https://file.aiquickdraw.com/mathbot/file/1729607771304oj6ltdia.jpg

The system given in the image appears to represent a **state-space model** for a dynamical system. Let's break it down into its components:

1. **State equation** (the first equation):
   \[
   \dot{x} = A x + B u
   \]
   Where:
   - \( \dot{x} \) is the derivative of the state vector \( x \) with respect to time.
   - \( A \) is the state matrix, which defines the system's dynamics.
   - \( B \) is the input matrix, which defines how the input \( u \) affects the system.

   From the image:
   \[
   A = \begin{bmatrix}
   -4 & 5 & -6 & -4 \\
   -15 & 15 & -17 & -11 \\
   4 & -4 & 7 & 4 \\
   -16 & 15 & -21 & -12
   \end{bmatrix},
   \quad
   B = \begin{bmatrix}
   1 \\
   0 \\
   0 \\
   0
   \end{bmatrix}
   \]

   Therefore, the system equation is:
   \[
   \dot{x} = A x + B u
   \]
   where \( u \) is a scalar input, and \( x \) is a 4-dimensional state vector.

2. **Output equation** (the second equation):
   \[
   y = C x
   \]
   Where:
   - \( y \) is the system output.
   - \( C \) is the output matrix, which maps the state vector \( x \) to the output.

   From the image:
   \[
   C = \begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix}
   \]
   
   This means that the output \( y \) is simply the first state variable \( x_1 \) in the state vector \( x \).

### Summary of the system:
- **State equation**: 
  \[
  \dot{x} = A x + B u
  \]
  where \( A \) and \( B \) are defined above.
  
- **Output equation**:
  \[
  y = C x
  \]
  where \( C \) is defined above.

Would you like further details on how to analyze the stability, controllability, or observability of this system? Let me know if you want to explore specific aspects of this model!

### Related questions:
1. How do we analyze the stability of this state-space system?
2. What is the system's controllability matrix, and how do we determine if the system is controllable?
3. How do we calculate the system's observability matrix?
4. How would we design a controller for this system using state feedback?
5. How do we simulate the time response of this system for a given input \( u(t) \)?

### Tip:
In state-space models, controllability and observability are key concepts for determining whether the system can be controlled or fully observed from its inputs and outputs.

Given the state-space model: ẋ = Ax + Bu, y = Cx, where A, B, C matrices are provided. Analyze the system.

This problem involves analyzing a linear system given in state-space form with matrices A, B, and C. Learn how to set up the system's state equation and output equation, and explore key concepts like controllability and observability.

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