control-systems-ai999x — offline, verified control systems solver with step-by-step answers

Desktop app & command line · Compute in your browser · Install (Python) · Guides · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Engineering · Tasks: Question Answering, Sentence Similarity, Feature Extraction

control-systems-ai999x is an offline, deterministic Python solver that answers control systems questions with verified results from 97 certified capabilities, gives 686 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

control-systems-ai999x is a compact, deterministic control systems engine for Python. It reads a question in plain English (or a typed call), picks a certified control systems capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Run it your way. control-systems-ai999x comes as the AI999X desktop app and as the command-line program control-systems for Windows and Linux, runs in your browser as WebAssembly, and installs as a Python package. The desktop app, the command line and the browser run the same compiled Rust program on your own device: offline, no Python needed.

Get it

Way to run it	For	Get it
AI999X Control Systems desktop app 1.2.0	Windows, Linux — this model in its own app: search, ask in words, forms, live figures, explanations, offline	Windows (x86-64) · Linux (x86-64)
AI999X desktop app 1.2.0	Windows, Linux — every AI999X model in one app: search, ask in words, forms, live figures, explanations, offline	Windows (x86-64) · Linux (x86-64)
Command line (control-systems)	Windows, Linux — one program per model, offline, no Python	Windows (x86-64) · Linux (x86-64)
In your browser	any device — nothing to install	Compute on the model page
Python	Python 3.10+	pip install control-systems-ai999x

Checksums: SHA256SUMS · every version: https://www.ai999x.com/downloads/desktop/latest/ · install guide: desktop & command line

Rust versions

Control Systems also comes as native programs compiled from Rust, version 1.2.0 of the AI999X desktop app and command line: they run offline on your computer and need no Python. Each file is in this repository's rust/ folder, on the GitHub release v1.0.0 and on the website.

Program	System	File	Size	Download	Use
Desktop app (Control Systems)	Windows x86-64	ai999x-control-systems-1.2.0-windows-x86_64.zip	7.4 MB	Hub · GitHub · website	unzip, then double-click ai999x-control-systems.exe (no installer)
Desktop app (Control Systems)	Linux x86-64	ai999x-control-systems-1.2.0-linux-x86_64.tar.gz	8.4 MB	Hub · GitHub · website	extract, then run ./ai999x-control-systems (X11 or Wayland desktop)
Command line (control-systems)	Windows x86-64	ai999x-control-systems-cli-1.2.0-windows-x86_64.zip	3.2 MB	Hub · GitHub · website (all models)	unzip, then control-systems.exe ask "<question>" or control-systems.exe solve <capability> name=value
Command line (control-systems)	Linux x86-64	ai999x-control-systems-cli-1.2.0-linux-x86_64.tar.gz	3.1 MB	Hub · GitHub · website (all models)	extract, then ./control-systems ask "<question>" or ./control-systems solve <capability> name=value

The command-line archive holds control-systems with a README of its commands and examples, the licence and the notice. Checksums: rust/SHA256SUMS (sha256sum -c SHA256SUMS).

Quick start

Desktop app

Download AI999X Control Systems for your system (above), unpack it and start it; type a question in words or pick a capability, fill in the form and press Compute. Figures and an explanation of the answer appear beside it. The AI999X desktop app has the same, with Control Systems among all AI999X models.

Linux, from a terminal:

curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/ai999x-control-systems-1.2.0-linux-x86_64.tar.gz
curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/SHA256SUMS
sha256sum -c --ignore-missing SHA256SUMS
tar xzf ai999x-control-systems-1.2.0-linux-x86_64.tar.gz
./ai999x-control-systems-1.2.0-linux-x86_64/ai999x-control-systems

Command line

Linux:

curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/ai999x-cli-1.2.0-linux-x86_64.tar.gz
curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/SHA256SUMS
sha256sum -c --ignore-missing SHA256SUMS
mkdir -p ~/.local/share && tar xzf ai999x-cli-1.2.0-linux-x86_64.tar.gz -C ~/.local/share
echo 'export PATH="$HOME/.local/share/ai999x-cli-1.2.0-linux-x86_64:$PATH"' >> ~/.bashrc && source ~/.bashrc
control-systems --version

Windows (PowerShell):

$F = "ai999x-cli-1.2.0-windows-x86_64.zip"
Invoke-WebRequest "https://www.ai999x.com/downloads/desktop/1.2.0/$F" -OutFile $F
Invoke-WebRequest "https://www.ai999x.com/downloads/desktop/1.2.0/SHA256SUMS" -OutFile SHA256SUMS
(Get-FileHash $F -Algorithm SHA256).Hash.ToLower() -eq ((Select-String -SimpleMatch $F SHA256SUMS).Line -split ' ')[0]
Expand-Archive $F -DestinationPath "$env:LOCALAPPDATA\AI999X" -Force
$D = "$env:LOCALAPPDATA\AI999X\ai999x-cli-1.2.0-windows-x86_64"; $env:Path += ";$D"
[Environment]::SetEnvironmentVariable("Path", [Environment]::GetEnvironmentVariable("Path", "User") + ";$D", "User")
control-systems --version

Every command below was run as shown; the text under it is what the program printed.

Ask in words — Write the question as you would say it: the program reads the quantities and their units, picks the capability and answers.

$ control-systems ask "stability margins: num = [10], den = [1, 3, 2, 0]"
Read as: Gain and phase margins
gm = 0.6
gm_db = -4.436974992
w_pc = 1.414213562 rad/s
pm_deg = -12.99720802
w_gc = 1.802203305 rad/s
  by: Gain and phase margins  (crossovers as positive real roots of |N|^2 - |D|^2 and Im(N(jw) conj D(jw)), Newton-polished)
  assumes: exactly one gain crossover and one phase crossover

What a capability needs — The equation or method, every input and output with its SI unit. Read this before solve: it tells you the input names.

$ control-systems info ctrl_stability_margins
Gain and phase margins
  crossovers as positive real roots of |N|^2 - |D|^2 and Im(N(jw) conj D(jw)), Newton-polished
  in   num      numerator of the open loop L(s) (vector) []
  in   den      denominator of L(s) (vector) []
  out  gm       gain margin (ratio) []
  out  gm_db    gain margin in dB []
  out  w_pc     phase-crossover frequency [rad/s]
  out  pm_deg   phase margin in degrees []
  out  w_gc     gain-crossover frequency [rad/s]

Solve — Inputs as name=value, each with any unit of the right kind. The answer lists every output, the method used, each input converted to SI, and the conditions the answer assumes.

