systems.builtin.deterministic.continuous.SymbolicPendulum

systems.builtin.deterministic.continuous.SymbolicPendulum(*args, **kwargs)

Simple inverted pendulum system - first-order state-space formulation.

Physical System:

A point mass attached to a massless rigid rod, free to rotate about a fixed pivot. The pendulum experiences: - Gravitational torque (proportional to sin(θ)) - Viscous damping (proportional to angular velocity) - External control torque

State Space:

State: x = [θ, θ̇] - θ (theta): Angular position from upward vertical [rad] * θ = 0: upright (unstable equilibrium) * θ = π: hanging down (stable equilibrium) - θ̇ (theta_dot): Angular velocity [rad/s]

Control: u = [τ] - τ (torque): Applied torque at pivot [N⋅m]

Output: y = [θ] - Measures only the angle (partial observation)

Dynamics:

The equations of motion are: θ̇ = θ̇ θ̈ = -(β/I)θ̇ + (g/l)sin(θ) + τ/I

where I = ml² is the moment of inertia.

Rewritten as first-order system: dx/dt = [θ̇, -(β/ml²)θ̇ + (g/l)sin(θ) + τ/(ml²)]ᵀ

Parameters:

m : float, default=1.0 Mass of the bob [kg]. Larger mass → more inertia, slower response. l : float, default=1.0 Length of the rod [m]. Longer rod → more gravity torque, slower dynamics. beta : float, default=1.0 Damping coefficient [N⋅m⋅s/rad]. Larger β → more energy dissipation. g : float, default=9.81 Gravitational acceleration [m/s²].