4.1.1. Bounds on the Uncertain Birth and Death Rate

The bounds of the unknown birth and death rate are given by

$$\begin{array}{l} \overline{D} = \phi^{+}(k)\overline{w}\_{\mathfrak{n}-1} + \phi^{-}(k)\underline{w}\_{\mathfrak{n}-1} \\ \underline{D} = \phi^{+}(k)\underline{w}\_{\mathfrak{n}-1} + \phi^{-}(k)\overline{w}\_{\mathfrak{n}-1} \end{array} \tag{10}$$

,

where *wn*−<sup>1</sup> <sup>∈</sup> <sup>R</sup>12×<sup>1</sup> and *wn*−<sup>1</sup> <sup>∈</sup> <sup>R</sup>12×<sup>1</sup> denote *<sup>n</sup>* <sup>−</sup> 1 bound concatenation on the uncertain birth and death rate defined by

> *wn*−<sup>1</sup> <sup>=</sup> *<sup>w</sup> <sup>w</sup> <sup>w</sup> <sup>T</sup>* , *wn*−<sup>1</sup> <sup>=</sup> *<sup>w</sup> <sup>w</sup> <sup>w</sup> <sup>T</sup>*

with

$$
\overline{w} = \begin{bmatrix}
\ \overline{u}(k) & 0 & 0 & 0
\end{bmatrix}^T, \quad \underline{w} = \begin{bmatrix}
\ \underline{a}(k) & 0 & 0 & 0
\end{bmatrix}^T.
$$

Furthermore,

$$\phi^+(k) = \max\{0, \ \phi(k)\}, \ \phi^+(k) = \max\{0, \ -\phi(k)\}, \ \phi(k) = \Sigma\_A - \Delta\_y(k)\Sigma\_{CA}$$

where

$$
\Sigma\_A = \begin{bmatrix}
\mathbb{\tilde{\zeta}}\_{11} & 0 & \mathbb{\tilde{\zeta}}\_{13} & \mathbb{\tilde{\zeta}}\_{14} & \mathbb{\tilde{\zeta}}\_{15} & 0 & 0 & \mathbb{\tilde{\zeta}}\_{18} \\
0 & \mathbb{\tilde{\zeta}}\_{22} & 0 & 0 & 0 & \mathbb{\tilde{\zeta}}\_{26} & 0 & 0 \\
0 & \mathbb{\tilde{\zeta}}\_{32} & \mathbb{\tilde{\zeta}}\_{33} & 0 & 0 & \mathbb{\tilde{\zeta}}\_{36} & \mathbb{\tilde{\zeta}}\_{37} & 0 & \mathbb{\mathbb{I}}\_{4} \\
0 & \mathbb{\tilde{\zeta}}\_{42} & \mathbb{\tilde{\zeta}}\_{43} & \mathbb{\tilde{\zeta}}\_{44} & 0 & 0 & \mathbb{\tilde{\zeta}}\_{47} & \mathbb{\tilde{\zeta}}\_{48}
\end{bmatrix}, \tag{11a}
$$

and

$$\begin{array}{l} \xi\_{11} = (1 - a(k\_1))(1 - a(k\_2)), \\ \xi\_{13} = b(k\_1)d(k\_2), \\ \xi\_{14} = b(k\_2)(1 - a(k\_1)) + b(k\_1)(1 - a(k\_2) - b(k\_2)), \\ \xi\_{15} = (1 - a(k\_2)), \\ \xi\_{18} = b(k\_2), \\ \xi\_{22} = (1 - a(k\_1) - c(k\_1))(1 - a(k\_2) - c(k\_2)), \\ \xi\_{23} = c(k\_2)(1 - a(k\_1) - c(k\_1)) + c(k\_1)(1 - a(k\_2) - c(k\_2)), \\ \xi\_{31} = c(k\_2)(1 - a(k\_1) - d(k\_1))(1 - a(k\_2) - d(k\_2)), \\ \xi\_{33} = c(k\_2), \\ \xi\_{34} = c(k\_2), \\ \xi\_{42} = d(k\_1)c(k\_2), \\ \xi\_{43} = d(k\_1)c(k\_2), \\ \xi\_{44} = (1 - a(k\_1) - d(k\_1)) + d(k\_1)(1 - a(k\_2) - d(k\_2)), \\ \xi\_{44} = (1 - a(k\_1) - b(k\_1))((1 - a(k\_2) - b(k\_2)), \\ \xi\_{45} = d(k\_2), \\ \xi\_{46} = (1 - a(k\_2) - b(k\_2)), \end{array}$$

