"""
Classes for modeling nonlinear systems.
This module provides a class for representing and simulating discrete-time
nonlinear systems using a state-space representation.
"""
from typing import Callable
import numpy as np
[docs]
class NonlinearSystem:
"""
A class representing a nonlinear dynamical system.
The system is defined by its dynamics and output functions, in the form:
.. math::
x(k+1) = f(x(k), u(k)) = f_0(x(k)) + B * u(k)
y(k) = h(x(k), u(k)) = h_0(x(k)) + D * u(k)
Attributes:
f (Callable[[np.ndarray, np.ndarray], np.ndarray]): The function
representing the system's dynamics.
h (Callable[[np.ndarray, np.ndarray], np.ndarray]): The function
representing the system's output.
n (int): The number of system states.
m (int): The number of inputs to the system.
p (int): The number of outputs of the system.
eps_max (float): The upper bound of the system measurement noise.
x (np.ndarray): The internal state vector of the system.
"""
[docs]
def __init__(
self,
f: Callable[[np.ndarray, np.ndarray], np.ndarray],
h: Callable[[np.ndarray, np.ndarray], np.ndarray],
n: int,
m: int,
p: int,
eps_max: float = 0.0,
):
"""
Initialize a nonlinear dynamical system with a dynamics function `f`
and an output function `h`.
Args:
f (Callable[[np.ndarray, np.ndarray], np.ndarray]): The function
representing the system's dynamics.
h (Callable[[np.ndarray, np.ndarray], np.ndarray]): The function
representing the system's output.
n (int): The number of system states.
m (int): The number of inputs to the system.
p (int): The number of outputs of the system.
eps_max (float): The upper bound of the system measurement noise.
Defaults to 0.0.
"""
self.f = f
self.h = h
self.n = n
self.m = m
self.p = p
self.eps_max = eps_max
# System state
self.x = np.zeros(self.n)
[docs]
def simulate_step(self, u: np.ndarray, w: np.ndarray) -> np.ndarray:
"""
Simulate a single time step of the nonlinear system with a given
input `u`.
The system simulation follows the state-space equations:
.. math::
x(k+1) = f(x(k), u(k)) = f_0(x(k)) + B * u(k)
y(k) = h(x(k), u(k)) = h_0(x(k)) + D * u(k)
Args:
u (np.ndarray): The input vector of shape `(m,)` at the current
time step, where `m` is the number of inputs.
w (np.ndarray): The measurement noise vector of shape `(p,)` at
the current time step, where `p` is the number of outputs.
Returns:
np.ndarray: The output vector `y` of shape `(p,)` at the current
time step, where `p` is the number of outputs.
Note:
This method updates the `x` attribute, which represents the
internal state vector of the system, after simulation.
"""
# Compute output using the output function
y = self.h(self.x, u) + w
# Update state based on the dynamics function
self.x = self.f(self.x, u)
return y
[docs]
def simulate(self, U: np.ndarray, W: np.ndarray, steps: int) -> np.ndarray:
"""
Simulate the nonlinear system over multiple time steps.
Args:
U (np.ndarray): An input matrix of shape `(steps, m)` where
`steps` is the number of time steps and `m` is the number of
inputs.
W (np.ndarray): A noise matrix of shape `(steps, p)` where `steps`
is the number of time steps and `p` is the number of outputs.
steps (int): The number of simulation steps.
Returns:
np.ndarray: The output matrix `Y` of shape `(steps, p)` containing
the simulated system outputs at each time step.
Note:
This method updates the `x` attribute, which represents the
internal state vector of the system, after each simulation step.
"""
# Initialize system output
Y = np.zeros((steps, self.p))
for k in range(steps):
Y[k, :] = self.simulate_step(U[k, :], W[k, :])
return Y
