The problem you shared is a simulation of a closed-loop control system with a state-space representation. Let's break it down and analyze what is given and asked:

Given system dynamics:

$\dot{e}(t) = A e(t) + B u(t)$ Where:

• $e(t) = [e_1(t) \, e_2(t) \, e_3(t) \, e_4(t)]^T$
• $A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -a_0 & -a_1 & -a_2 & -a_3 \end{bmatrix}$
• $B = [0 \, 0 \, 0 \, 1]^T$
• $u(t) = K_r r(t) - K e(t)$
• Parameters: $r(t) = 1.5$, $a_0 = 11.3$, $a_1 = 23.5$, $a_2 = 17.0$, $a_3 = 5.0$, $K = [k_1 \, k_2 \, k_3 \, k_4]$

Initial Conditions:

• $e(0) = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}$

The Problem:

You need to find the value of $K$ for which the characteristic equation is: $s^4 + 8s^3 + 24s^2 + 32s + 16 = 0$

Steps:

1. Characteristic Equation of the System: The characteristic equation of a state-space system is given by: $\text{det}(sI - (A - BK)) = 0$ The goal is to match this with the desired characteristic equation provided.

2. Solve for $K$: You need to calculate the feedback gains $K = [k_1, k_2, k_3, k_4]$ such that the closed-loop characteristic equation matches the given one.

Would you like me to proceed with detailed calculations and solve for the gains $K$?