CtrlSolve — Interactive Bode, Nyquist, and Response Plots

Preview: interactive control plots in the browser

CtrlSolve is a small, self-hosted control-systems helper: type a transfer function G(s), then generate key plots and response curves instantly.

It’s designed for quick sanity checks (stability, margins), lightweight design exploration, and classroom-style demonstrations without heavy tooling.

Plots: Bode (magnitude in dB + phase), Nyquist, Nichols, pole–zero map, root locus.
Time-domain responses: step, impulse, and ramp response curves.
Stability summary: stable/unstable, gain margin (GM) and phase margin (PM).
Extra details: zeros/poles list and a state-space realization (A, B, C, D).
Accepts LaTeX-style input and automatically simplifies/expands many common expressions.

Transfer function G(s)

\frac{10}{s^2 + 2s + 10}

Closed-loop analysis (Feedback)

Controller Design

Controller type Kp Ki Kd Filter tau Gain Zero Pole Time range Omega range

Time Domain

Step Response

Impulse Response

Ramp Response

Frequency Domain

Bode Plot

Nyquist Plot

Nichols Plot

Stability & Design

Pole-Zero Map

Root Locus

Named I/O Interconnect

Build closed-loop systems from named blocks and connection tables.

Minimal SISO Closed-Loop Wizard

Plant P(s) num Plant P(s) den Controller C(s) num Controller C(s) den Feedback sign

r → Σ − → C(s)1 / 1 → P(s)1 / 1, 1 → y

feedback -y

Systems

Num Den

Connections

External Channels

Input Output Analyze

Validation

State Space

Enter A, B, C, D matrices and prepare state-feedback design workflows.

Matrices

A B C D

Model

Name Sample time Input names Output names

Analysis

Input channel Output channel Time range Omega range

Design

Method Q R

LQR / Kalman Closed-Loop Noise Simulation

LQR Q LQR R Kalman G Kalman QN Kalman RN Kalman NN Initial state Initial estimate State reference Input feedforward Duration Samples Process noise std Measurement noise std Random seed

Nonlinear

Explore phase-plane behavior for curated nonlinear system templates.

Model

Template Parameters Constant input Streamline direction

Phase Plane

Window Vector grid Initial states Duration Samples

Time Series

Initial state Duration Samples

Bifurcation

Parameter Parameter range Initial state State index Duration Transient Samples

Poincare

Initial state Duration Transient Samples State index Section value Direction Return state index

Lyapunov Scan

Parameter Parameter range Initial state Duration Transient Samples Perturbation

Validation

Virtual Experiments

Compare overhead-crane motion plans and anti-sway controllers using trajectories computed by the Go backend.

r(t) Target position

x_d, v_d, a_d Motion plan

ZV / ZVD Input shaper

PD / LQR Anti-sway control

ẋ = f(x, a) Physical model

y(t), J Key metrics

x, v, θ, ω Feedback

Experiment preset

Physical model

Actual cable length m

Estimated cable length m

Trolley mass kg Load mass kg Initial position m Initial velocity m/s Initial angle deg Initial angular velocity deg/s

Motion plan

Target position m Trajectory Maximum velocity m/s Maximum acceleration m/s² Maximum jerk m/s³ Duration s Output samples

Anti-sway control

Input shaper Feedback

Advanced parameters

Gravity m/s² Angular damping N·m·s/rad Track friction N·s/m Position Kp Position Kd Sway Kp Sway Kd Estimated damping ratio LQR Q weights · x LQR Q weights · v LQR Q weights · θ LQR Q weights · ω LQR R

Crane animation

0.000 s

Time

3D Tower-crane Anti-sway Laboratory

GPU · waiting

Inspect a spherical-pendulum load whose paid-out suspended length is changed by the hoist while the Go backend coordinates trolley travel, slewing, hoisting, and anti-sway control.

r_d(t), ψ_d(t), L_d(t) Lift command

S(t) Synchronized motion

ZV / ZVD Input shaper

PD / K_LQR MIMO anti-sway

q̈ = f(q, p̈_s, u) Spherical pendulum

y(t), J, M_b Engineering metrics

r, ṙ, ψ, ψ̇, θ_r, θ_t State feedback

Operating scenario

Anti-sway strategy

Parameters

Crane geometry & load

SI

Tower height m Jib length m Counter-jib length m Payload mass kg Trolley mass kg Initial actual cable length m

Estimated cable length m

Trolley, slewing & hoist task

Payload start & target coordinates World XYZ · Z up

Start point r = 9.000 m · ψ = 0.000°

X m Y m Z m

Target point r = 24.000 m · ψ = 70.000°

X m Y m Z m

Start and target Z are independent: the Go backend converts them into initial and target cable lengths and generates a limit-compliant ideal hoist trajectory. Manual Cartesian edits start from rest with zero sway; advanced initial sway displays the resulting actual start point.

