ltv-mpc

Installation | Documentation | Usage | Contributing

Linear time-variant model predictive control in Python.

This module is designed for prototyping. If you need performance, check out one of the related libraries below.

Installation

pip install ltv-mpc

Usage

The module defines two types:

• Problem defines the model predictive control problem, that is:
  • Constrained linear time-variant system
  • Cost function to optimize
  • Initial (and goal) state
• Solution holds the solution to a given problem.

The solve_mpc function takes a Problem as input and outputs the corresponding Solution.

Example

Let us define a triple integrator:

    import numpy as np

    horizon_duration = 1.0
    nb_timesteps = 16
    T = horizon_duration / nb_timesteps
    A = np.array([[1.0, T, T ** 2 / 2.0], [0.0, 1.0, T], [0.0, 0.0, 1.0]])
    B = np.array([T ** 3 / 6.0, T ** 2 / 2.0, T]).reshape((3, 1))

Suppose for the sake of example that acceleration is the main constraint acting on our system. We thus define an acceleration constraint |acceleration| <= max_accel:

    max_accel = 3.0  # [m] / [s] / [s]
    accel_from_state = np.array([0.0, 0.0, 1.0])
    ineq_matrix = np.vstack([+accel_from_state, -accel_from_state])
    ineq_vector = np.array([+max_accel, +max_accel])

This leads us to the following linear MPC problem:

    from ltv_mpc import Problem

    initial_pos = 0.0
    goal_pos = 1.0
    problem = ltv_mpc.Problem(
        transition_state_matrix=A,
        transition_input_matrix=B,
        ineq_state_matrix=ineq_matrix,
        ineq_input_matrix=None,
        ineq_vector=ineq_vector,
        initial_state=np.array([initial_pos, 0.0, 0.0]),
        goal_state=np.array([goal_pos, 0.0, 0.0]),
        nb_timesteps=nb_timesteps,
        terminal_cost_weight=1.0,
        stage_state_cost_weight=None,
        stage_input_cost_weight=1e-6,
    )

We can solve it with:

    from ltv_mpc import solve_mpc

    solution = solve_mpc(problem, mpc)

The solution holds complete state and input trajectories as stacked vectors. For instance, we can extract and plot positions, velocities and accelerations as follows:

    import pylab

    t = np.linspace(0.0, horizon_duration, nb_timesteps + 1)
    X = solution.stacked_states
    positions, velocities, accelerations = X[:, 0], X[:, 1], X[:, 2]
    pylab.ion()
    pylab.plot(t, positions)
    pylab.plot(t, velocities)
    pylab.plot(t, accelerations)
    pylab.grid(True)

🏗️ Work in progress

This module is still under development and its API might change. Future works may include:

• Complete documentation
• Complete test coverage
• General linear stage cost functions

See also