pykoop.dynamic_models.DuffingOscillator

class DuffingOscillator(alpha=1, beta=-1, delta=0.1)

Bases: ContinuousDynamicModel

Duffing oscillator model.

Equation is \ddot{x} + \delta \dot{x} + \beta x + \alpha x^3 = u(t) where usually u(t) = a \cos(\omega t).

Parameters:

• alpha (float) –

• beta (float) –

• delta (float) –

__init__(alpha=1, beta=-1, delta=0.1)

Instantiate DuffingOscillator.

Parameters:

• alpha (float) – Coefficient of cubic term.

• beta (float) – Coefficient of linear term.

• delta (float) – Coefficient of first derivative.

Return type:

None

Methods

__init__([alpha, beta, delta])	Instantiate DuffingOscillator.
f(t, x, u)	Implement differential equation.
g(t, x)	Implement output equation.
simulate(t_range, t_step, x0, u, **kwargs)	Simulate the model using numerical integration.

f(t, x, u)

Implement differential equation.

Parameters:

• t (float) – Time (s).

• x (np.ndarray) – State.

• u (np.ndarray) – Input.

Returns:

Time derivative of state.

Return type:

np.ndarray

g(t, x)

Implement output equation.

Parameters:

• t (float) – Time (s).

• x (np.ndarray) – State.

Returns:

Measurement of state.

Return type:

np.ndarray

simulate(t_range, t_step, x0, u, **kwargs)

Simulate the model using numerical integration.

Parameters:

• t_range (Tuple[float, float]) – Start and stop times in a tuple.

• t_step (float) – Timestep of output data.

• x0 (np.ndarray) – Initial condition, shape (n, ).

• u (Callable[[float], np.ndarray]) – Input function of time.

• **kwargs (dict) – Keyword arguments for integrate.solve_ivp().

Returns:

Time and state at every timestep. Each timestep is one row.

Return type:

Tuple[np.ndarray, np.ndarray]