How to run a panel 01 Pick a panel from the tab bar or open its route directly. 02 Enter the input. Control takes numerator and denominator coefficients in descending powers (1 and 1 1 0 for 1/(s(s+1))); Filter takes family, section count and Fs/Fc; Optimization takes a text model with one constraint per line. 03 Read the cards, not only the curve. Stability, margins, poles and zeros sit above the plot; step metrics appear on the Step tab; LP status, objective and variables appear above the feasible region. 04 Switch the view tabs — Bode, Nyquist, Root Locus, Step and Impulse for a loop; Heat 1-D, Wave 1-D and Laplace for a PDE. 05 Send the result onward. A transfer function sent from Filter or Control Design fills the Control coefficient fields. Worked examples All readouts below come from the live panels.
Panel
Input
Readout
Control
G(s) = 1 / 1 1 0, K = 1 (defaults)
Stable ✓ ; phase margin 51.83° @ 0.786 rad/s ; gain margin ∞ (provably no −180° crossing) ; closed-loop poles -0.5 + 0.866j and -0.5 − 0.866j ; zeros —
Control · Step
the same closed loop
OS 16.3% , rise 1.63s , settle 8.08s , final 1
Filter Design · IIR
butterworth lowpass, 2 sections, Fs 48000 Hz, Fc 1000 Hz (defaults)
spec IIR butterworth lowpass, 2 biquad(s), Fs=48000, Fc=1000 ; first section b0 0.00407407, b1 0.00814814, b2 0.00407407, a1 -1.88856, a2 0.904852; sweep row 2 reads 996.189 Hz
Optimization
max: 3x + 2y · x + y <= 10 · x <= 8 · y <= 6Optimal ✓ , objective 28 , x 8 , y 2 ; binding x + y <= 10 ; x <= 8
ODE · Phase portrait
x' = y, y' = -2*sin(x) (pendulum preset)
(-6.283, 0) center, (-3.142, 0) saddle, (4.424e-21, 0) center (a numerical zero of sin x), (3.142, 0) saddle, (6.283, 0) center
The default open loop crosses 0 dB at 0.786 rad/s , giving the 51.83° phase margin and a closed-loop step that overshoots 16.3% before settling in 8.08s .
Which loop the cards describe Bode, Nyquist and the margin cards use the open loop K·G(s); the stability card and pole list describe the unit-feedback closed loop, and Step/Impulse default to that loop with an Open/Closed toggle. That is why 1/(s(s+1)) reads Stable ✓ although the plant has a pole at the origin. K scales the open loop; an optional PID block places Kp, Ki and Kd ahead of the same analysis.
Limits to keep in mind
Numeric readouts carry method limits. Margins come from a swept Bode response, and the panel says when a crossing is provably absent — the default gain margin reads ∞ . Poles and zeros are numerical roots; step metrics warn when the tail has not settled.
The optimiser is a linear solver. Objectives and constraints must be linear, and integers are declared with int x y. Variables are treated as non-negative: max: -x under x <= 5 returns objective 0 at x = 0 , so signed quantities need a reparameterisation. Optimal, infeasible and unbounded describe the model, not the real process. What it does not do: circuits, units or symbolic work. It consumes coefficient lists and plain numbers; it does not simulate a schematic, convert units or derive formulas. Circuits, RLC impedance, Smith charts and FFT live in the Science Lab ; symbolic calculus, matrices and step-by-step algebra in Math Tools ; a quick expression is the scientific calculator . The biquad count is not the filter order The signature trap is the section count: Filter Design's order field counts second-order sections, and its label says so — Biquads (order = 2×). The default 2 gives a 4th-order filter, not a 2nd-order one. The spec line reads IIR butterworth lowpass, 2 biquad(s), Fs=48000, Fc=1000 , the table lists two sections, and the sweep row for 2 reads 996.189 Hz. Type 1 and the spec shrinks to 1 biquad(s) ; type 13 and the panel designs 12 sections, warning Order 13 exceeds the limit of 12; designed with 12 . Compare the spec line, not the number you typed, when matching a design against a textbook table.
Where it fits Checking a control tutorial For G(s) = 1/(s(s+1)) at K = 1 the margin card reads 51.83° and the step card reads 16.3% overshoot — the standard lightly damped second-order pair. Root locus and Nyquist show where the poles travel as K changes.
Prototyping a digital filter Design a lowpass at Fs = 48000 Hz and Fc = 1000 Hz, read the biquad sections and the 1–12 cutoff sweep, then copy the C or Python coefficient block or send H(z) onward with the sample rate attached.
Models, plans and dynamics The default LP returns x = 8 , y = 2 , objective 28 ; the integer preset returns x = 3 , y = 2 , objective 5 . The ODE panel classifies a pendulum's equilibria without manual linearisation, and the fitting and PDE panels cover least-squares fitting, system identification, heat, wave and Laplace problems.
Privacy The control, filter, signal, ODE, optimisation, fit, PDE and MATLAB panels compute inside your browser and upload nothing; the Python / ML and R kernels download their runtimes from the network only when you open them.
References
Wikipedia, Bode plot , en.wikipedia.org （访问日期：2026-10-01）— gain and phase margins.
Wikipedia, Butterworth filter , en.wikipedia.org （访问日期：2026-10-01）— −3 dB cutoff.
Wikipedia, Digital biquad filter , en.wikipedia.org （访问日期：2026-10-01）— second-order sections.
Wikipedia, Linear programming , en.wikipedia.org （访问日期：2026-10-01）— non-negative variables.
Wikipedia, Phase plane , en.wikipedia.org （访问日期：2026-10-01）— equilibria.
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Updated 2026-10-01