A root locus plot traces how the closed-loop poles of a transfer function migrate through the complex plane as a system parameter (typically gain K) varies from 0 to infinity. It is a fundamental tool in classical control theory for analyzing system stability and designing controllers. The plot reveals critical information about pole trajectories, stability boundaries, and gain margins.
#' anyplot.ai
#' root-locus-basic: Root Locus Plot for Control Systems
#' Library: ggplot2 3.5.1 | R 4.4.1
#' Quality: 86/100 | Created: 2026-06-18
library(ggplot2)
library(dplyr)
library(ragg)
set.seed(42)
# Theme tokens
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 <- c(
"#009E73", # 1 — brand green
"#C475FD", # 2 — lavender
"#4467A3", # 3 — blue
"#BD8233", # 4 — ochre
"#AE3030" # 5 — matte red
)
# Data: third-order system G(s) = K / (s(s+2)(s+4))
# Open-loop poles: 0, -2, -4. No finite zeros.
# Root locus computed numerically by sweeping K.
gain_vals <- c(seq(0, 2, length.out = 300),
seq(2, 10, length.out = 400),
seq(10, 40, length.out = 300))
gain_vals <- sort(unique(gain_vals))
# Characteristic polynomial: s^3 + 6s^2 + 8s + K = 0
# Roots via polyroot for each K
root_locus_rows <- lapply(gain_vals, function(K) {
# coefficients: s^3 + 6s^2 + 8s + K
r <- polyroot(c(K, 8, 6, 1))
# Sort by imaginary part to keep branch assignment consistent
r <- r[order(Im(r))]
list(
gain = rep(K, 3),
branch = c("Branch 1", "Branch 2", "Branch 3"),
re = Re(r),
im = Im(r)
)
})
df <- bind_rows(lapply(root_locus_rows, function(x) {
data.frame(gain = x$gain, branch = x$branch, re = x$re, im = x$im,
stringsAsFactors = FALSE)
}))
# Keep only points within a reasonable window
df <- df[abs(df$re) <= 8 & abs(df$im) <= 8, ]
# Open-loop poles (K=0)
poles <- data.frame(re = c(0, -2, -4), im = c(0, 0, 0))
# Imaginary axis crossing: K at which Routh criterion gives jw roots
# For s^3+6s^2+8s+K: Routh row 1=[1,8], row 2=[6,K], pivot=(48-K)/6
# Crossing when K=48, auxiliary eq 6s^2+48=0 -> s=+/-j*sqrt(8)
crossing <- data.frame(re = c(0, 0), im = c(sqrt(8), -sqrt(8)))
# Arrow data: show direction of increasing gain along each branch
arrow_k_vals <- c(6, 18, 6) # one per branch at a distinctive location
branch_names <- c("Branch 1", "Branch 2", "Branch 3")
arrow_df <- bind_rows(lapply(seq_along(branch_names), function(i) {
k0 <- arrow_k_vals[i]
dk <- 0.6
seg_start <- df[abs(df$gain - k0) < 0.15 & df$branch == branch_names[i], ][1, ]
seg_end <- df[abs(df$gain - (k0 + dk)) < 0.15 & df$branch == branch_names[i], ][1, ]
if (!is.na(seg_start$re) && !is.na(seg_end$re)) {
data.frame(x = seg_start$re, y = seg_start$im,
xend = seg_end$re, yend = seg_end$im,
branch = branch_names[i])
}
}))
# Damping ratio reference lines: radial rays from origin at angle arccos(zeta)
# from the negative real axis. A pole with damping ratio zeta lies at angle
# (pi - arccos(zeta)) from the positive real axis => direction (-zeta, sqrt(1-zeta^2)).
zeta_vals <- c(0.2, 0.4, 0.6, 0.8)
plot_re_max <- 7
plot_im_max <- 6
zeta_df <- bind_rows(lapply(zeta_vals, function(z) {
re_end <- -plot_re_max * z
im_end <- plot_re_max * sqrt(1 - z^2)
# Clip to plot bounds
if (abs(im_end) > plot_im_max) {
sc <- plot_im_max / abs(im_end)
re_end <- re_end * sc
im_end <- im_end * sc
}
rbind(
data.frame(x = 0, y = 0, xend = re_end, yend = im_end),
data.frame(x = 0, y = 0, xend = re_end, yend = -im_end)
)
}))
# Natural frequency circles: concentric circles at omega_n = 1..5
omega_df <- bind_rows(lapply(1:5, function(w) {
theta <- seq(0, 2 * pi, length.out = 200)
data.frame(re = w * cos(theta), im = w * sin(theta), grp = paste0("wn", w))
}))
# Title
plot_title <- "root-locus-basic · r · ggplot2 · anyplot.ai"
title_size <- 12
# Plot
p <- ggplot(df, aes(x = re, y = im, color = branch, group = branch)) +
# Natural frequency circles (dashed, muted — behind locus)
geom_path(
data = omega_df,
aes(x = re, y = im, group = grp),
color = INK_MUTED, linewidth = 0.3, linetype = "dashed",
alpha = 0.55, inherit.aes = FALSE
) +
# Damping ratio rays (dashed, muted — behind locus)
geom_segment(
data = zeta_df,
aes(x = x, y = y, xend = xend, yend = yend),
color = INK_MUTED, linewidth = 0.3, linetype = "dashed",
alpha = 0.55, inherit.aes = FALSE
) +
# Axis reference lines
geom_hline(yintercept = 0, color = INK_SOFT, linewidth = 0.4) +
geom_vline(xintercept = 0, color = INK_SOFT, linewidth = 0.4) +
# Root locus branches
geom_path(linewidth = 1.0, alpha = 0.85) +
# Direction arrows
geom_segment(
data = arrow_df,
aes(x = x, y = y, xend = xend, yend = yend, color = branch),
arrow = arrow(length = unit(0.18, "cm"), type = "closed"),
linewidth = 1.2,
inherit.aes = FALSE
) +
# Imaginary axis crossings (stability boundary)
geom_point(
data = crossing,
aes(x = re, y = im),
shape = 18, size = 4, color = IMPRINT_PALETTE[5],
inherit.aes = FALSE
) +
# Open-loop poles
geom_point(
data = poles,
aes(x = re, y = im),
shape = 4, size = 4, stroke = 1.5, color = INK,
inherit.aes = FALSE
) +
# Color scale (3 branches)
scale_color_manual(
values = IMPRINT_PALETTE[1:3],
name = "Branch"
) +
# Equal axes to preserve geometry
coord_equal(xlim = c(-7, 2), ylim = c(-6, 6)) +
labs(
title = plot_title,
x = "Real Axis",
y = "Imaginary Axis"
) +
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 = 0.15),
panel.grid.minor = element_line(color = INK_MUTED, linewidth = 0.08),
panel.border = element_blank(),
axis.line = element_line(color = INK_SOFT, linewidth = 0.4),
axis.title = element_text(color = INK, size = 10),
axis.text = element_text(color = INK_SOFT, size = 8),
plot.title = element_text(color = INK, size = title_size,
margin = margin(b = 8)),
legend.background = element_rect(fill = ELEVATED_BG, color = INK_SOFT,
linewidth = 0.3),
legend.text = element_text(color = INK_SOFT, size = 8),
legend.title = element_text(color = INK, size = 9),
legend.position = "right",
plot.margin = margin(12, 14, 10, 10)
)
# Save
ggsave(
filename = sprintf("plot-%s.png", THEME),
plot = p,
device = ragg::agg_png,
width = 8,
height = 4.5,
units = "in",
dpi = 400
)
Part of Root Locus Plot for Control Systems on anyplot.ai.