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

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

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

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

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

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

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

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