CALCULATORS / CONTROL SYSTEMS AND DYNAMIC RESPONSE

Scalar state-space equilibrium and pole calculator

Scalar continuous state-space DC equilibrium and pole; not a matrix eigensolver.

controls-state-space.napkin / editable

SCENARIO COMPARISON

Compare the numbers.

This source describes an analytical model. Its quantities and checks are the view.

Scenario	equilibrium	timeConstants
	0.5	0.5
	0.625	0.625

All calculated quantities are listed with the inputs. Switch to Source to inspect the equations.

SCENARIOS / CALCULATED VALUESSAME SOURCE

Change an input to see the consequences.

stable scalar pole-2.00 s^-1✓ passes

BASELINE INPUTS

stateCoefficient-2 s^-1inputCoefficient1 s^-1input1

CALCULATED

equilibrium0.5

timeConstant0.5 s

Edits change the baseline source.

THE SOURCE

Edits below update the same model.

# Small state-space simulation
# Scope: Scalar continuous state-space DC equilibrium and pole; not a matrix eigensolver.
# Status: illustrative worked example; engineering review pending.
# Inputs are example values. Checks are editable example requirements, not standards.

stateCoefficient = -2 s^-1
inputCoefficient = 1 s^-1
input = 1

equilibrium := -inputCoefficient*input/stateCoefficient
timeConstant := -1/stateCoefficient [s]

check stable scalar pole: stateCoefficient < 0 s^-1

scenario alternative: stateCoefficient = -1.6 s^-1

Calculated from your source · not saved

Baseline · Baseline

stable scalar pole-2.00 s^-1 ✓ passes

alternative · alternative

stable scalar pole-1.60 s^-1 ✓ passes

Illustrative first-order example. Checks are editable requirements, not design standards. The broader topic may require additional equations, data, and validation.

Formula and units

This example uses the following equations. Input names and units match the editable source above.

equilibrium: equilibrium = -inputCoefficient*input/stateCoefficient
time constant: timeConstant = -1/stateCoefficient [s]

Original worked example

These are the baseline values in the downloadable source. Edits above let you test your own values.

Given

state coefficient: -2 s^-1
input coefficient: 1 s^-1
input: 1

Calculated

equilibrium: 0.5
time constant: 0.5 s

Assumptions and limits

Scalar continuous state-space DC equilibrium and pole; not a matrix eigensolver.

The requirement checks use the limits written in this example. Passing a check does not establish that the model covers every load, material behavior, or operating condition in a real design.

Download this .napkin example ↓

Take it further with your agent

Napkin keeps the inputs, equations, checks and assumptions together in one study. Your agent sets it up; you change the numbers.

Try with Claude ↗

codex 'Use Napkin (https://napkin.engineering/mcp, or the local Napkin plugin) to build a study: Small state-space simulation. Inputs: A, B, C, D matrices; inputs; initial state. Outputs: state and output trajectories. Model scope: Declared linear or low-order nonlinear plant models. Read napkin_capabilities and the current snapshot. Ask me for the values that matter, label assumptions with units, author .napkin source, validate it and apply it with napkin_source using the current revision, then open the study. Add checks for the limits I supply. Keep unsupported analyses explicit.'

Claude needs the Napkin connector; Codex needs the Napkin plugin. Set up your agent ↗

The broader study

The topic brief below describes the broader scope. The worked example above states the particular calculation it implements.

Inputs: A, B, C, D matrices, inputs, initial state
Outputs: state and output trajectories
Modes: calculate, compare, sweep
Drawing: Response and stability plots

EXECUTABLE WORKED EXAMPLE · FIRST-ORDER MODEL · CHECK THE SCOPE & ASSUMPTIONS