1. EE 460 Advanced Control and System Integration
EE 460 Advanced Control and Sys Integration State Space Modeling The Linear Inverted Pendulum
Wed, Sept 9
EE 460 Advanced Control and System Integration

2. The Linear Inverted Pendulum
The Feedback Instruments Digital Pendulum
The input is a horizontal force (𝑓) generated by a DC servomotor
The outputs are the pendulum angle () and the cart position (𝑥)
Wed, Sept 9
EE 460 Advanced Control and System Integration

3. The Linear Inverted Pendulum Dynamic Model
𝑓: external force
𝑀: Mass of cart
𝑚: mass of ball
𝑙: length of beam
𝑏: viscous friction
Linear Forces:
For small angles
If  is “small” then  2 →0
𝑙𝑠𝑖𝑛() 𝑥 𝑚  𝑙
External force 𝑓 𝑀 𝑏
Viscous friction
Wed, Sept 9
EE 460 Advanced Control and System Integration

4. The Linear Inverted Pendulum Dynamic Model
𝑑: rotary viscous friction
𝐼: moment of inertia of the beam
Rotational Forces (torque):
Linear force  distance
For small angles
𝑚 𝑥 𝑐𝑜𝑠()
𝑚 𝑥 𝑚 
𝑚𝑔𝑠𝑖𝑛() 𝑙 𝑚𝑔
Rotational
friction 𝑑 𝑀
Wed, Sept 9
EE 460 Advanced Control and System Integration

5. The Linear Inverted Pendulum Dynamic Model  State Space Model
Two second order ODEs, hence, a forth order ss-model
State vector
Position:
Velocity:
Angle:
Angular speed:
Wed, Sept 9
EE 460 Advanced Control and System Integration

6. The Linear Inverted Pendulum Dynamic Model  State Space Model
The output measurements are the pendulum angle () and the cart position (𝑥)
Wed, Sept 9
EE 460 Advanced Control and System Integration

7. The Linear Inverted Pendulum Derivation for acceleration
Substitute for 𝜃
Wed, Sept 9
EE 460 Advanced Control and System Integration

8. The Linear Inverted Pendulum Derivation for angular acceleration
Substitute for 𝑥
Wed, Sept 9
EE 460 Advanced Control and System Integration