$ control-systems solve ctrl_stability_margins "num=[10]" "den=[1, 3, 2, 0]"
gm = 0.6
gm_db = -4.436974992
w_pc = 1.414213562 rad/s
pm_deg = -12.99720802
w_gc = 1.802203305 rad/s
  by: Gain and phase margins  (crossovers as positive real roots of |N|^2 - |D|^2 and Im(N(jw) conj D(jw)), Newton-polished)
  assumes: exactly one gain crossover and one phase crossover

Many questions at once — One JSON request per line in, one JSON answer per line out, in order: for scripts, notebooks and agents (the program starts once). In PowerShell: Get-Content cases.jsonl | control-systems batch

cases.jsonl:

{"key": "ctrl_stability_margins", "inputs": {"num": "[10]", "den": "[1, 3, 2, 0]"}}
{"key": "ctrl_routh_hurwitz", "inputs": {"coefficients": "[1, 2, 3, 4, 5]"}}

$ control-systems batch < cases.jsonl
{"outcome":"SOLVED","message":"","value":-12.9972080155,"unit":"","solved_for":"pm_deg","values":[["gm",0.6,""],["gm_db",-4.43697499233,""],["w_pc",1.41421356237,"rad/s"],["pm_deg",-12.9972080155,""],["w_gc",1.80220330461,"rad/s"]],"target":"pm_deg","phrase":"gain margin (ratio)","assumptions":["exactly one gain crossover and one phase crossover"],"name":"Gain and phase margins","equation":"crossovers as positive real roots of |N|^2 - |D|^2 and Im(N(jw) conj D(jw)), Newton-polished"}
{"outcome":"SOLVED","message":"","value":2.0,"unit":"","solved_for":"rhp_roots","values":[["first_column",[1.0,2.0,1.0,-6.0,5.0],""],["rhp_roots",2,""],["stable",false,""]],"target":"rhp_roots","phrase":"first column of the Routh array","assumptions":["no zero in the first column (special cases are refused)"],"name":"Routh-Hurwitz stability test","equation":"Routh array in exact fractions; sign changes of the first column"}

In your browser

Open Control Systems on ai999x.com, choose a capability and press Compute: the model runs on your own device as WebAssembly; nothing is sent anywhere. Change any input (any unit of the right kind) and the answer, its explanation and the figures follow.

In your own page: control_systems_ai999x.wasm and the loader ax-wasm.js (https://www.ai999x.com/compute/rust/ax-wasm.js).

Python

pip install control-systems-ai999x

Worked examples in every form

The same worked example from the command line, from Python and in the browser. Each command-line output is what the program printed.

Lag compensator placement

C(s) = (s + z)/(s + p) with z = w_c/ratio (default a decade below the crossover) and p = z/beta: zero, pole, DC gain beta, and the phase lag and gain it adds at w_c.

Command line:

$ control-systems solve ctrl_lag_design beta=10 w_c=1
zero = 0.1
pole = 0.01
dc_gain = 10
phase_deg = -5.13765444
magnitude_db = 0.04277946506
  by: Lag compensator placement  (zero a decade below crossover, pole beta times lower)
  beta = 10
  w_c = 1
  assumes: beta > 1; ratio >= 2

Python:

import control_systems_ai999x as m

a = m.solve("ctrl_lag_design", beta="10", w_c="1")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

Lead compensator design for a required phase lead

alpha = (1 - sin phi)/(1 + sin phi) and T = 1/(w_m sqrt(alpha)) so that C(s) = (T s + 1)/(alpha T s + 1) adds the phase lead phi at w_m; its zero, pole and gain at w_m.

Command line:

$ control-systems solve ctrl_lead_design phi_max_deg=30 w_m=1
alpha = 0.3333333333
T = 1.732050808 s
zero = 0.5773502692
pole = 1.732050808
gain_db = 4.771212547
  by: Lead compensator design for a required phase lead  (sin phi_max = (1 - alpha)/(1 + alpha), w_m = 1/(T sqrt(alpha)))
  phi_max_deg = 30
  w_m = 1
  assumes: 0 < phi < 90 degrees; w_m > 0

Python:

import control_systems_ai999x as m

a = m.solve("ctrl_lead_design", phi_max_deg="30", w_m="1")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

Maximum phase lead of a lead compensator

For C(s) = (s + z)/(s + p) with z < p: the maximum phase lead asin((p - z)/(p + z)), the frequency sqrt(z p) where it occurs, alpha = z/p and the gain there, 10 log10(z/p) dB.

Command line:

$ control-systems solve ctrl_lead_max_phase z=1 p=9
phi_max_deg = 53.13010235
w_max = 3 rad/s
alpha = 0.1111111111
magnitude_db = -9.542425094
  by: Maximum phase lead of a lead compensator  (stationary point of the phase)
  z = 1
  p = 9
  assumes: 0 < z < p

Python:

import control_systems_ai999x as m

a = m.solve("ctrl_lead_max_phase", z="1", p="9")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

Desired closed-loop poles from overshoot and settling time

zeta from the overshoot, sigma = 4/Ts from the 2 % settling time, wn = sigma/zeta, the poles -sigma +/- j wd and the desired characteristic polynomial.

Command line:

$ control-systems solve ctrl_specs_to_poles OS=0.1630335348 Ts=2
zeta = 0.5
wn = 4 rad/s
sigma = 2
wd = 3.464101615 rad/s
char_poly = [1.0,4.0,15.9999999982]
  by: Desired closed-loop poles from overshoot and settling time  (zeta from the overshoot formula; sigma = 4/Ts)
  OS = 0.1630335348
  Ts = 2
  assumes: 0 < OS < 100 %; Ts > 0

Python:

import control_systems_ai999x as m

a = m.solve("ctrl_specs_to_poles", OS="0.1630335348", Ts="2")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

Documentation

• Rust user guide — the command line, every capability with a command and what it printed: PDF · Markdown
• Python guide — every capability with a Python example: PDF · Markdown
• Desktop app & command line — downloads, install, ask and search, figures, offline use, checksums: https://www.ai999x.com/desktop-cli/
• Model page — compute in your browser, every capability explained: https://www.ai999x.com/models/control-systems-ai999x/

Highlights

• Verified, not generated. Every answer is computed by a certified control systems capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 686 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Native and in the browser. Every capability also computes natively in Rust: the desktop app and the control-systems command line for Windows and Linux, and WebAssembly on the model page — no Python, no network.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: transfer functions, time and frequency response, stability margins, root locus, state space, LQR, Kalman filters, discretization and PID tuning.

