4.1.3. Bounds on Measurement Noise Vector

The bounds for the measurement noise vector are described by

$$\begin{array}{l}\overline{V} = \boldsymbol{\varrho}\_{\boldsymbol{v}}^{+}(\boldsymbol{k})\overline{\boldsymbol{v}}\_{\boldsymbol{n}} + \boldsymbol{\varrho}\_{\boldsymbol{v}}^{-}(\boldsymbol{k})\underline{\boldsymbol{v}}\_{\boldsymbol{n}\prime} \\\underline{V} = \boldsymbol{\varrho}\_{\boldsymbol{v}}^{+}(\boldsymbol{k})\underline{\boldsymbol{v}}\_{\boldsymbol{n}} + \boldsymbol{\varrho}\_{\boldsymbol{v}}^{-}(\boldsymbol{k})\overline{\boldsymbol{v}}\_{\boldsymbol{n}\prime} \end{array} \tag{1.3}$$

where *vn*, *vn* <sup>∈</sup> <sup>R</sup>4×1, respectively, denote the *<sup>n</sup>* concatenation of *<sup>v</sup>* <sup>∈</sup> <sup>R</sup> and *<sup>v</sup>* <sup>∈</sup> <sup>R</sup>, given by

$$
\overline{\boldsymbol{\upsilon}}\_{\rm tr} = \begin{bmatrix} \overline{\boldsymbol{\upsilon}} \\ \overline{\boldsymbol{\upsilon}} \\ \overline{\boldsymbol{\upsilon}} \\ \overline{\boldsymbol{\upsilon}} \end{bmatrix}, \quad \underline{\boldsymbol{\upsilon}}\_{\rm tr} = \begin{bmatrix} \underline{\underline{\boldsymbol{\upsilon}}} \\ \underline{\underline{\boldsymbol{\upsilon}}} \\ \underline{\underline{\boldsymbol{\upsilon}}} \end{bmatrix},
$$

and

$$\varphi\_v^+(k) = \max\{0, \, \varphi\_v(k)\}, \, \phi\_v^-(k) = \max\{0, \, -\varphi\_v(k)\}$$

with

$$
\varphi\_v(k) = -\Delta\_y(k).
$$

**Theorem 1.** *When Assumption 1 is satisfied for the given SEIR model (4), the interval state estimator given by (9) yields the following relations:*

$$
\underline{\mathbf{x}}(k) \le \mathbf{x}(k) \le \overline{\mathbf{x}}(k), \quad \forall k \ge 3,\tag{14}
$$

*provided that x*(0) ≤ *x*(0) ≤ *x*(0)*.*

**Proof of Theorem 1.** The solution to the SEIR model (4) at any time instant *k* for *x*(0) ∈ [*x*(0), *x*(0)] and *w*(*k*) ∈ [*w*, *w*] can be obtained as

$$\mathbf{x}(k) = \prod\_{\ell=1}^{k} A(k-\ell)\mathbf{x}(0) + E\mathbb{S}(k-1) + w(k-1) + \sum\_{m=0}^{k-2} \left\{ \prod\_{\ell=1}^{k-m-1} A(k-\ell) \right\} \{E\mathbb{S}(m) + w(m)\}.\tag{15}$$

The given SEIR model is a 4th order system i.e., *n* = 4. Therefore, the states *x*(*k*) can be determined at any time *k* using previous state values at *k* − 3 as follows:

$$\begin{array}{l} \mathbf{x}(k) = A(k-1)A(k-2)A(k-3)\mathbf{x}(k-3) + A(k-1)A(k-2)\{E\Im(k-3) + w(k-3)\} \\ \quad + A(k-1)\{E\Im(k-2) + w(k-2)\} + E\Im(k-1) + w(k-1). \end{array} \tag{16}$$

Now using theory of interval analysis [39] for all *x*(*k* − 3) ∈ [*x*(*k* − 3)], (*ϑ*) ∈ [ (*ϑ*)] and *w*(*ϑ*) ∈ [*w*(*ϑ*)] with *ϑ* ∈ {*k* − 3, *k* − 2, *k* − 1}, the state vector *x*(*k*) ∈ [*x*(*k*)] is given by

$$\mathbf{x}[\mathbf{x}(k)] = A(k-1)A(k-2)A(k-3)[\mathbf{x}(k-3)] + \Sigma\_A \{ E\_{n-1}[\odot(k)] + [w\_{n-1}] \},\tag{17}$$

As a result, utilizing the past input/output values and the observability matrix, the following set inversion formula is obtained to get the state enclosure [*x*(*k* − 3)]:

$$\mathbb{E}\left[\mathbf{x}(k-\mathfrak{A})\right] = \bigcirc^{-1}\left\{\left[\Psi(k-\mathfrak{A}:k)\right] - \Sigma\_{\mathbf{C}A}E\_{n-1}\left[\mathfrak{A}(k)\right]\right\} + \bigcirc^{-1}\Sigma\_{\mathbf{C}A}[w\_{n-1}],\tag{18}$$

where

$$\begin{bmatrix} \Psi(k-3:k) \end{bmatrix} = \begin{bmatrix} y(k-3) \\ y(k-2) \\ y(k-1) \\ y(k) \end{bmatrix} - \begin{bmatrix} v\_n \end{bmatrix}.$$

Consequently, by combing (17) and (18), one gets

$$\begin{array}{c} \left[ \mathbf{x}(k) \right] = A(k-1)A(k-2)A(k-3) \bigcirc^{-1} \left( \left[ \mathbf{Y}(k-3:k) \right] - \Sigma\_{\complement A} \left\{ E\_{n-1} \left[ \mathbf{S}(k) \right] + \left[ w\_{n-1} \right] \right\} \right) \\ + \Sigma\_{A} \left\{ E\_{n-1} \left[ \mathbf{S}(k) \right] + \left[ w\_{n-1} \right] \right\}, \end{array} \tag{19}$$

$$\begin{array}{c} \left[ \mathbf{x}(k) \right] = \Delta\_{\mathcal{Y}}(k) \{ \left[ \Psi(k - 3:k) \right] - \left[ v\_{n} \right] \} - \Delta\_{\mathcal{Y}}(k) \Sigma\_{CA} \{ E\_{n-1} \left[ \Im(k) \right] + \left[ w\_{n-1} \right] \} \\ + \Sigma\_{A} \{ E\_{n-1} \left[ \Im(k) \right] + \left[ w\_{n-1} \right] \}, \end{array} \tag{20}$$

$$\Delta\_y[\mathbf{x}(k)] = \Delta\_y(k)[\Psi(k-3:k)] - \Delta\_y(k)[\upsilon\_n] + (\Sigma\_A - \Delta\_y(k)\Sigma\_{CA})\{E\_{n-1}[\odot(k)] + [w\_{n-1}]\},\tag{21}$$

$$
\Phi[\mathbf{x}(k)] = \Delta\_{\mathcal{Y}}(k)[\Psi(k-3:k)] + \Phi(k)E\_{n-1}[\odot(k)] + \Phi(k)[w\_{n-1}] + \varphi\_{\mathcal{V}}(k)[\upsilon\_{n}],\tag{22}
$$

$$[\mathbf{x}(k)] = \hat{\mathbf{x}}(k) + [\boldsymbol{\Lambda}] + [\boldsymbol{D}] + [\boldsymbol{V}],\tag{23}$$

where [*x*(*k*)] = [*x*, *x*], [Λ]=[Λ, Λ], [*D*]=[*D*, *D*] and [*V*]=[*V*, *V*]. This completes the proof of Theorem 1.
