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: