SciPy与Python Control库lsim函数行为不一致问题排查

问题根源 #

• SciPy的lsim函数在处理**无输入（u=None）**的连续系统仿真时存在逻辑疏漏：当输入为None且系统B矩阵为列向量（如示例中的(2,1)）时，会错误地将连续系统按离散时间系统的逻辑处理，而非执行连续时间的状态积分。
• Python Control库的lsim对无输入场景的处理符合控制工程预期，会严格按照连续时间状态方程$\dot{x}=Ax$进行数值积分。

解决方法 #

要让SciPy的lsim正确仿真无输入连续系统，需显式传入与时间序列长度匹配的零输入矩阵，而非使用None。

修正后的完整MWE #

"""
Minimal working example (MWE) fixing the discrepancy between lsim for the SciPy module and control module
"""

# Import modules
import numpy as np
import scipy.signal as sgnl
import matplotlib.pyplot as plt
import control as ctrl
import control.matlab as ctrl_mtl

# Number of samples
N_k = 100    # Total number of outputs

# Continous-time (unstable) open loop system dynamics
A = np.array([[0, 1],
              [-2, 3]])
B = np.array([[0],
              [1]])
n_x = A.shape[0]
n_u = B.shape[1]

# Initial state
x_0 = np.array([[1], [1]], dtype=float)

# Sampling period
h = 0.5

# Controller (full-state feedback)
K = np.array([[0, -4]])

# Closed-loop (stable) continous-time system
sys_cl_scipy = sgnl.StateSpace(A + B @ K, np.zeros((n_x, 1)), np.eye(n_x), np.zeros((n_x, 1)))
sys_cl_ctrl = ctrl.StateSpace(A + B @ K, np.zeros((n_x, 1)), np.eye(n_x), np.zeros((n_x, 1)))

# Simulate dynamics
# 修正：显式生成零输入序列，替代None
u_scipy = np.zeros((N_k, n_u))
_, _, x_scipy = sgnl.lsim(sys_cl_scipy, u_scipy, np.linspace(0, h * (N_k - 1), N_k), X0=np.ravel(x_0), interp=False)
_, _, x_ctrl = ctrl_mtl.lsim(sys_cl_ctrl, T=np.linspace(0, h * (N_k - 1), N_k), X0=x_0)
x_scipy = x_scipy.T
x_ctrl = x_ctrl.T

# Plot the state through state-space (verification)
fig_verify, ax = plt.subplots()
ax.plot(np.ravel(x_scipy[0, :]), np.ravel(x_scipy[1, :]), label="x SciPy", color="green")
ax.plot(np.ravel(x_ctrl[0, :]), np.ravel(x_ctrl[1, :]), label="x control", color="blue")
plt.legend()
plt.title("State trajectory")

# Display all plots
plt.show()

补充说明 #

• 当系统存在非零输入时，SciPy的lsim行为与Control库一致；仅在无输入且B矩阵为列向量的场景下会触发该问题。
• 显式传入零输入序列可绕过SciPy内部的逻辑判断漏洞，确保连续系统的仿真逻辑正确执行。

内容的提问来源于stack exchange，提问作者Bart Wolleswinkel