Model overview

Property	Value
Package	control-systems-ai999x 1.0.0 (import control_systems_ai999x)
Type	Deterministic neuro-symbolic control systems engine (rule-based reading + certified symbolic capabilities)
Capabilities	97 certified procedures
Knowledge pack	version 1.0.0
Golden answers	686 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Python quick start

pip install control-systems-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/control-systems-ai999x/resolve/main/dist/control_systems_ai999x-1.0.0-py3-none-any.whl

import control_systems_ai999x as control_systems

control_systems.ask("stability margins: num = [10], den = [1, 3, 2, 0]").show()
# -> SOLVED — gm = 0.6, gm_db = -4.43697, w_pc = 1.41421, pm_deg = -12.9972  (ctrl_stability_margins)
control_systems.ask("Routh-Hurwitz: coefficients = [1, 2, 3, 4, 5]").show()
# -> SOLVED — first_column = [1.0, 2.0, 1.0, -6.0, 5.0], rhp_roots = 2, stable = 0  (ctrl_routh_hurwitz)
control_systems.solve("ctrl_percent_overshoot", zeta=0.5).show()
# -> SOLVED — PO = 16.3034, Mp = 0.163034, pole_angle_deg = 60  (ctrl_percent_overshoot)
control_systems.solve("ctrl_lqr", A=[[0, 1], [0, 0]], B=[0, 1], Q=[[1, 0], [0, 1]], R=1).show()
# -> SOLVED — K = [1.0, 1.73205080757], P = [[1.73205080757, 1.0], [1.0, 1.73205080757]], poles = [(-0.866025403784-0.5j), (-0.866025403784+0.5j)]  (ctrl_lqr)
control_systems.ask("Tustin discretization: num = [1], den = [1, 1], T = 0.1").show()
# -> SOLVED — numz = [0.047619047619, 0.047619047619], denz = [1.0, -0.904761904762]  (ctrl_tustin)
control_systems.ask("Who created you?").message
# -> control-systems-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

The Python package also installs a Python command, control-systems-ai999x "<question>"; for the native program use control-systems (above). The outputs above were produced by running this exact code on control-systems-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Python worked examples, with their steps

Each question was put to control-systems-ai999x; the answer is the one it returned, with the steps it took.

Gain and phase margins · Stability

stability margins: num = [10], den = [1, 3, 2, 0]

→ gm = 0.6, gm_db = -4.436974992, w_pc = 1.414213562 rad/s, pm_deg = -12.99720802, w_gc = 1.802203305 rad/s

Steps

• Gain and phase margins: num = [10], den = [1, 3, 2, 0] → gm = 0.6, gm_db = -4.436974992, w_pc = 1.414213562 rad/s, pm_deg = -12.99720802, w_gc = 1.802203305 rad/s

Routh-Hurwitz stability test · Stability

Routh-Hurwitz: coefficients = [1, 2, 3, 4, 5]

→ first_column = [1, 2, 1, -6, 5], rhp_roots = 2, stable = False

Steps

• Routh-Hurwitz stability test: coefficients = [1, 2, 3, 4, 5] → first_column = [1, 2, 1, -6, 5], rhp_roots = 2, stable = False

Percent overshoot from the damping ratio · Time response

m.solve("ctrl_percent_overshoot", zeta=0.5)

→ PO = 16.30335348, Mp = 0.1630335348, pole_angle_deg = 60

Steps

• Percent overshoot from the damping ratio: zeta = 0.5 → PO = 16.30335348, Mp = 0.1630335348, pole_angle_deg = 60

Continuous-time LQR gain (Riccati equation) · Optimal control

m.solve("ctrl_lqr", A=[[0, 1], [0, 0]], B=[0, 1], Q=[[1, 0], [0, 1]], R=1)

→ K = [1, 1.732050808], P = [[1.732050808, 1], [1, 1.732050808]], poles = [-0.8660254038-0.5i, -0.8660254038+0.5i]

Steps

• Continuous-time LQR gain (Riccati equation): A = [0 1; 0 0], B = [0, 1], Q = [1 0; 0 1], R = 1 → K = [1, 1.732050808], P = [[1.732050808, 1], [1, 1.732050808]], poles = [-0.8660254038-0.5i, -0.8660254038+0.5i]

Tustin (bilinear) discretization · Discretization

Tustin discretization: num = [1], den = [1, 1], T = 0.1

→ numz = [0.04761904762, 0.04761904762], denz = [1, -0.9047619048]

Steps

• Tustin (bilinear) discretization: num = [1], den = [1, 1], T = 0.1 → numz = [0.04761904762, 0.04761904762], denz = [1, -0.9047619048]

Lag compensator placement · Compensators

lag compensator design: beta = 10, w_c = 1

→ zero = 0.1, pole = 0.01, dc_gain = 10, phase_deg = -5.13765444, magnitude_db = 0.04277946506

Steps

• Lag compensator placement: beta = 10, w_c = 1 → zero = 0.1, pole = 0.01, dc_gain = 10, phase_deg = -5.13765444, magnitude_db = 0.04277946506

Figures

Every point was computed by control-systems-ai999x.

Lag compensator placement

phase_deg (phase added at w_c in degrees (negative)) against beta (required low-frequency gain increase (> 1)), with w_c = 1. 31 points computed by control-systems-ai999x; marked: beta = 10, phase_deg = -5.13765.

Desired closed-loop poles from overshoot and settling time

wn (natural frequency) against OS (maximum overshoot (write 10% or 0.1)), with Ts = 2. 31 points computed by control-systems-ai999x; marked: OS = 0.163034, wn = 4 rad/s.

Percent overshoot from the damping ratio

PO (percent overshoot (%)) against zeta (damping ratio (0 <= zeta < 1)). 31 points computed by control-systems-ai999x; marked: zeta = 0.5, PO = 16.3034.

Mapping an s-plane pole to the z-plane (z = e^{sT})

z_mag (|z| = e^{s_re T}) against s_re (real part of the s-plane pole (1/s)), with T = 0.1, s_im = 0. 31 points computed by control-systems-ai999x; marked: s_re = -2, z_mag = 0.818731.

Capabilities

97 certified procedures. A few of them:

• Procedures: -3 dB bandwidth of a transfer function; Backward Euler (backward difference) discretization; Bandwidth of a standard second-order system; Bode magnitude and phase at one frequency; Closed-loop poles of a PID loop; Closed-loop poles under state feedback u = -K x; Closed-loop transfer function of a feedback loop; Cohen-Coon tuning of a first-order-plus-dead-time process; Continuous Lyapunov equation; Continuous equivalent of a discrete pole (s = ln z / T); Continuous-time LQR gain (Riccati equation); Controllability Gramian; Controllability matrix and rank; Controllable canonical form of a transfer function; DC gain and final value of a discrete transfer function; DC gain and final value of a step response; DC gain of a state-space model; Damping ratio from percent overshoot; Damping ratio from the logarithmic decrement; Deadbeat state feedback (discrete time); Desired closed-loop poles from overshoot and settling time; Discrete Lyapunov equation; Discrete PID in velocity (incremental) form; Discrete-time LQR gain (discrete Riccati equation) …

control_systems_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 97 formulas

Show every formula, by topic

PID tuning / Cohen-Coon

Cohen-Coon tuning of a first-order-plus-dead-time process

$$\text{PID: }{K}_p=\frac{\tau }{K\theta }(\frac{4}{3}+\frac{\theta }{4\tau }),T_i=\theta \frac{32+6\theta \mathrm{/}\tau }{13+8\theta \mathrm{/}\tau },T_d=\frac{4\theta }{11+2\theta \mathrm{/}\tau }\backslash text\{PID:\; \}\; K\_p\; =\; \backslash frac\{\backslash tau\}\{K\backslash theta\}\backslash left(\backslash frac\{4\}\{3\}\; +\; \backslash frac\{\backslash theta\}\{4\backslash tau\}\backslash right),\backslash ;\; T\_i\; =\; \backslash theta\backslash frac\{32+6\backslash theta/\backslash tau\}\{13+8\backslash theta/\backslash tau\},\backslash ;\; T\_d\; =\; \backslash frac\{4\backslash theta\}\{11+2\backslash theta/\backslash tau\}$$