$$
\Sigma\_{CA} = \begin{bmatrix}
 0\_{4\times 1} & 0\_{4\times 1} & 0\_{4\times 1} \\
 \mathcal{C} & 0\_{4\times 1} & 0\_{4\times 1} \\
 \mathcal{C}A(k\_1) & \mathcal{C} & 0\_{4\times 1} \\
 \mathcal{C}A(k\_1)A(k\_2) & \mathcal{C}A(k\_1) & \mathcal{C}
\end{bmatrix},
$$

$$
\begin{aligned}
 \begin{bmatrix}
 0 & 0 & 0 & 0 \\
 1 & 0 & 0 & 0 \\
 1 - a(k\_1) & 0 & 0 & b(k\_1) \\
 \{1 - a(k\_1)\} & 0 & b(k\_1) & -b(k\_1)b(k\_2) + \\
 \{1 - a(k\_2)\} & 0 & d(k\_1) & b(k\_1)\{1 - a(k\_1)\} + \\
 \{1 - a(k\_2)\} & 0 & b(k\_2)\{1 - a(k\_1)\} \\
 \mathcal{C}A(k\_2) & b(k\_2)\{1 - a(k\_1)\} \\
 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 \\
\end{bmatrix},
\\
\begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
 1 & 0 & 0 & 0 & 0 & 0 \\
\end{bmatrix},
\end{aligned}
$$

$$
\begin{bmatrix}
1 - a(k\_1) & 0 & 0 & b(k\_1) & 1 & 0 & 0 & 0
\end{bmatrix},
$$

with *k*<sup>1</sup> = *k* − 1, *k*<sup>2</sup> = *k* − 2.

4.1.2. Bounds on Uncertain Input

The bounds on the uncertain input are represented as

$$
\begin{array}{l}
\overline{\Lambda}(k) = (\phi E\_{n-1})^{+} \overline{\hspace{0.1cm}}(k) - (\phi E\_{n-1})^{-} \overline{\hspace{0.1cm}}(k), \\
\underline{\Lambda}(k) = (\phi E\_{n-1})^{+} \underline{\hspace{0.1cm}}(k) - (\phi E\_{n-1})^{-} \overline{\hspace{0.1cm}}(k).
\end{array}
\tag{12}
$$

Using (3) and (4), one can write

$$\begin{aligned} y(k+1) &= \mathbb{C}x(k+1) + v(k+1), \\ &= \mathbb{C}A(k)x(k) + \mathbb{C}E\mathbb{S}(k) + \mathbb{C}w(k) + v(k+1), \\\\ -\mathbb{C}E\mathbb{S}(k) &= \mathbb{C}A(k)x(k) + \mathbb{C}w(k) - y(k+1) + v(k+1). \end{aligned}$$

Moreover, from (1), (3) and (4), we derive that

$$CE = -1,\text{ }Cw(k) = a(k).$$

This results in

$$
\mathbb{G}(k) = \mathbb{C}A(k)\mathbf{x}(k) + a(k) - y(k+1) + \upsilon(k+1),
$$

with

$$\begin{array}{l} \overline{\mathfrak{J}}(k) = (\mathsf{C}A)^{+} \overline{\mathfrak{x}}(k) - (\mathsf{C}A)^{-} \underline{\mathfrak{x}}(k) + | - \underline{\mathfrak{y}}(k+1)| + \overline{\mathfrak{v}} + \overline{\mathfrak{a}} \\ \underline{\mathfrak{J}}(k) = (\mathsf{C}A)^{+} \underline{\mathfrak{x}}(k) - (\mathsf{C}A)^{-} \overline{\mathfrak{x}}(k) + | - \underline{\mathfrak{y}}(k+1)| - \overline{\mathfrak{v}} + \underline{\mathfrak{a}} \end{array}$$

such that [ (*k*)] = [ (*k*), (*k*)] and *En*−<sup>1</sup> = [*EEE*] *T*.
