This repository contains the simulation code accompanying the paper "Enhanced Sampled-Data Model Predictive Control via Nonlinear Lifting."

If you use this code in your research, please cite:

@article{gerdpratoom2025enhanced,
  title={Enhanced Sampled-Data Model Predictive Control via Nonlinear Lifting},
  author={Gerdpratoom, Nuthasith and Matsuzaki, Fumiya and Yamamoto, Yutaka and Yamamoto, Kaoru},
  journal={International Journal of Robust and Nonlinear Control},
  year={2025},
  publisher={Wiley Online Library}
}

Available: Wiley Online Library

Before running the simulation, make sure the following are installed:

• Python 3.x with the following packages:
  • numpy
  • matplotlib
  • casadi
  • opengen
• Rust (required by OpEn for solver compilation)
• OpEn (Optimization Engine)

Please follow the installation instructions in the official OpEn documentation.

Before running any simulations, you must first generate the optimizer solvers using OpEn. This compiles Rust-based optimization solvers for each controller type.

cd OpEn/singlerate

# Generate standard NMPC optimizer
python gen.py

# Generate lifted NMPC optimizer
python gen_lifted.py

You need to generate optimizers for three different sampling periods (0.1s, 0.25s, 0.5s). Edit the sampling_time parameter in each file before running:

cd OpEn/multirate

# Generate for sampling_time = 0.1
# Edit gen.py, gen_lifting.py, and gen_lifting_multirate.py to set sampling_time = 0.1
python gen.py
python gen_lifting.py
python gen_lifting_multirate.py

# Generate for sampling_time = 0.25
# Edit the files to set sampling_time = 0.25
python gen.py
python gen_lifting.py
python gen_lifting_multirate.py

# Generate for sampling_time = 0.5
# Edit the files to set sampling_time = 0.5
python gen.py
python gen_lifting.py
python gen_lifting_multirate.py

Note: Each optimizer generation may take several minutes as OpEn compiles the Rust solver code.

After generating all optimizers, you can run the main simulation scripts:

python main_singlerate.py

This runs two controllers (NMPC and Lifted NMPC) and displays:

• Cart position and velocity trajectories
• Control input sequences
• State norms
• Solver computation times
• Performance metrics (RMS values)

python main_multirate.py

This runs nine simulations (3 sampling periods and 3 controller types) and compares:

• Standard NMPC
• Lifted NMPC
• Lifted NMPC with multi-rate control

across sampling periods of 0.1s, 0.25s, and 0.5s.

Edit the parameters in main_singlerate.py or main_multirate.py:

# Simulation time step
sampling_time_sim = 0.001

# Sampling period
sampler_period = 0.05  # for single-rate

# Initial condition
x_state_0 = [0, np.pi, 0, 0]  # [position, angle, velocity, angular_velocity]

# Cost weights
Q = [2.5, 10, 0.01, 0.01]   # State weights
Qt = [3.0, 30, 0.1, 0.02]   # Terminal weights
R = [0.1]                    # Control weight

Edit the physical parameters when creating the model:

model = CartPoleModel(
    gravity_acceleration=9.8,
    length=1.0,
    mass_cart=1.0,
    mass_pole=0.2
)