PID tuning / SIMC

SIMC (Skogestad) PI tuning

$$K_p=\frac{\tau }{K({\tau }_c+\theta )},T_i=\min (\tau ,4({\tau }_c+\theta ))K\_p\; =\; \backslash frac\{\backslash tau\}\{K(\backslash tau\_c+\backslash theta)\},\backslash ;\; T\_i\; =\; \backslash min(\backslash tau,\; 4(\backslash tau\_c+\backslash theta))$$

PID tuning / Ziegler-Nichols

Ziegler-Nichols tuning from a plant model

$$K_u=\frac{1}{\mathrm{\mid }G(j{\omega }_{180})\mathrm{\mid }},P_u=\frac{2\pi }{{\omega }_{180}}K\_u\; =\; \backslash frac\{1\}\{\backslash vert\; G(j\backslash omega\_\{180\})\backslash vert\; \},\backslash quad\; P\_u\; =\; \backslash frac\{2\backslash pi\}\{\backslash omega\_\{180\}\}$$

Ziegler-Nichols tuning from the process reaction curve

$$\text{PID: }{K}_p=\frac{1.2T}{KL},T_i=2L,T_d=0.5L\backslash text\{PID:\; \}\; K\_p\; =\; \backslash frac\{1.2\backslash ,T\}\{KL\},\backslash ;\; T\_i\; =\; 2L,\backslash ;\; T\_d\; =\; 0.5L$$

Ziegler-Nichols tuning from the ultimate gain and period

$$\text{PID: }{K}_p=0.6K_u,T_i=P_u\mathrm{/}2,T_d=P_u\mathrm{/}8\backslash text\{PID:\; \}\; K\_p\; =\; 0.6K\_u,\backslash ;\; T\_i\; =\; P\_u/2,\backslash ;\; T\_d\; =\; P\_u/8$$

PID tuning / closed loop

Closed-loop poles of a PID loop

$$sD(s)+(K_d{s}^2+K_{p}s+K_i)N(s)=0sD(s)\; +\; (K\_ds^2\; +\; K\_ps\; +\; K\_i)N(s)\; =\; 0$$

PID tuning / forms

PID: parallel gains to standard form

$$T_i=K_p\mathrm{/}{K}_i,T_d=K_d\mathrm{/}{K}_{p}T\_i\; =\; K\_p/K\_i,\backslash quad\; T\_d\; =\; K\_d/K\_p$$

PID: standard form to parallel gains

$$K_i=K_p\mathrm{/}{T}_i,K_d=K_p{T}_{d}K\_i\; =\; K\_p/T\_i,\backslash quad\; K\_d\; =\; K\_pT\_d$$

PID tuning / pole placement

PI gains placing the closed-loop poles of a first-order plant

$$K_p=\frac{2\zeta {\omega }_n\tau -1}{K},K_i=\frac{{\omega }_n^2\tau }{K}K\_p\; =\; \backslash frac\{2\backslash zeta\backslash omega\_n\backslash tau\; -\; 1\}\{K\},\backslash quad\; K\_i\; =\; \backslash frac\{\backslash omega\_n^2\backslash tau\}\{K\}$$

Compensators / lag

Lag compensator placement

$$C(s)=\frac{s+z}{s+z\mathrm{/}\beta },z={\omega }_c\mathrm{/}10C(s)\; =\; \backslash frac\{s\; +\; z\}\{s\; +\; z/\backslash beta\},\backslash quad\; z\; =\; \backslash omega\_c/10$$

Compensators / lead

Lead compensator design for a required phase lead

$$\alpha =\frac{1-\sin {\varphi }_m}{1+\sin {\varphi }_m},T=\frac{1}{{\omega }_m\sqrt{\alpha }}\backslash alpha\; =\; \backslash frac\{1-\backslash sin\backslash phi\_m\}\{1+\backslash sin\backslash phi\_m\},\backslash quad\; T\; =\; \backslash frac\{1\}\{\backslash omega\_m\backslash sqrt\backslash alpha\}$$

Maximum phase lead of a lead compensator

$$\sin {\varphi }_{max}=\frac{p-z}{p+z}=\frac{1-\alpha }{1+\alpha },{\omega }_m=\sqrt{zp}\backslash sin\backslash phi\_\{max\}\; =\; \backslash frac\{p-z\}\{p+z\}\; =\; \backslash frac\{1-\backslash alpha\}\{1+\backslash alpha\},\backslash quad\; \backslash omega\_m\; =\; \backslash sqrt\{zp\}$$

Design / specifications

Desired closed-loop poles from overshoot and settling time

$$\zeta =\frac{-\ln OS}{\sqrt{{\pi }^2+{\ln }^{2}OS}},\sigma =\frac{4}{T_s},{\omega }_n=\frac{\sigma }{\zeta }\backslash zeta\; =\; \backslash frac\{-\backslash ln\; OS\}\{\backslash sqrt\{\backslash pi^2+\backslash ln^2\; OS\}\},\backslash quad\; \backslash sigma\; =\; \backslash frac\{4\}\{T\_s\},\backslash quad\; \backslash omega\_n\; =\; \backslash frac\{\backslash sigma\}\{\backslash zeta\}$$

Discretization / Euler

Backward Euler (backward difference) discretization

$$s\leftarrow \frac{z-1}{Tz}s\; \backslash leftarrow\; \backslash frac\{z-1\}\{Tz\}$$

Forward Euler (forward difference) discretization

$$s\leftarrow \frac{z-1}{T}s\; \backslash leftarrow\; \backslash frac\{z-1\}\{T\}$$

Discretization / FOH

First-order-hold (triangle-hold) discretization

$$B_d={\mathrm{\Gamma }}_1-{\mathrm{\Gamma }}_2+\mathrm{\Phi }{\mathrm{\Gamma }}_2,D_d=D+C{\mathrm{\Gamma }}_{2}B\_d\; =\; \backslash Gamma\_1\; -\; \backslash Gamma\_2\; +\; \backslash Phi\backslash Gamma\_2,\backslash quad\; D\_d\; =\; D\; +\; C\backslash Gamma\_2$$

Discretization / Tustin

Tustin (bilinear) discretization

$$s\leftarrow \frac{2}{T}\frac{z-1}{z+1}s\; \backslash leftarrow\; \backslash frac\{2\}\{T\}\backslash ,\backslash frac\{z-1\}\{z+1\}$$

Tustin discretization with frequency prewarping

$$s\leftarrow \frac{\omega }{\tan (\omega T\mathrm{/}2)}\frac{z-1}{z+1}s\; \backslash leftarrow\; \backslash frac\{\backslash omega\}\{\backslash tan(\backslash omega\; T/2)\}\backslash ,\backslash frac\{z-1\}\{z+1\}$$

