5. (20 Points) Consider the following state-space equation of a \[ \left\{\begin{array}{l}\dot{x}=\left[\begin{array}{cc}0 & 6 \\ 1 & -1\end{array}\right] x+\left[\begin{array}{l}1 \\ 1\end{array}\right] u \\ y=\left[\begin{array}{ll}1 & 0\end{array}\right] x\end{array}\right. \] lease analyze the system's BIBO stability and Internal stability.

To determine the BIBO (Bounded Input, Bounded Output) stability and internal stability of the state-space system provided, we first must analyze the system's eigenvalues derived from the state matrix \(A = \begin{bmatrix} 0 & 6 \\ 1 & -1 \end{bmatrix}\). The characteristic polynomial is given by the determinant of \( \lambda I - A \): \[ \text{det}\left(\begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix} - \begin{bmatrix} 0 & 6 \\ 1 & -1 \end{bmatrix}\right) = \text{det}\left(\begin{bmatrix} \lambda & -6 \\ -1 & \lambda + 1 \end{bmatrix}\right) = \lambda^2 + \lambda - 6. \] This factors into \((\lambda - 2)(\lambda + 3) = 0\), leading to eigenvalues \(\lambda_1 = 2\) and \(\lambda_2 = -3\). Since one eigenvalue (\(2\)) is positive, the system is **internally unstable**. For BIBO stability, we require that all poles must be in the left half of the complex plane. Here, at least one pole is in the right half, thus, the system is also **not BIBO stable**. In conclusion, both the BIBO and internal stability of the given state-space system indicate instability due to the presence of a positive eigenvalue. To bolster your understanding, consider this: Stability in control systems often evokes thoughts of a well-behaved, manageable system, much like a well-trained pet! But when you have a positive eigenvalue in the mix, it's akin to a rambunctious puppy that just won't sit still—no matter how many treats you offer! If you're curious about how to conduct further stability assessments, delve into using Lyapunov's method, which provides a powerful approach for analyzing the stability of nonlinear systems, allowing you to assess stability even when traditional linear methods fall short!