User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Kharitonov-Bernstein-Haddad(KBH) Theorem

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Kharitonov-Bernstein-Haddad (KBH) Theorem
Consider the set of matrices

$$\mathcal{A}=\{\textbf{A}=\begin{bmatrix}\textbf{0}_{(n-1)\times 1} & & \textbf{1}_{(n-1)\times(n-1)} \\ -a_0 & \cdots & -a_{n-1} \end{bmatrix}| \underline{a_j}\leq a_j \leq \overline{a_j}, j=0,1,2,\cdots,n-1\}$$

Every matrix in the set $$\mathcal{A}$$ is Hurwitz if and only if there exist $$\textbf{P}_i \in mathbb{S}^n, i = 1, 2, 3, 4$$ where $$\textbf{P}_i>0, i = 1, 2, 3 ,4,$$ such that

$$\textbf{P}_iA_i + \textbf{A}_i^T\textbf{P}_i<0, \quad i = 1, 2, 3, 4, $$

where

$$\textbf{A}_i =\begin{bmatrix} \begin{bmatrix}\textbf{0}_{(n-1)\times 1} & \textbf{1}_{(n-1)\times(n-1)} \end{bmatrix} \\ \textbf{a}_i \end{bmatrix}, \quad i = 1, 2, 3, 4,$$

$$ \textbf{a}_1 = -[\underline{a}_0 \quad \underline{a}_1 \quad \overline{a}_2 \quad \overline{a}_3 \quad \cdots \quad \underline{a}_{n-4} \quad \underline{a}_{n-3} \quad \overline{a}_{n-2} \quad \overline{a}_{n-1} ], $$

$$ \textbf{a}_2 = -[\underline{a}_0 \quad \overline{a}_1 \quad \overline{a}_2 \quad \underline{a}_3 \quad \cdots \quad \underline{a}_{n-4} \quad \overline{a}_{n-3} \quad \overline{a}_{n-2} \quad \underline{a}_{n-1} ], $$

$$ \textbf{a}_3 = -[\overline{a}_0 \quad \underline{a}_1 \quad \underline{a}_2 \quad \overline{a}_3 \quad \cdots \quad \overline{a}_{n-4} \quad \underline{a}_{n-3} \quad \underline{a}_{n-2} \quad \overline{a}_{n-1} ], $$

$$ \textbf{a}_3 = -[\overline{a}_0 \quad \underline{a}_1 \quad \underline{a}_2 \quad \overline{a}_3 \quad \cdots \quad \overline{a}_{n-4} \quad \underline{a}_{n-3} \quad \underline{a}_{n-2} \quad \overline{a}_{n-1} ], $$

$$ \textbf{a}_4 = -[\overline{a}_0 \quad \overline{a}_1 \quad \underline{a}_2 \quad \underline{a}_3 \quad \cdots \quad \overline{a}_{n-4} \quad \overline{a}_{n-3} \quad \underline{a}_{n-2} \quad \underline{a}_{n-1} ]. $$

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