Discretization / ZOH

Zero-order-hold discretization of a first-order system

$$G(z)=\frac{K(1-e^{-T\mathrm{/}\tau })}{z-e^{-T\mathrm{/}\tau }}G(z)\; =\; \backslash frac\{K(1\; -\; e^\{-T/\backslash tau\})\}\{z\; -\; e^\{-T/\backslash tau\}\}$$

Zero-order-hold discretization of a transfer function

$$A_d=e^{AT},B_d={\int }_0^T{e}^{A\tau }d\tau BA\_d\; =\; e^\{AT\},\backslash ;\; B\_d\; =\; \backslash int\_0^T\; e^\{A\backslash tau\}d\backslash tau\backslash ,B$$

Discretization / digital PID

Discrete PID in velocity (incremental) form

$$\mathrm{\Delta }{u}_k=(K_p+K_{i}T+\frac{K_d}{T})e_k-(K_p+\frac{2K_d}{T})e_{k-1}+\frac{K_d}{T}{e}_{k-2}\backslash Delta\; u\_k\; =\; (K\_p\; +\; K\_iT\; +\; \backslash tfrac\{K\_d\}\{T\})e\_k\; -\; (K\_p\; +\; \backslash tfrac\{2K\_d\}\{T\})e\_\{k-1\}\; +\; \backslash tfrac\{K\_d\}\{T\}e\_\{k-2\}$$

Discretization / discrete systems

DC gain and final value of a discrete transfer function

$$y(\mathrm{\infty })=AG(1)y(\backslash infty)\; =\; A\backslash ,G(1)$$

Step response of a discrete transfer function at sample k

$$y[k]=\sum _i{b}_{i}u[k-i]-\sum _{i\ge 1}{a}_{i}y[k-i]y[k]\; =\; \backslash sum\_\{i\}\; b\_i\; u[k-i]\; -\; \backslash sum\_\{i\backslash ge\; 1\}\; a\_i\; y[k-i]$$

Discretization / pole mapping

Continuous equivalent of a discrete pole (s = ln z / T)

$$s=\frac{\ln z}{T}s\; =\; \backslash frac\{\backslash ln\; z\}\{T\}$$

Mapping an s-plane pole to the z-plane (z = e^{sT})

$$z=e^{sT}=e^{\sigma T}\mathrm{\angle }\omega Tz\; =\; e^\{sT\}\; =\; e^\{\backslash sigma\; T\}\backslash angle\backslash ,\backslash omega\; T$$

Estimation / Kalman filter

One step of a scalar Kalman filter

$$K=\frac{P^{-}}{P^{-}+r},\widehat{x}={\widehat{x}}^{-}+K(z-{\widehat{x}}^{-}),P=(1-K)P^{-}K\; =\; \backslash frac\{P^-\}\{P^-\; +\; r\},\backslash quad\; \backslash hat\; x\; =\; \backslash hat\; x^-\; +\; K(z\; -\; \backslash hat\; x^-),\backslash quad\; P\; =\; (1-K)P^-$$

Steady-state Kalman-Bucy filter gain

$$AP+P{A}^T-P{C}^T{V}^{-1}CP+W=0,L=P{C}^T{V}^{-1}AP\; +\; PA^T\; -\; PC^TV^\{-1\}CP\; +\; W\; =\; 0,\backslash quad\; L\; =\; PC^TV^\{-1\}$$

Steady-state discrete Kalman filter gain

$$P=AP{A}^T-AP{C}^T(CP{C}^T+R{)}^{-1}CP{A}^T+Q,K=P{C}^T(CP{C}^T+R{)}^{-1}P\; =\; APA^T\; -\; APC^T(CPC^T\; +\; R)^\{-1\}CPA^T\; +\; Q,\backslash quad\; K\; =\; PC^T(CPC^T\; +\; R)^\{-1\}$$

Steady-state gain of a scalar Kalman filter

$$P=a^{2}P-\frac{a^2{c}^2{P}^2}{c^{2}P+r}+q,K=\frac{Pc}{c^{2}P+r}P\; =\; a^2P\; -\; \backslash frac\{a^2c^2P^2\}\{c^2P\; +\; r\}\; +\; q,\backslash quad\; K\; =\; \backslash frac\{Pc\}\{c^2P\; +\; r\}$$

Frequency response / Bode

Bode magnitude and phase at one frequency

$$\mathrm{\mid }G(j\omega ){\mathrm{\mid }}_{dB}=20{\log }_{10}\mathrm{\mid }G(j\omega )\mathrm{\mid }\backslash vert\; G(j\backslash omega)\backslash vert\; \_\{dB\}\; =\; 20\backslash log\_\{10\}\backslash vert\; G(j\backslash omega)\backslash vert$$

Frequency response of a first-order lag

$$\mathrm{\mid }G(j\omega )\mathrm{\mid }=\frac{K}{\sqrt{1+{\omega }^2{\tau }^2}},\mathrm{\angle }G=-\arctan (\omega \tau )\backslash vert\; G(j\backslash omega)\backslash vert\; =\; \backslash frac\{K\}\{\backslash sqrt\{1+\backslash omega^2\backslash tau^2\}\},\backslash quad\; \backslash angle\; G\; =\; -\backslash arctan(\backslash omega\backslash tau)$$

Frequency response of a standard second-order system

$$\mathrm{\mid }G\mathrm{\mid }=\frac{{\omega }_n^2}{\sqrt{({\omega }_n^2-{\omega }^2{)}^2+(2\zeta {\omega }_n\omega {)}^2}}\backslash vert\; G\backslash vert\; =\; \backslash frac\{\backslash omega\_n^2\}\{\backslash sqrt\{(\backslash omega\_n^2-\backslash omega^2)^2\; +\; (2\backslash zeta\backslash omega\_n\backslash omega)^2\}\}$$

Frequency response / bandwidth

-3 dB bandwidth of a transfer function

$$\mathrm{\mid }G(j{\omega }_B)\mathrm{\mid }=\frac{\mathrm{\mid }G(0)\mathrm{\mid }}{\sqrt{2}}\backslash vert\; G(j\backslash omega\_B)\backslash vert\; =\; \backslash frac\{\backslash vert\; G(0)\backslash vert\; \}\{\backslash sqrt\; 2\}$$

Bandwidth of a standard second-order system

$${\omega }_B={\omega }_n\sqrt{1-2{\zeta }^2+\sqrt{4{\zeta }^4-4{\zeta }^2+2}}\backslash omega\_B\; =\; \backslash omega\_n\backslash sqrt\{1-2\backslash zeta^2+\backslash sqrt\{4\backslash zeta^4-4\backslash zeta^2+2\}\}$$

Frequency response / norms

H-infinity norm (peak gain) of a stable transfer function

$$vertGver{t}_{\mathrm{\infty }}=\underset{\omega }{\sup }\mathrm{\mid }G(j\omega )\mathrm{\mid }\backslash \backslash vert\; G\backslash \backslash vert\; \_\backslash infty\; =\; \backslash sup\_\backslash omega\; \backslash vert\; G(j\backslash omega)\backslash vert$$

