Inverted Pendulum on a Cart -- Nonlinear State Space equations

In summary, the conversation is about finding the general Inverted Pendulum on a cart nonlinear state space equations with two degrees of freedom. This includes displacement, velocity, pendulum angle from vertical, angular velocity and parameters such as y-state, mass of pendulum, mass of cart, length of pendulum line, gravity, friction, and force on cart. The person is looking for the proper functionality and is unsure about the derived equations. They have found some equations but are unsure if they are accurate. They are seeking help and thanking the recipient in advance.
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elecboss
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Hi good day. I am trying to find the general Inverted Pendulum on a cart nonlinear state space equations with two degrees of freedom with x, x_dot, theta, theta_dot which represents displacement, velocity, pendulum angle from vertical, angular velocity. However from research, I am seeing different state space equations. In the image attached is Matlab code I found that gives the proper functionality but I am not sure how it was derived.
y-state
m-mass of the pendulum
M- mass of the cart
L- length of pendulum line
g- gravity
d- friction
u- force on cart
The other image shows the closest set of equations I found but I am unsure. Please help thank you.

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1. What is an inverted pendulum on a cart?

An inverted pendulum on a cart is a classic control system problem in which a pendulum is attached to a cart that can move horizontally along a track. The goal is to control the movement of the cart in order to keep the pendulum balanced in an upright position.

2. What are nonlinear state space equations?

Nonlinear state space equations are mathematical models that describe the behavior of a dynamical system that cannot be represented by linear equations. In the case of an inverted pendulum on a cart, the nonlinear state space equations take into account the nonlinear dynamics of the pendulum and the cart.

3. How are the state space equations derived for an inverted pendulum on a cart?

The state space equations for an inverted pendulum on a cart are derived using Newton's laws of motion, along with equations for the dynamics of the pendulum and the cart. These equations are then converted into state space form, which includes the state variables and their derivatives.

4. What is the significance of nonlinear state space equations in control systems?

Nonlinear state space equations are important in control systems because they accurately model the nonlinear dynamics of many real-world systems. This allows for more precise control and better performance of the system compared to using linear equations, which may result in errors and instability.

5. How are the state space equations used to control an inverted pendulum on a cart?

The state space equations are used to design a controller that can manipulate the inputs of the system (such as the force applied to the cart) in order to achieve a desired output (such as keeping the pendulum balanced). This is done through the use of feedback control, in which the system's output is constantly monitored and compared to a reference value, and the controller adjusts the inputs accordingly to maintain stability.

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