Pole-zero plot

Python Code

"""Control Systems — Pole-zero plot (s-plane closed-loop + z-plane Butterworth)."""


import numpy as np
import plotly.graph_objects as go
from plotly.subplots import make_subplots
from scipy.signal import tf2zpk, butter


def generate():
    # s-plane: closed-loop at K=6: num=6, den=s^3+3s^2+2s+6
    z_s, p_s, k_s = tf2zpk([6], [1, 3, 2, 6])

    # z-plane: 2nd-order Butterworth, Wn=0.3 (digital)
    b, a = butter(2, 0.3)
    z_z, p_z, k_z = tf2zpk(b, a)

    fig = make_subplots(
        rows=1, cols=2,
        subplot_titles=["s-plane (Closed-loop K=6)", "z-plane (Butterworth, Wn=0.3)"],
        horizontal_spacing=0.12,
    )

    # ── s-plane ──────────────────────────────────────────────────────────────
    # Stability boundary: vertical line at Re=0
    fig.add_shape(
        type="line", x0=0, y0=-3, x1=0, y1=3,
        line=dict(color=GRID, width=1, dash="dash"),
        row=1, col=1,
    )

    # Poles
    fig.add_trace(go.Scatter(
        x=p_s.real, y=p_s.imag,
        mode="markers",
        name="Poles (s)",
        marker=dict(color=VIOLET, size=12, symbol="x", line=dict(width=2.5)),
        hovertemplate="s=(%{x:.3f}+j%{y:.3f})<extra>pole</extra>",
    ), row=1, col=1)

    # Zeros
    if len(z_s) > 0:
        fig.add_trace(go.Scatter(
            x=z_s.real, y=z_s.imag,
            mode="markers",
            name="Zeros (s)",
            marker=dict(color=PINK, size=12, symbol="circle-open", line=dict(width=2.5)),
            hovertemplate="s=(%{x:.3f}+j%{y:.3f})<extra>zero</extra>",
        ), row=1, col=1)

    # ── z-plane ──────────────────────────────────────────────────────────────
    # Unit circle
    theta_uc = np.linspace(0, 2 * np.pi, 360)
    fig.add_trace(go.Scatter(
        x=np.cos(theta_uc), y=np.sin(theta_uc),
        mode="lines",
        name="Unit circle",
        line=dict(color=GRID, width=1, dash="dash"),
        showlegend=True,
        hoverinfo="skip",
    ), row=1, col=2)

    # Poles
    fig.add_trace(go.Scatter(
        x=p_z.real, y=p_z.imag,
        mode="markers",
        name="Poles (z)",
        marker=dict(color=TEAL, size=12, symbol="x", line=dict(width=2.5)),
        hovertemplate="z=(%{x:.3f}+j%{y:.3f})<extra>pole</extra>",
    ), row=1, col=2)

    # Zeros
    fig.add_trace(go.Scatter(
        x=z_z.real, y=z_z.imag,
        mode="markers",
        name="Zeros (z)",
        marker=dict(color=PINK, size=12, symbol="circle-open", line=dict(width=2.5)),
        hovertemplate="z=(%{x:.3f}+j%{y:.3f})<extra>zero</extra>",
    ), row=1, col=2)

    fig.update_layout(
        legend=dict(orientation="h", y=1.12),
        margin=dict(t=60, b=60, l=60, r=40),
        height=520,
    )

    # Equal aspect for both subplots
    fig.update_xaxes(
        zeroline=True, zerolinecolor=GRID, zerolinewidth=1,
        title_text="Real",
        row=1, col=1,
    )
    fig.update_yaxes(
        zeroline=True, zerolinecolor=GRID, zerolinewidth=1,
        title_text="Imaginary",
        scaleanchor="x", scaleratio=1,
        row=1, col=1,
    )
    fig.update_xaxes(
        zeroline=True, zerolinecolor=GRID, zerolinewidth=1,
        title_text="Real",
        range=[-1.5, 1.5],
        row=1, col=2,
    )
    fig.update_yaxes(
        zeroline=True, zerolinecolor=GRID, zerolinewidth=1,
        title_text="Imaginary",
        range=[-1.5, 1.5],
        scaleanchor="x2", scaleratio=1,
        row=1, col=2,
    )

    return fig


if __name__ == "__main__":
    generate()

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