H2 norm of a stable transfer function

$$vertGver{t}_2^2=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{\mid }G(j\omega ){\mathrm{\mid }}^{2}d\omega =B^T{W}_{o}B\backslash \backslash vert\; G\backslash \backslash vert\; \_2^2\; =\; \backslash frac\{1\}\{2\backslash pi\}\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash vert\; G(j\backslash omega)\backslash vert\; ^2d\backslash omega\; =\; B^TW\_oB$$

Frequency response / resonance

Resonant peak and resonant frequency

$$M_r=\frac{1}{2\zeta \sqrt{1-{\zeta }^2}},{\omega }_r={\omega }_n\sqrt{1-2{\zeta }^2}M\_r\; =\; \backslash frac\{1\}\{2\backslash zeta\backslash sqrt\{1-\backslash zeta^2\}\},\backslash quad\; \backslash omega\_r\; =\; \backslash omega\_n\backslash sqrt\{1-2\backslash zeta^2\}$$

Identification / FOPDT

First-order-plus-dead-time fit by Smith's two-point method

$$\tau =1.5(t_{63}-t_{28}),\theta =t_{63}-\tau \backslash tau\; =\; 1.5(t\_\{63\}\; -\; t\_\{28\}),\backslash quad\; \backslash theta\; =\; t\_\{63\}\; -\; \backslash tau$$

Identification / second order

Damping ratio from the logarithmic decrement

$$\delta =\frac{1}{n}\ln \frac{x_1}{x_{n+1}},\zeta =\frac{\delta }{\sqrt{4{\pi }^2+{\delta }^2}}\backslash delta\; =\; \backslash frac\{1\}\{n\}\backslash ln\backslash frac\{x\_1\}\{x\_\{n+1\}\},\backslash quad\; \backslash zeta\; =\; \backslash frac\{\backslash delta\}\{\backslash sqrt\{4\backslash pi^2+\backslash delta^2\}\}$$

Second-order model from a step response

$${\omega }_n=\frac{\pi }{T_p\sqrt{1-{\zeta }^2}}\backslash omega\_n\; =\; \backslash frac\{\backslash pi\}\{T\_p\backslash sqrt\{1-\backslash zeta^2\}\}$$

Modelling / DC motor

Speed transfer function of a DC motor

$$\frac{\mathrm{\Omega }(s)}{V(s)}=\frac{K_t}{(Js+b)(Ls+R)+K_t^2}\backslash frac\{\backslash Omega(s)\}\{V(s)\}\; =\; \backslash frac\{K\_t\}\{(Js+b)(Ls+R)+K\_t^2\}$$

Optimal control / LQR

Continuous-time LQR gain (Riccati equation)

$$A^{T}P+PA-PB{R}^{-1}{B}^{T}P+Q=0,K=R^{-1}{B}^{T}PA^TP\; +\; PA\; -\; PBR^\{-1\}B^TP\; +\; Q\; =\; 0,\backslash quad\; K\; =\; R^\{-1\}B^TP$$

Discrete-time LQR gain (discrete Riccati equation)

$$P=A^{T}PA-A^{T}PB(R+B^{T}PB{)}^{-1}{B}^{T}PA+QP\; =\; A^TPA\; -\; A^TPB(R\; +\; B^TPB)^\{-1\}B^TPA\; +\; Q$$

LQR for a first-order system

$$2aP-\frac{b^2{P}^2}{r}+q=0,K=\frac{bP}{r}2aP\; -\; \backslash frac\{b^2P^2\}\{r\}\; +\; q\; =\; 0,\backslash quad\; K\; =\; \backslash frac\{bP\}\{r\}$$

Root locus / asymptotes

Root-locus asymptotes (centroid and angles)

$${\sigma }_a=\frac{\sum p_i-\sum z_j}{n-m},{\theta }_k=\frac{(2k+1){180}^{\circ }}{n-m}\backslash sigma\_a\; =\; \backslash frac\{\backslash sum\; p\_i\; -\; \backslash sum\; z\_j\}\{n-m\},\backslash quad\; \backslash theta\_k\; =\; \backslash frac\{(2k+1)180^\backslash circ\}\{n-m\}$$

Root locus / breakaway points

Root-locus breakaway and break-in points

$$\frac{dK}{ds}=0⟺N{D}^{\mathrm{\prime }}-N^{\mathrm{\prime }}D=0\backslash frac\{dK\}\{ds\}\; =\; 0\; \backslash iff\; N\; D\text{'}\; -\; N\text{'}\; D\; =\; 0$$

Root locus / departure and arrival angles

Root-locus angle of arrival at a complex zero

$${\theta }_a={180}^{\circ }+\sum \mathrm{\angle }(z-p_j)-\sum _{i\mathrm{\ne }}\mathrm{\angle }(z-z_i)\backslash theta\_a\; =\; 180^\backslash circ\; +\; \backslash sum\backslash angle(z\; -\; p\_j)\; -\; \backslash sum\_\{i\backslash ne\}\backslash angle(z\; -\; z\_i)$$

Root-locus angle of departure from a complex pole

$${\theta }_d={180}^{\circ }+\sum \mathrm{\angle }(p-z_i)-\sum _{j\mathrm{\ne }}\mathrm{\angle }(p-p_j)\backslash theta\_d\; =\; 180^\backslash circ\; +\; \backslash sum\backslash angle(p\; -\; z\_i)\; -\; \backslash sum\_\{j\backslash ne\}\backslash angle(p\; -\; p\_j)$$

Root locus / magnitude and angle criteria

Root-locus gain and angle at a test point

$$K=\frac{1}{\mathrm{\mid }G(s_0)\mathrm{\mid }},\mathrm{\angle }G(s_0)=\pm {180}^{\circ }K\; =\; \backslash frac\{1\}\{\backslash vert\; G(s\_0)\backslash vert\; \},\backslash quad\; \backslash angle\; G(s\_0)\; =\; \backslash pm\; 180^\backslash circ$$

Stability / Lyapunov

Continuous Lyapunov equation

$$A^{T}P+PA+Q=0A^TP\; +\; PA\; +\; Q\; =\; 0$$

Discrete Lyapunov equation

$$A^{T}PA-P+Q=0A^TPA\; -\; P\; +\; Q\; =\; 0$$

Stability / Nyquist

Nyquist criterion: encirclements of -1

$$Z=N+PZ\; =\; N\; +\; P$$

Stability / Routh-Hurwitz

Range of gain for closed-loop stability (Routh, j-omega crossing)

$$D(j\omega )+KN(j\omega )=0D(j\backslash omega)\; +\; K\; N(j\backslash omega)\; =\; 0$$

Routh-Hurwitz stability test

$$b_i=\frac{a_1{a}_{0,i+1}-a_0{a}_{1,i+1}}{a_1}b\_i\; =\; \backslash frac\{a\_1a\_\{0,i+1\}\; -\; a\_0a\_\{1,i+1\}\}\{a\_1\}$$

