**Problem 2.** Consider the inverted pendulum on a cart given in Figure 1.![Inverted pendulum schematic.](image)**Figure 1.** Inverted pendulum schematic.The state variables for this system can be defined to be: \( x_1(t) = \theta(t) \), \( x_2(t) = \dot{\theta}(t) \). Assume that the equation of motion for the pendulum can be written as follows:\[\begin{bmatrix}\dot{x}_1 \\\dot{x}_2\end{bmatrix}=\begin{bmatrix}x_2 \\\frac{g \sin(x_1)}{l \left(\frac{4}{3} - \frac{m \cos^2(x_1)}{m_c + m}\right)} + \frac{\cos(x_1)}{l \left(\frac{4}{3} - \frac{m \cos^2(x_1)}{m_c + m}\right)} u\end{bmatrix}\]**Explanation:**- **Diagram:** The figure shows a schematic of an inverted pendulum mounted on a cart. The cart moves horizontally, and the pendulum pivots about its base, depicting the angle \( \theta = x_1 \). - **Forces:** The forces acting on the pendulum include gravitational force \( mg \sin \theta \) and control input \( u \) which affects the motion of the cart.- **Variables:** - \( x_1(t) = \theta(t) \): Angle of the pendulum with the vertical. - \( x_2(t) = \dot{\theta}(t) \): Angular velocity of the pendulum.- **Equation of Motion:** Describes how the system evolves over time, considering both the gravitational force and external input \( u \). The complexity arises from the terms involving trigonometric functions \( \sin(x_1) \) and \( \cos(x_1) \), reflecting the pendulum's rotation dynamics. where \( g = 9.8 \, \text{m/s}^2 \), \( m_c \) is the cart's mass, \( m \) is the mass of the pole, \( l \) is the half length of the pole, and \( u \) is the applied force.a) Find a linearized state equation if the desired equilibrium value for each state is zero.b) Find a linearized state equation if it is desired to position the pendulum at a \( 45^\circ \).

Answered: Image /qna-images/answer/6ef426f9-c341-4e0d-a05f-3749bcf4058e.jpg

where g=9.8 m/s, mc is the cart's mass, m is the mass of the pole, I is the half length of the pole, and u is the applied force. a) b) Find a linearized state equation if the desired equilibrium value for each state is zero. Find a linearized state equation if it is desired to position the pendulum at a 45°.

where g=9.8 m/s, mc is the cart's mass, m is the mass of the pole, I is the half length of the pole, and u is the applied force. a) b) Find a linearized state equation if the desired equilibrium value for each state is zero. Find a linearized state equation if it is desired to position the pendulum at a 45°.

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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