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The cart-pole system under LQR control is shown below. Determine the state space model (A,B,C,D)

The image shows a control system diagram for a **nonlinear cart-pole system** under Linear Quadratic Regulator (LQR) control, with **full state feedback (FSF)**. To determine the state-space model (A, B, C, D) of this system, we need to define the equations governing the dynamics of the cart-pole system, which can be linearized around the operating point.

In this case, the **state-space representation** has the following form:
\[
\dot{x} = Ax + Bu
\]
\[
y = Cx + Du
\]
Where:
- **x** represents the state vector (e.g., \( [x, \dot{x}, \theta, \dot{\theta}]^T \)).
- **u** is the input (control force applied to the cart).
- **y** is the output (which may be a subset of the state).
- **A** is the system matrix, **B** is the input matrix, **C** is the output matrix, and **D** is the feedthrough matrix.

### 1. Define the State Variables
From the diagram:
- \( x \): position of the cart.
- \( \dot{x} \): velocity of the cart.
- \( \theta \): angle of the pole.
- \( \dot{\theta} \): angular velocity of the pole.

Let’s consider the state vector as:
\[
x = \begin{bmatrix} x \\ \dot{x} \\ \theta \\ \dot{\theta} \end{bmatrix}
\]
These are the variables used in both the system and feedback (FSF block).

### 2. Dynamics of the Cart-Pole System
For a **cart-pole system**, the nonlinear equations can be linearized around the vertical position \( \theta = 0 \) (upright position). The linearized dynamics of the system result in matrices \( A \) and \( B \).

The typical linearized state-space matrices for the cart-pole system (simplified) are given by:
\[
A = \begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & -\frac{mg}{M} & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & \frac{(M+m)g}{Ml} & 0
\end{bmatrix}, \quad
B = \begin{bmatrix}
0 \\ \frac{1}{M} \\ 0 \\ -\frac{1}{Ml}
\end{bmatrix}
\]
Where:
- \( M \) is the mass of the cart.
- \( m \) is the mass of the pole.
- \( l \) is the length of the pole.
- \( g \) is the acceleration due to gravity.

### 3. Feedback and LQR Gains
The LQR controller calculates the control input \( u = -Kx \), where \( K \) are the **LQR gains** shown in the diagram. The gains are derived from solving the Algebraic Riccati Equation for the system.

### 4. Output Equations
From the diagram, the outputs can be considered as the full state feedback (FSF), meaning the output is:
\[
y = \begin{bmatrix} x \\ \dot{x} \\ \theta \\ \dot{\theta} \end{bmatrix} = Cx
\]
Thus, **C = I** (identity matrix), and typically **D = 0** for systems without direct feedthrough.

Thus, in general:
\[
C = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \quad D = 0
\]

### Summary:
The state-space model would be:
\[
A = \begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & -\frac{mg}{M} & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & \frac{(M+m)g}{Ml} & 0
\end{bmatrix}, \quad
B = \begin{bmatrix}
0 \\ \frac{1}{M} \\ 0 \\ -\frac{1}{Ml}
\end{bmatrix}, \quad
C = I, \quad D = 0
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you derive the state-space model for the cart-pole system from first principles?
2. What are the steps to linearize a nonlinear system like the cart-pole?
3. How is the LQR gain matrix \( K \) calculated?
4. What role does full state feedback play in controlling the system?
5. How can the system's stability be analyzed using the eigenvalues of matrix \( A \)?

### Tip:
When linearizing nonlinear systems, always check the validity of the operating point to ensure the linear approximation is accurate for small deviations.

The cart-pole system under LQR control is shown below. Determine the state space model (A,B,C,D).

Explore the state-space model (A, B, C, D) of a nonlinear cart-pole system under LQR control with full state feedback. Learn how to linearize the dynamics of the system, understand the role of LQR gains, and apply control theory concepts to stabilize the system.

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