Stability / discrete

Jury (Schur-Cohn) stability test for a discrete polynomial

Stability / margins

Gain and phase margins

$$GM=\frac{1}{\mathrm{\mid }L(j{\omega }_{pc})\mathrm{\mid }},PM={180}^{\circ }+\mathrm{\angle }L(j{\omega }_{gc})GM\; =\; \backslash frac\{1\}\{\backslash vert\; L(j\backslash omega\_\{pc\})\backslash vert\; \},\backslash quad\; PM\; =\; 180^\backslash circ\; +\; \backslash angle\; L(j\backslash omega\_\{gc\})$$

Stability / poles

Poles and stability of a characteristic polynomial

Stability / robustness

Peak sensitivity Ms and modulus margin

$$M_s=\underset{\omega }{\max }\frac{1}{\mathrm{\mid }1+L(j\omega )\mathrm{\mid }}M\_s\; =\; \backslash max\_\backslash omega\; \backslash frac\{1\}\{\backslash vert\; 1\; +\; L(j\backslash omega)\backslash vert\; \}$$

State space / DC gain

DC gain of a state-space model

$$G(0)=D-C{A}^{-1}BG(0)\; =\; D\; -\; CA^\{-1\}B$$

State space / Gramians

Controllability Gramian

$$A{W}_c+W_c{A}^T+B{B}^T=0AW\_c\; +\; W\_cA^T\; +\; BB^T\; =\; 0$$

Observability Gramian

$$A^T{W}_o+W_{o}A+C^{T}C=0A^TW\_o\; +\; W\_oA\; +\; C^TC\; =\; 0$$

State space / controllability

Controllability matrix and rank

$$\mathcal{C}=[BAB\cdots A^{n-1}B]\backslash mathcal\{C\}\; =\; [B\backslash ;AB\backslash ;\backslash cdots\backslash ;A^\{n-1\}B]$$

State space / conversions

Controllable canonical form of a transfer function

Transfer function of a state-space model

$$G(s)=C(sI-A{)}^{-1}B+D=\frac{C\mathrm{a}\mathrm{d}\mathrm{j}(sI-A)B}{\det (sI-A)}+DG(s)\; =\; C(sI-A)^\{-1\}B\; +\; D\; =\; \backslash frac\{C\backslash ,\backslash mathrm\{adj\}(sI-A)\backslash ,B\}\{\backslash det(sI-A)\}\; +\; D$$

State space / observability

Observability matrix and rank

$$\mathcal{O}=\left[\begin{array}{c}C\\ CA\\ \mathrm{⋮}\\ C{A}^{n-1}\end{array}\right]\backslash mathcal\{O\}\; =\; \backslash begin\{bmatrix\}C\backslash \backslash \; CA\backslash \backslash \; \backslash vdots\backslash \backslash \; CA^\{n-1\}\backslash end\{bmatrix\}$$

State space / observers

Observer (estimator) gain by pole placement

$$L=\alpha (A){\mathcal{O}}^{-1}\left[\begin{array}{c}0\\ \mathrm{⋮}\\ 1\end{array}\right]L\; =\; \backslash alpha(A)\backslash ,\backslash mathcal\{O\}^\{-1\}\backslash begin\{bmatrix\}0\backslash \backslash \; \backslash vdots\backslash \backslash \; 1\backslash end\{bmatrix\}$$

State space / pole placement

Deadbeat state feedback (discrete time)

$$K=[0\cdots 1]{\mathcal{C}}^{-1}{A}^{n}K\; =\; [0\backslash cdots\; 1]\backslash ,\backslash mathcal\{C\}^\{-1\}A^n$$

Pole placement by Ackermann's formula

State space / stability

Eigenvalues (poles) of a state matrix

$$\det (sI-A)=0\backslash det(sI\; -\; A)\; =\; 0$$

State space / state feedback

Closed-loop poles under state feedback u = -K x

$$\det (sI-A+BK)=0\backslash det(sI\; -\; A\; +\; BK)\; =\; 0$$

State space / time response

State response to an initial state and a constant input

$$x(t)=e^{At}{x}_0+{\int }_0^t{e}^{A\tau }d\tau Bux(t)\; =\; e^\{At\}x\_0\; +\; \backslash int\_0^t\; e^\{A\backslash tau\}d\backslash tau\backslash ,Bu$$

State-transition matrix e^{At}

$$\mathrm{\Phi }(t)=e^{At}=\sum _k\frac{(At{)}^k}{k!}\backslash Phi(t)\; =\; e^\{At\}\; =\; \backslash sum\_k\; \backslash frac\{(At)^k\}\{k!\}$$

Steady-state error

System type, static error constant and steady-state error

$$e_{ss}=\frac{A}{1+K_p},\frac{A}{K_v},\frac{A}{K_a}e\_\{ss\}\; =\; \backslash frac\{A\}\{1+K\_p\},\backslash ;\; \backslash frac\{A\}\{K\_v\},\backslash ;\; \backslash frac\{A\}\{K\_a\}$$

Time delay / Pade

Pade approximation of a time delay

$$e^{-\theta s}\approx \frac{\sum _k{c}_k(-\theta s{)}^k}{\sum _k{c}_k(\theta s{)}^k}e^\{-\backslash theta\; s\}\; \backslash approx\; \backslash frac\{\backslash sum\_k\; c\_k(-\backslash theta\; s)^k\}\{\backslash sum\_k\; c\_k(\backslash theta\; s)^k\}$$

Time response / closed loop

Proportional control of a first-order plant

$$T(s)=\frac{K{K}_p}{\tau s+1+K{K}_p}T(s)\; =\; \backslash frac\{KK\_p\}\{\backslash tau\; s\; +\; 1\; +\; KK\_p\}$$

Time response / dead time

Step response of a first-order-plus-dead-time process

$$y(t)=KA(1-e^{-(t-\theta )\mathrm{/}\tau })u(t-\theta )y(t)\; =\; KA\backslash left(1\; -\; e^\{-(t-\backslash theta)/\backslash tau\}\backslash right)\backslash ,u(t-\backslash theta)$$

Time response / final and initial values

Final value theorem

$$y(\mathrm{\infty })=\underset{s\to 0}{\lim }sY(s)y(\backslash infty)\; =\; \backslash lim\_\{s\backslash to\; 0\}\; sY(s)$$

Initial value theorem (value and slope at t = 0+)

$$y(0^{+})=\underset{s\to \mathrm{\infty }}{\lim }sY(s)y(0^+)\; =\; \backslash lim\_\{s\backslash to\backslash infty\}\; sY(s)$$

Time response / first order

Step response of a first-order system

$$y(t)=KA(1-e^{-t\mathrm{/}\tau })y(t)\; =\; K\; A\backslash left(1\; -\; e^\{-t/\backslash tau\}\backslash right)$$

