Double Inverted Pendulum on a Cart

Example

DoubleInvertedPendulum/

Where you can find this pendulum

A. Bogdanov, “Optimal control of a double inverted pendulum on a cart,” Oregon Health and Science University, Tech. Rep. CSE-04-006, OGI School of Science and Engineering, Beaverton, OR, 2004.

Problem Description

Swing-up control of a double inverted pendulum on a cart is a benchmark problem for NMPC algorithms due to its high nonlinearity. The pendulum we want to swing up is:

• The state vector of the system is $x = [\theta_0,\dot{\theta}_0 , \theta_1 ,\dot{\theta}_1 ,\theta_2 ,\dot{\theta}_2]^{T}$, where $\theta_0$ is the displacement of the cart, and where $\theta_1$, $\theta_2$ are the pendulum angles. The displacement is bounded by $-0.1 \leq \theta_0 \leq 0.1$.

• The control force $u$ is constrained with $|u|\leq 10$.

• The system is modeled as $D(\theta)\ddot{\theta}+C(\theta,\dot{\theta})\dot{\theta}+G(\theta)=Hu$ with $\theta=[\theta_0,\theta_1,\theta_2]^T$.

The task is to swing up the pendulum from the initial state $x_0 = [0,\pi,\pi,0,0,0]^{T}$.

OCP in ParNMPC

• The state constraints are softened by introducing a slack variable $s\geq 0$ with a quadratic penalty on $s$.

• A terminal penalty function is imposed to swing up the pendulum.

Closed-loop Simulation using ParNMPC

Step 1. NMPC problem formulation

See Workflow of ParNMPC > NMPC Problem Formulation.

Example

DoubleInvertedPendulum/NMPC_Problem_Formulation.m

Step 2. Code generation and deployment in Simulink

See Workflow of ParNMPC > Code Generation and Deployment > Simulink.

1. Code generation

   Example

   DoubleInvertedPendulum/Simu_Simulink_Setup.m

2. Deployment

   Example

   DoubleInvertedPendulum/Simu_Simulink.slx