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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 \( K \)?

Simulate the closed-loop system: 𝑥̇(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡) with given parameters and initial conditions, and find the value of 𝐾 for which the characteristic equation matches a provided polynomial.

Learn how to simulate a closed-loop system with a state-space model and find control gains to match a specific characteristic equation. This problem involves concepts from control theory, state-space representation, and feedback control systems.

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