Time and frequency specifications of a first-order system

$$t_r=\tau \ln 9,t_{s,2\mathrm{\%}}=\tau \ln 50,{\omega }_B=1\mathrm{/}\tau t\_r\; =\; \backslash tau\backslash ln\; 9,\backslash quad\; t\_\{s,2\backslash \%\}\; =\; \backslash tau\backslash ln\; 50,\backslash quad\; \backslash omega\_B\; =\; 1/\backslash tau$$

Time response / general

Step response of a transfer function at a given time

$$y(t)=C{\int }_0^t{e}^{A\tau }d\tau B{A}_u+D{A}_{u}y(t)\; =\; C\backslash int\_0^t\; e^\{A\backslash tau\}d\backslash tau\backslash ,B\backslash ,A\_u\; +\; D\; A\_u$$

Time response / second order

Damping ratio from percent overshoot

$$\zeta =\frac{-\ln (\mathrm{\%}OS\mathrm{/}100)}{\sqrt{{\pi }^2+{\ln }^2(\mathrm{\%}OS\mathrm{/}100)}}\backslash zeta\; =\; \backslash frac\{-\backslash ln(\backslash \%OS/100)\}\{\backslash sqrt\{\backslash pi^2\; +\; \backslash ln^2(\backslash \%OS/100)\}\}$$

Exact settling time of a second-order step response

$$T_s=\max \{t:\mathrm{\mid }y(t)-1\mathrm{\mid }=\text{band}\}T\_s\; =\; \backslash max\backslash \{t\; :\; \backslash vert\; y(t)\; -\; 1\backslash vert\; =\; \backslash text\{band\}\backslash \}$$

Mass-spring-damper natural frequency and damping ratio

$${\omega }_n=\sqrt{k\mathrm{/}m},\zeta =\frac{c}{2\sqrt{km}}\backslash omega\_n\; =\; \backslash sqrt\{k/m\},\backslash quad\; \backslash zeta\; =\; \backslash frac\{c\}\{2\backslash sqrt\{km\}\}$$

Natural frequency and damping ratio of a second-order denominator

$$s^2+2\zeta {\omega }_{n}s+{\omega }_n^{2}s^2\; +\; 2\backslash zeta\backslash omega\_n\; s\; +\; \backslash omega\_n^2$$

Peak time of a second-order step response

$$T_p=\frac{\pi }{{\omega }_n\sqrt{1-{\zeta }^2}}T\_p\; =\; \backslash frac\{\backslash pi\}\{\backslash omega\_n\backslash sqrt\{1-\backslash zeta^2\}\}$$

Percent overshoot from the damping ratio

$$\mathrm{\%}OS=100e^{-\zeta \pi \mathrm{/}\sqrt{1-{\zeta }^2}}\backslash \%OS\; =\; 100\backslash ,e^\{-\backslash zeta\backslash pi/\backslash sqrt\{1-\backslash zeta^2\}\}$$

Rise time of an underdamped second-order system

$$T_{r,0-100}=\frac{\pi -{\cos }^{-1}\zeta }{{\omega }_d}T\_\{r,0-100\}\; =\; \backslash frac\{\backslash pi\; -\; \backslash cos^\{-1\}\backslash zeta\}\{\backslash omega\_d\}$$

Second-order specifications from the pole location

$${\omega }_n=\sqrt{{\sigma }^2+{\omega }_d^2},\zeta =\sigma \mathrm{/}{\omega }_n,\mathrm{\%}OS=100e^{-\sigma \pi \mathrm{/}{\omega }_d}\backslash omega\_n\; =\; \backslash sqrt\{\backslash sigma^2+\backslash omega\_d^2\},\backslash quad\; \backslash zeta\; =\; \backslash sigma/\backslash omega\_n,\backslash quad\; \backslash \%OS\; =\; 100e^\{-\backslash sigma\backslash pi/\backslash omega\_d\}$$

Settling time of a second-order system (rule and envelope)

$$T_s\approx \frac{4}{\zeta {\omega }_n}(2\mathrm{\%}),T_s\approx \frac{3}{\zeta {\omega }_n}(5\mathrm{\%})T\_s\; \backslash approx\; \backslash frac\{4\}\{\backslash zeta\backslash omega\_n\}\backslash ;(2\backslash \%),\backslash qquad\; T\_s\; \backslash approx\; \backslash frac\{3\}\{\backslash zeta\backslash omega\_n\}\backslash ;(5\backslash \%)$$

Step response of a standard second-order system

$$y(t)=1-e^{-\zeta {\omega }_{n}t}(\cos {\omega }_{d}t+\frac{\zeta }{\sqrt{1-{\zeta }^2}}\sin {\omega }_{d}t)y(t)\; =\; 1\; -\; e^\{-\backslash zeta\backslash omega\_n\; t\}\backslash left(\backslash cos\backslash omega\_d\; t\; +\; \backslash frac\{\backslash zeta\}\{\backslash sqrt\{1-\backslash zeta^2\}\}\backslash sin\backslash omega\_d\; t\backslash right)$$

Transfer functions / DC gain

DC gain and final value of a step response

$$y(\mathrm{\infty })=AG(0)y(\backslash infty)\; =\; A\backslash ,G(0)$$

Transfer functions / block diagrams

Closed-loop transfer function of a feedback loop

$$T(s)=\frac{G}{1+GH}=\frac{N_G{D}_H}{D_G{D}_H+N_G{N}_H}T(s)\; =\; \backslash frac\{G\}\{1\; +\; GH\}\; =\; \backslash frac\{N\_GD\_H\}\{D\_GD\_H\; +\; N\_GN\_H\}$$

Parallel connection of two transfer functions

$$G_1+G_2=\frac{N_1{D}_2+N_2{D}_1}{D_1{D}_2}G\_1\; +\; G\_2\; =\; \backslash frac\{N\_1D\_2\; +\; N\_2D\_1\}\{D\_1D\_2\}$$

Series (cascade) connection of two transfer functions

$$G(s)=G_1(s)G_2(s)=\frac{N_1{N}_2}{D_1{D}_2}G(s)\; =\; G\_1(s)G\_2(s)\; =\; \backslash frac\{N\_1N\_2\}\{D\_1D\_2\}$$

Transfer functions / evaluation

Value of a transfer function at a complex point

$$G(s_0)=\frac{N(s_0)}{D(s_0)}G(s\_0)\; =\; \backslash frac\{N(s\_0)\}\{D(s\_0)\}$$

Transfer functions / poles and zeros

Poles, zeros and gain of a transfer function

$$G(s)=k\frac{\prod (s-z_i)}{\prod (s-p_j)}G(s)\; =\; k\backslash ,\backslash frac\{\backslash prod\; (s\; -\; z\_i)\}\{\backslash prod\; (s\; -\; p\_j)\}$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
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Limitations

• It answers what its certified capabilities cover (transfer functions, time and frequency response, stability margins, root locus, state space, LQR, Kalman filters, discretization and PID tuning); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{control_systems_ai999x_1_0_0,
  title  = {control-systems-ai999x: an offline, verified control systems solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/control-systems-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/control-systems-ai999x/discussions or visit https://ai999x.com.

Support the project

control-systems-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com