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Let's tackle this control systems problem by analyzing the provided block diagram step by step.

### Part (a): **Tìm hàm truyền kín của hệ thống** (Find the closed-loop transfer function)

1. **Identify the components:**
   - \( K_1 \frac{s + 20}{s} \): A gain block with a transfer function.
   - Two integrators \( \frac{1}{s} \): Represented by two blocks in series.
   - Feedback path: \( \frac{K_3(s + K_4)}{s + K_5} \) and a constant gain \( K_2 \).

2. **Find the open-loop transfer function** \( G(s) \):
   The forward path includes the series of:
   - \( K_1 \frac{s+20}{s} \)
   - Two integrators \( \frac{1}{s} \times \frac{1}{s} = \frac{1}{s^2} \)
   
   Thus, the open-loop transfer function before feedback is:
   \[
   G(s) = K_1 \frac{s+20}{s} \times \frac{1}{s^2} = \frac{K_1 (s+20)}{s^3}
   \]

3. **Feedback transfer function** \( H(s) \):
   The feedback path is:
   \[
   H(s) = K_2 \times \frac{K_3 (s + K_4)}{s + K_5}
   \]

4. **Closed-loop transfer function**:
   The closed-loop transfer function is given by:
   \[
   T(s) = \frac{G(s)}{1 + G(s)H(s)}
   \]
   Substituting \( G(s) \) and \( H(s) \) into this equation:
   \[
   T(s) = \frac{\frac{K_1 (s+20)}{s^3}}{1 + \frac{K_1 (s+20)}{s^3} \times K_2 \times \frac{K_3 (s + K_4)}{s + K_5}}
   \]
   Simplify this to find the exact closed-loop transfer function.

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### Part (b): **Từ hàm truyền kín tìm phương trình trạng thái mô tả hệ thống** (Find the state-space representation from the closed-loop transfer function)

To find the state-space representation:
1. Express the system's dynamics from the transfer function by finding its differential equation in the time domain.
2. From the differential equation, you can define the state variables (such as position, velocity, etc.).
3. Formulate the state-space representation:
   - State equation: \( \dot{x}(t) = A x(t) + B u(t) \)
   - Output equation: \( y(t) = C x(t) + D u(t) \)

The state-space matrices \( A \), \( B \), \( C \), and \( D \) can be derived from the transfer function.

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Do you need further details on how to compute the closed-loop transfer function and state-space representation? Here are five related questions you can explore further:

1. How does feedback influence the stability of the control system?
2. How do the parameters \( K_1 \), \( K_2 \), \( K_3 \), \( K_4 \), and \( K_5 \) affect the overall system response?
3. What is the significance of the integrators \( \frac{1}{s} \) in the system?
4. How can we apply root-locus or frequency response techniques to analyze this system?
5. How does the transfer function translate into poles and zeros, and what do they represent?

**Tip**: Always double-check if the feedback path is positive or negative when calculating closed-loop transfer functions!

Tìm hàm truyền kín của hệ thống và tìm phương trình trạng thái từ hàm truyền kín

Solve for the closed-loop transfer function of a control system based on a block diagram, and derive the state-space representation. This problem includes concepts such as Laplace transforms, state-space equations, and closed-loop systems. Suitable for undergraduate control systems students.

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