This document presents a simulation of controlling an inverted pendulum. It includes equations of motion to model the system state based on the pendulum angle and position over time. A sliding control method is used to control the pendulum angle and ride height, choosing control inputs to minimize errors between the actual and desired states. Simulation results are shown controlling the pendulum velocity and angle to stabilize the system. An animation demonstrates the full simulated control of the inverted pendulum.

1. Simulation of Inverted Pendulum
Presented to: Dr. SH. JAVADI
Presented by:
Pourya parsa
January 2018
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2. System Model
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3. Equations of Motion
𝑠𝑡𝑎𝑡𝑒: 𝑥 𝜃 𝑙 𝑥 𝜃 𝑙
𝑇
𝑠𝑡𝑎𝑡𝑒: 𝑥 𝜃 𝑦 𝑥 𝜃 𝑦
𝑇
𝑧 = 𝑓 𝑧, 𝑑, 𝑑, 𝑑 + 𝐵 𝑧 𝑢
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4. Controlling Pendulum Angle &
Ride Height: Sliding Control
𝑞 = 𝜃 𝑦 𝑇
, 𝑞 = 𝜃 𝑦
𝑇
, 𝑞 = 𝑞 − 𝑞 𝑑,
𝑠 = 𝑞 + 𝜆 𝑞
𝑞 = 𝑓 𝑞, 𝑑, 𝑑, 𝑑 + 𝐵(𝑞)𝑢
𝑢 = 𝐵−1
𝑞 𝑑 − 𝑓 − 𝜆 𝑞
𝑢 = 𝑢 + 𝐹 + 𝜂 𝑠𝑎𝑡(
𝑠
Φ
)
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5. Controlling Pendulum Velocity
𝜃 𝑑 = −𝐾 𝑃 𝑥 − 𝑥 𝑑 − 𝐾𝐼
0
𝑡
𝑥 − 𝑥 𝑑 𝑑𝑡
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6. Results
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7. Results
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8. Animation
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