Nyquist Plot for Control Systems — ggplot2

A Nyquist plot maps a system's open-loop frequency response onto the complex plane by plotting the imaginary part against the real part of the transfer function as frequency varies from zero to infinity. It is the primary tool for applying the Nyquist stability criterion to determine whether a closed-loop control system is stable. The plot visually reveals gain and phase margins and is widely used in classical control theory and electronic circuit design.

Nyquist Plot for Control Systems rendered with ggplot2

R source (ggplot2)

#' anyplot.ai
#' nyquist-basic: Nyquist Plot for Control Systems
#' Library: ggplot2 3.5.1 | R 4.4.1
#' Quality: 90/100 | Created: 2026-06-17

library(ggplot2)
library(dplyr)
library(ragg)
library(grid)

# Theme tokens (Imprint palette, theme-adaptive chrome)
THEME       <- Sys.getenv("ANYPLOT_THEME", "light")
PAGE_BG     <- if (THEME == "light") "#FAF8F1" else "#1A1A17"
ELEVATED_BG <- if (THEME == "light") "#FFFDF6" else "#242420"
INK         <- if (THEME == "light") "#1A1A17" else "#F0EFE8"
INK_SOFT    <- if (THEME == "light") "#4A4A44" else "#B8B7B0"
INK_MUTED   <- if (THEME == "light") "#6B6A63" else "#A8A79F"

# Imprint palette — brand green always first
IMPRINT_PALETTE <- c(
  "#009E73",  # 1 — brand green (first series)
  "#C475FD",  # 2 — lavender
  "#4467A3",  # 3 — blue
  "#BD8233",  # 4 — ochre
  "#AE3030",  # 5 — matte red (semantic anchor: critical/error)
  "#2ABCCD",  # 6 — cyan
  "#954477",  # 7 — rose
  "#99B314"   # 8 — lime
)

# Data: open-loop transfer function G(s) = 2 / (s*(s+1)*(s+2))
# Phase crossover at ω = sqrt(2) ≈ 1.414 rad/s, where G(jω) = -1/3 (stable)
# Gain crossover (|G| = 1) near ω ≈ 0.77 rad/s
set.seed(42)
n_pts <- 800
omega <- exp(seq(log(0.05), log(100), length.out = n_pts))

G_jw <- function(w) {
  s <- 1i * w
  2 / (s * (s + 1) * (s + 2))
}

Gjw     <- G_jw(omega)
re_vals <- Re(Gjw)
im_vals <- Im(Gjw)

df <- data.frame(re = re_vals, im = im_vals, omega = omega)

# Unit circle (reference: gain = 1 boundary)
theta       <- seq(0, 2 * pi, length.out = 300)
unit_circle <- data.frame(x = cos(theta), y = sin(theta))

# Frequency annotation points: key control-theory landmarks
# pc = phase crossover (Im = 0, curve crosses real axis at ω = sqrt(2))
label_omegas  <- c(0.3, 0.7, 1.0, sqrt(2), 2.5)
label_idx     <- sapply(label_omegas, function(w) which.min(abs(omega - w)))
label_df      <- data.frame(
  re    = re_vals[label_idx],
  im    = im_vals[label_idx],
  label = c("ω=0.3", "ω=0.7", "ω=1.0", "ω=√2 (pc)", "ω=2.5"),
  nx    = c( 0.10,    0.10,    0.10,    -0.28,         0.20),
  ny    = c( 0.13,    0.13,   -0.18,     0.20,          0.16)
)

# Directional arrows showing direction of increasing ω along the curve
arrow_steps <- c(185, 255, 330, 415, 515)
arrow_end   <- arrow_steps + 22
arrow_df    <- data.frame(
  x    = re_vals[arrow_steps],
  y    = im_vals[arrow_steps],
  xend = re_vals[arrow_end],
  yend = im_vals[arrow_end]
)
# Keep only arrows within the display window
visible  <- abs(arrow_df$x) < 3.2 & abs(arrow_df$y) < 3.2 &
            abs(arrow_df$xend) < 3.2 & abs(arrow_df$yend) < 3.2
arrow_df <- arrow_df[visible, ]

# Title
plot_title <- "nyquist-basic · r · ggplot2 · anyplot.ai"

# Plot
p <- ggplot() +
  # Unit circle: gain = 1 reference
  geom_path(
    data = unit_circle, aes(x = x, y = y),
    color = INK_MUTED, linewidth = 0.55, linetype = "dashed"
  ) +
  # Reference axes through origin
  geom_hline(yintercept = 0, color = INK_SOFT, linewidth = 0.35) +
  geom_vline(xintercept = 0, color = INK_SOFT, linewidth = 0.35) +
  # Nyquist curve (brand green — first series)
  geom_path(
    data = df, aes(x = re, y = im),
    color = IMPRINT_PALETTE[1], linewidth = 1.15
  ) +
  # Directional arrows showing increasing ω
  geom_segment(
    data  = arrow_df,
    aes(x = x, y = y, xend = xend, yend = yend),
    color = IMPRINT_PALETTE[1], linewidth = 1.15,
    arrow = arrow(length = unit(0.22, "cm"), type = "closed", angle = 22)
  ) +
  # Frequency annotation dots (blue — second categorical series)
  geom_point(
    data = label_df, aes(x = re, y = im),
    color = IMPRINT_PALETTE[3], size = 2.5, shape = 19
  ) +
  geom_text(
    data = label_df, aes(x = re, y = im, label = label),
    color = INK_SOFT, size = 2.7, nudge_x = label_df$nx, nudge_y = label_df$ny
  ) +
  # Critical point (-1, 0): matte red X (semantic anchor for instability boundary)
  annotate(
    "point", x = -1, y = 0,
    color = IMPRINT_PALETTE[5], size = 6, shape = 4, stroke = 2.8
  ) +
  annotate(
    "text", x = -1.0, y = -0.24, label = "(-1, 0)",
    color = IMPRINT_PALETTE[5], size = 3.0, hjust = 0.5
  ) +
  # Axis labels and title
  labs(x = "Real", y = "Imaginary", title = plot_title) +
  # 1:1 aspect ratio — required so unit circle appears circular
  coord_fixed(ratio = 1, xlim = c(-3, 1.5), ylim = c(-3, 1.5)) +
  theme_minimal(base_size = 8) +
  theme(
    plot.background  = element_rect(fill = PAGE_BG, color = PAGE_BG),
    panel.background = element_rect(fill = PAGE_BG, color = NA),
    panel.grid.major = element_line(color = INK_MUTED, linewidth = 0.2),
    panel.grid.minor = element_blank(),
    panel.border     = element_rect(color = INK_MUTED, fill = NA, linewidth = 0.3),
    axis.title       = element_text(color = INK, size = 10),
    axis.text        = element_text(color = INK_SOFT, size = 8),
    plot.title       = element_text(color = INK, size = 12),
    plot.margin      = margin(20, 20, 20, 20, unit = "pt"),
    legend.position  = "none"
  )

# Save — square canvas (2400 × 2400 px: width=6, height=6 in, dpi=400)
ggsave(
  filename = sprintf("plot-%s.png", THEME),
  plot     = p,
  device   = ragg::agg_png,
  width    = 6,
  height   = 6,
  units    = "in",
  dpi      = 400
)

Part of Nyquist Plot for Control Systems on anyplot.ai.

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