Derived trolley, slew & hoist parameters

Initial radius m Target radius m Initial slew angle deg Target slew angle deg Target cable length m

Trajectory Duration s Trolley speed limit m/s Slew-rate limit deg/s Hoist speed limit m/s Output samples

Anti-sway controller

Input shaper Feedback

Dynamics & actuator limits

Gravity m/s² Sway damping N·m·s/rad Trolley friction N·s/m Slew inertia kg·m² Slew damping N·m·s/rad Trolley acceleration m/s² Trolley jerk m/s³ Slew acceleration deg/s² Slew jerk deg/s³ Hoist acceleration m/s² Hoist jerk m/s³ Initial trolley velocity m/s Initial slew rate deg/s Initial radial sway deg Initial radial sway rate deg/s Initial tangential sway deg Initial tangential sway rate deg/s

Controller gains & LQR weights

Position Kp Position Kd Sway Kp Sway Kd Estimated damping ratio Q · e_r Q · ė_r Q · e_t Q · ė_t Q · θ_r Q · θ̇_r Q · θ_t Q · θ̇_t R · a_r R · a_t

Wind disturbance

Wind speed m/s Wind direction deg Gust starts s Gust duration s Air density kg/m³ Drag coefficient Projected load area m²

Model assumptions

The world frame is right-handed and Z-up; slew angle is measured from +X toward +Y.
The load is a point mass on a massless, taut, inextensible rope; the hoist changes its paid-out suspended length L(t), while a three-dimensional unit vector represents its direction.
Trolley radial and slew angular accelerations are ideal actuator inputs; cable length follows the backend-generated L_d(t) hoist trajectory subject to speed, acceleration, and jerk limits, without modeling a closed-loop hoist motor.
angular_damping is an isotropic rotational viscous damping coefficient in N·m·s/rad applied to both spherical-pendulum modes.
The wind input is a finite one-cosine speed pulse; drag force is 0.5·ρ·Cd·A·v² and acts only on the load.
Drive force, slew torque, hoist power and energy, total power and energy, and base moment are engineering load estimates; tower flexibility, gearbox efficiency, and structural wind loads are omitted.
The nonlinear plant is integrated with DOP853 using rtol 1e-9, atol 1e-11, a maximum step no larger than 0.02 s, and explicit stabilization of the unit rope-direction constraint.
LQR uses an eight-state, two-input small-angle model fixed at estimated_cable_length; hoisting is feedforward, while the simulated spatial plant remains nonlinear and time-varying.
settling_time is absolute simulation time after motion_end_time, requiring trolley, tangential, and cable-length position errors ≤ 0.03 m, low joint and hoist rates, resultant sway ≤ 0.01 rad, and sway-rate norm ≤ 0.02 rad/s for the remainder of the record.

Digital twin

GPU crane viewport

0.000 s

Time

3D scene ready on first visit Run a simulation to animate the backend trajectory.

Payload center of mass · live

X — Y — Z — m L —m v_L —m/s

Selected object center ? No component selected

X — Y — Z — m

Middle mouse rotate Right mouse pan Wheel zoom Shift + ↑↓←→ rotate; arrow keys pan

Run summary

Control & load KPIs

Residual sway—

Maximum sway—

Settling time—

Final load error—

Peak trolley force—

Peak slew torque—

Cable tension—

Energy—

Base moment—

Control effort—

Maximum hoist speed—

Maximum hoist acceleration—

Additional engineering indicators

Motion end time: —
RMS sway: —
Maximum load offset: —
Peak mechanical power: —
Maximum hook force: —
Maximum wind force: —
Minimum load height: —
Rope-constraint error: —
Minimum cable length: —
Maximum cable length: —
Peak hoist power: —
Hoist energy consumption: —

Controller diagnostics

—: —

Warnings

—

Data

Preview CSV and FRD-style data before identification or frequency analysis.

Import

t,u,y
0,0,0
0.1,1,0.05
0.2,1,0.14
0.3,1,0.26
0.4,1,0.39
0.5,1,0.52

Mapping

Delimiter Has header Preview rows Time Inputs Outputs

Identification

Method Input column Output column

Preview

Report

Collect model inputs, summaries, plots, and notes into a shareable analysis report.

Report Preview

CtrlSolve analysis

Run a classic analysis or save a workspace draft and the latest summary will appear here.

Preview: interactive control plots in the browser

Controller Design

Design Result

State Space

Zeros & Poles

Sensitivity & Robust Performance

Step Response

Impulse Response

Ramp Response

Bode Plot

Nyquist Plot

Nichols Plot

Pole-Zero Map

Root Locus

Named I/O Interconnect

Minimal SISO Closed-Loop Wizard

Systems

Connections

External Channels

Validation

Analysis Result

Step Response

Impulse Response

Ramp Response

Bode Plot

Nyquist Plot

Nichols Plot

Pole-Zero Map

Root Locus

State Space

Matrices

Model

Analysis

Design

Analysis Result

Structural Diagnostics

Model Reduction

Reduction Result

Step Response Comparison

Bode Comparison

Design Result

LQR / Kalman Closed-Loop Noise Simulation

Closed-Loop Simulation Result

Step Response

Impulse Response

Ramp Response

Bode Plot

Pole-Zero Map

Nonlinear

Model

Phase Plane

Time Series

Bifurcation

Poincare

Lyapunov Scan

Validation

Analysis Result

Phase Portrait

States over Time

Parameter response

Poincare Section

Largest Lyapunov Exponent

Virtual Experiments

Crane animation

Position

Sway angle

Acceleration

Cable tension

3D Tower-crane Anti-sway Laboratory

GPU crane viewport

Synchronized plots

Plan-view trajectory

Trolley & slewing motion

Hoist motion

Data

Import

Mapping

Identification

Preview

Identification Result