ltv-mpc

Installation | Documentation | Example | Contributing

Linear time-variant (LTV) model predictive control in Python. Solve a quadratic program of the form:

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

Installation

pip install ltv-mpc

Usage

This module defines a one-stop shop function:

solve_mpc(problem: Problem) -> Solution

The Problem type defines the model predictive control problem (LTV system, LTV constraints, initial state and cost function to optimize) while the Solution holds the resulting state and input trajectories.

Example

Let us define a triple integrator:

    import numpy as np

    horizon_duration = 1.0  # [s]
    N = 16  # number of discretization steps
    T = horizon_duration / N
    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])
    C = np.vstack([+accel_from_state, -accel_from_state])
    e = np.array([+max_accel, +max_accel])

This leads us to the following linear MPC problem:

    from ltv_mpc import Problem

    x_init = np.array([0.0, 0.0, 0.0])
    x_goal = np.array([1.0, 0.0, 0.0])
    problem = Problem(
        transition_state_matrix=A,
        transition_input_matrix=B,
        ineq_state_matrix=C,
        ineq_input_matrix=None,
        ineq_vector=e,
        initial_state=x_init,
        goal_state=x_goal,
        nb_timesteps=N,
        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, solver="quadprog")

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

    import pylab

    t = np.linspace(0.0, horizon_duration, N + 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)
    pylab.legend(("position", "velocity", "acceleration"))

This example produces the following trajectory:

The behavior is a weighted compromise between reaching the goal state (weight 1.0) and keeping reasonable finite jerk inputs (weight 1e-6). The latter mitigate bang-bang accelerations but prevent fully reaching the goal within the horizon. See the examples folder for more examples.

🏗️ 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

This module is designed for prototyping. If you need performance, check out one of the following libraries, and open a PR if you know other relevant ones: