$$

*where f*(*x*)*,* ˜ *f*(*x*, *x*) *and* ˜ *f*(*x*, *x*) *satisfy Properties 3 and 4.*

*The functions mentioned in Property 3 are defined as a*(*x*) = *sin*(*x*) + *x and b*(*x*) = *x. a*(*x*) *and b*(*x*) *are both obviously increasing functions. Then,* ˜ *<sup>f</sup>*(*x*, *<sup>x</sup>*) <sup>−</sup> ˜ *f*(*x*, *x*) *can be transformed into*

$$\begin{aligned} &\tilde{f}(\overline{\mathbf{x}},\underline{\mathbf{x}}) - \tilde{f}(\mathbf{x},\mathbf{x}) \\ &= \sin(\overline{\mathbf{x}}) + \overline{\mathbf{x}} - \underline{\mathbf{x}} - \sin(\mathbf{x}) \\ &= (\sin(\overline{\mathbf{x}}) - \sin(\mathbf{x})) + (\overline{\mathbf{x}} - \mathbf{x}) + (\mathbf{x} - \underline{\mathbf{x}}) \\ &\le 2(\overline{\mathbf{x}} - \mathbf{x}) + (\mathbf{x} - \underline{\mathbf{x}}). \end{aligned} \tag{14}$$

*From Property 5, we have* ˜ *<sup>f</sup>*(*x*, *<sup>x</sup>*) <sup>−</sup> ˜ *f*(*x*, *x*) ≤ *F*1*e* + *F*2*e. Combining* (14) *with Property 5, 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.*

*From Example 1, Property 5 is feasible, and the result from Formulas* (9)*–*(12) *stands.*

For System (7), nonlinear function *f*(*xi*(*t*)) was assumed to satisfy Properties 3–5; then, the interval observer for (7) was designed with

$$\begin{cases} D\_l^a \overline{\boldsymbol{x}}\_i(t) = A \overline{\boldsymbol{x}}\_i(t) + L(\boldsymbol{y}\_i(t) - \mathbb{C} \overline{\boldsymbol{x}}\_i(t)) + \gamma M \sum\_{i=1}^N a\_{ij} (\overline{\boldsymbol{x}}\_j(t) - \overline{\boldsymbol{x}}\_i(t)) + \tilde{f}(\overline{\boldsymbol{x}}\_i, \underline{\boldsymbol{x}}\_i), \\ D\_l^a \underline{\boldsymbol{x}}\_i(t) = A \underline{\boldsymbol{x}}\_i(t) + L(\boldsymbol{y}\_i(t) - \mathbb{C} \underline{\boldsymbol{x}}\_i(t)) + \gamma M \sum\_{i=1}^N a\_{ij} (\underline{\boldsymbol{x}}\_j(t) - \underline{\boldsymbol{x}}\_i(t)) + \tilde{f}(\underline{\boldsymbol{x}}\_i, \overline{\boldsymbol{x}}\_i), \end{cases} \tag{15}$$

where *M* and *L* are interval observer gains.

The error dynamics of the interval observer is

$$\begin{cases} D\_t^a \overline{\boldsymbol{\varepsilon}}\_i(t) = D\_t^a \overline{\boldsymbol{\pi}}\_i(t) - D\_t^a \boldsymbol{\pi}\_i(t) \\ \qquad = (A - LC - \gamma M \sum\_{j=1}^N \mathcal{L}\_{ij}) \overline{\boldsymbol{\varepsilon}}\_i(t) + \overline{f}(\overline{\boldsymbol{\boldsymbol{x}}}\_i, \underline{\boldsymbol{\boldsymbol{x}}}\_i) - \overline{f}(\mathbf{x}\_i, \mathbf{x}\_i), \\ D\_t^a \underline{\boldsymbol{\varepsilon}}\_i(t) = D\_t^a \underline{\boldsymbol{\varepsilon}}\_i(t) - D\_t^a \overline{\boldsymbol{\pi}}\_i(t) \\ \qquad = (A - LC - \gamma M \sum\_{j=1}^N \mathcal{L}\_{ij}) \underline{\boldsymbol{\varepsilon}}\_i(t) + \overline{f}(\mathbf{x}\_i, \mathbf{x}\_i) - \overline{f}(\underline{\boldsymbol{\underline{x}}}\_i, \overline{\mathbf{x}}\_i). \end{cases} \tag{16}$$

Then, *e*(*t*), ˜ *f*(*x*, *x*), ˜ *f*(*x*, *x*) and ˜ *f*(*x*, *x*) are defined as

$$\mathbf{c}(t) = \begin{bmatrix} \varepsilon\_1 \\ \vdots \\ \varepsilon\_N \end{bmatrix}, \bar{f}(\overline{\mathbf{x}}, \underline{\mathbf{x}}) = \begin{bmatrix} \bar{f}(\overline{\mathbf{x}}\_1, \underline{\mathbf{x}}\_1) \\ \vdots \\ \bar{f}(\overline{\mathbf{x}}\_N, \underline{\mathbf{x}}\_N) \end{bmatrix}, \bar{f}(\underline{\mathbf{x}}, \overline{\mathbf{x}}) = \begin{bmatrix} \bar{f}(\underline{\mathbf{x}}\_1, \overline{\mathbf{x}}\_1) \\ \vdots \\ \bar{f}(\underline{\mathbf{x}}\_N, \overline{\mathbf{x}}\_N) \end{bmatrix}, \bar{f}(\mathbf{x}, \mathbf{x}) = \begin{bmatrix} \bar{f}(\mathbf{x}\_1, \mathbf{x}\_1) \\ \vdots \\ \bar{f}(\mathbf{x}\_N, \mathbf{x}\_N) \end{bmatrix}.$$

System (16) can be written in the following form:

$$\begin{cases} D\_t^a \overline{\mathfrak{z}}(t) = (I\_N \otimes (A - LC) - \gamma(\mathcal{L} \otimes M)\overline{\mathfrak{z}}(t) + \overline{f}(\mathfrak{x}, \underline{\mathfrak{x}}) - \overline{f}(\mathfrak{x}, \mathbf{x}), \\ D\_t^a \underline{\mathfrak{z}}(t) = (I\_N \otimes (A - LC) - \gamma(\mathcal{L} \otimes M))\underline{\mathfrak{z}}(t) + \overline{f}(\mathfrak{x}, \mathbf{x}) - \overline{f}(\underline{\mathfrak{x}}, \overline{\mathfrak{x}}). \end{cases} \tag{17}$$
